Evaluation of a Non-Parametric Penalized Kaplan–Meier Estimator Under Interval-Censored Survival Data

Abstract

Interval-censored survival data arise frequently in biomedical and epidemiological studies where event times are observed only within observation intervals. Classical non-parametric estimators, such as the Kaplan–Meier (KM) estimator under imputation and the Turnbull estimator, often suffer from instability, irregular fluctuations, and overfitting when sample sizes are small or when the prevalence rate is low. Recent methodological developments, which include smoothed and penalized approaches, have been proposed to improve stability and reduce estimation error in such settings. This study evaluates and benchmarks the finite-sample performance of a nonparametric penalized likelihood KM estimator under interval-censored data. The method is compared with the classical KM estimator using four imputation strategies, that is, midpoint, regression, uniform, and multiple imputation. From a symmetry perspective, midpoint and uniform imputation preserve interval symmetry through deterministic and probabilistic mechanisms, respectively, whereas regression and multiple imputation intentionally introduce structural asymmetry to reflect data-driven risk heterogeneity and distributional uncertainty. To assess and benchmark the performance of the penalized KM estimator, an extensive Monte Carlo (MC) simulation study was conducted across varying sample sizes and prevalence rates using error-based metrics. The MC simulation results revealed that the nonparametric penalized KM estimator consistently outperforms the classical KM estimator in small samples across all prevalence rates. The gains are more pronounced under low prevalence rates where the penalized KM estimator is superior for small to relatively moderate samples of $n\approx$ 40–100. Among the imputation techniques, regression and multiple imputation generally exhibited superior performance. Real data application further confirms these findings, demonstrating that the nonparametric penalized KM estimator yields more stable and accurate survival curves than the classical KM estimator in small samples.

1. Introduction

Interval-censored data arise when a failure time, denoted by T, cannot be observed exactly, but is only known to lie within an interval determined by a sequence of examination times [1,2,3]. Thus, only the lower and upper bounds of the interval, L and U, are known, while the exact event time remains unknown. Mathematically, this is represented as $L<T\le U$, or in set notation, $T\in (L,U]$. A fundamental assumption in survival analysis is that the censoring mechanism is independent of, or non-informative about, the failure time of interest. Formally, for an interval $I=(L,U]$,

$$P(T\le t\mid L=l,U=u,L<T\le U)=P(T\le t\mid l<T\le u),$$

which implies that the interval $(L,U]$ (or equivalently, its endpoints) does not provide further information about T beyond the fact that the event occurs within the observed bounds. Such data often arise in biomedical and reliability studies, especially in clinical trials and longitudinal research with periodic follow-up visits [1,3,4]. In many practical settings, the analysis of interval-censored survival data requires estimation of the underlying survival function [1,5,6,7,8]. Treating interval-censored survival times as exact event times can lead to biased estimates and, consequently, results and conclusions that lack reliability [9,10,11]. Because the exact event times are unobserved, interval-censored data may naturally be viewed as a missing-data problem. As a result, a substantial body of research has focused on addressing this challenge through the development of imputation techniques and nonparametric procedures for the estimation of the survival function under interval censoring.

Classical nonparametric maximum likelihood estimators (NPMLEs), including the Kaplan-Meier (KM) estimator for right-censored data [12] and Turnbull (TB) for interval-censored data [5], form the basis of modern survival analysis. Despite their wide adoption, these methods face well-documented limitations when applied to interval-censored data under finite-sample conditions [13,14]. Since the exact time is unknown, applying the classical KM estimator on interval censored data is not straightforward, as it requires exact event times. To address this, several imputation-based approaches [15], such as midpoint [16], regression, uniform, and multiple imputation [17], are frequently used to convert interval-censored data into imputed event times before applying the KM estimator. Midpoint and uniform imputation preserve interval symmetry through deterministic and probabilistic mechanisms, respectively, while regression and multiple imputation incorporate structural asymmetry to account for data-driven risk heterogeneity and distributional uncertainty. However, these imputation-based KM methods exhibit considerable instability in small samples. They often produce irregular survival curves and suffer from overfitting, especially when the censoring intervals are wide or the prevalence of events is low [1]. Similarly, the classical TB, though appropriate for interval-censored data, tends to generate irregular multi-modal survival functions due to the inherent undersmoothing [18]. These limitations can result in inflated estimation errors and poor representation of the underlying survival process.

In general, NPMLEs, which includes the KM and the TB estimators for incomplete data [19], commonly suffer from overfitting when applied to small samples, which can lead to poor estimation performance [20]. Several approaches have been proposed to address this limitation, including semiparametric frameworks based on finite mixture models [21], spline-based methods [22,23,24], and smoothing techniques. Despite their flexibility, semiparametric procedures often face challenges related to parameter tuning, computational complexity, and nonconvergence, whereas smoothing methods are generally regarded as more practical. Kernel density estimation is a fundamental nonparametric smoothing technique that represents each observation as a localized “pump” of density with a common functional form [25,26,27]. Using these kernel densities, the extent of smoothing is primarily determined by the choice of kernel and the window width, commonly referred to as the bandwidth. In the context of the KM and TB estimators, applying smoothing provides a natural and straightforward approach for reducing excessive variability of the density estimate. More broadly, improving estimation efficiency under limited or incomplete data has received considerable attention across different statistical frameworks. For example, recent studies have shown that carefully designed sampling schemes, such as ranked set sampling and its extensions, can substantially enhance the finite-sample performance of parameter estimators for parametric distributions [28,29,30]. These developments highlight a growing emphasis on methodological strategies that mitigate small-sample inefficiency, which also motivates the present focus on smoothing and imputation-based approaches in nonparametric survival analysis.

Focusing on the classical KM estimator, several researchers have proposed smoothing techniques for KM estimation. Wang [31] considered kernel smoothing approaches to reduce variability in the KM curve, Efron [32] developed penalized likelihood formulations, while Babyak [33] proposed constrained estimators to control overfitting. On the other hand, recently, Wu and Kolassa [34] proposed a non-parametric approach to reduce the overestimation of the KM estimator when the event and censoring times are independent. These approaches share a common goal of reducing overfitting while preserving the nonparametric flexibility of the KM estimator. A notable contribution is the smoothed nonparametric penalized likelihood KM estimator proposed by Tubbs et al. [35]. This estimator applies kernel smoothing and a Bayesian Information Criterion (BIC)-type penalty to reduce overfitting by controlling curve complexity. The simulation results of Tubbs et al. [35] have shown that this estimator has demonstrated substantial improvements in reducing the estimation error and producing smoother and more reliable survival curves, especially when the sample sizes are small. Their proposed procedure advances nonparametric survival analysis by improving estimation accuracy and predictive performance, particularly in settings where survival data are constrained by censoring, small sample sizes, low prevalence, or a limited number of repeated measurements per subject. Tubbs et al. [35] demonstrated that smoothing NPMLEs through complexity-based penalized likelihood provides a simple yet effective strategy for enhancing survival and density estimation, as well as for improving exact time-to-event prediction through imputation.

Motivated by these developments, this study addresses a critical gap in the literature, namely, the absence of a systematic comparison between the improved nonparametric smoothed penalized KM estimator proposed by Tubbs et al. [35] and the classical KM estimator implemented using commonly applied imputation strategies under interval-censored settings. Through a comprehensive Monte Carlo (MC) simulation study, complemented by a convergence analysis, this work aims not only to provide clear evidence on the conditions under which smoothing enhances estimator performance, mitigates overfitting, and yields more reliable survival inference but also to identify the actual sample sizes at which the smoothed KM estimator demonstrably outperforms the classical KM. The findings are further validated using a real-world interval-censored dataset. Such insights are particularly valuable for practitioners analyzing interval-censored survival data, especially in applications where sample sizes are small and estimation stability is of primary importance. Accordingly, this study focuses exclusively on the KM estimator, given its widespread use and interpretability in survival analysis [12], as well as its central role in imputation-based and smoothing methodologies [31,32,35].

The remainder of this article is organized as follows. Section 2 presents the methodological framework, including the classical KM estimator, the smoothed penalized likelihood KM estimator, and the four imputation strategies examined in this study. Section 3 outlines the MC simulation design and describes the metrics used to evaluate the performance of the estimators under varying sample sizes and prevalence rates. It also presents the application of the proposed methods to a real interval-censored dataset. Finally, Section 4 provides concluding remarks and summarizes key insights from this study.

2. Materials and Methods

2.1. Kaplan–Meier Estimator

The KM estimator, introduced by Kaplan and Meier [12], is one of the most widely used non-parametric maximum likelihood estimators for the survival function from censored time-to-event data [36]. The KM estimator provides a product limit formulation that efficiently handles right-censored data without requiring strong distributional assumptions. The KM estimator was originally developed in the context of biomedical follow-up studies, where individuals can leave the study or remain event-free at the end of observation, resulting in right-censored event times. Although primarily intended for right-censored data, the same mathematical form also accommodates left-censored data with appropriate modifications. The KM estimator of the survival function is

$$\hat{S}\left(t\right)=\prod _{t_{\left(j\right)}\le t}\left(1-{\displaystyle \frac{d_j}{n_j}}\right),$$

where $t_{\left(1\right)}<t_{\left(2\right)}<\cdots <t_{\left(J\right)}$ denote the ordered distinct event times, $d_j$ the number of events at $t_{\left(j\right)}$, and $n_j$ the number at risk immediately prior to $t_{\left(j\right)}$ with $\hat{S}\left(0\right)=1$. This produces a step function that decreases only at observed event times while remaining constant between these times. The KM estimator is not, in its original form, designed for interval-censored data, in which the exact event time $T_i^{*}$ is unknown but is known to lie within an interval $(L_i,R_i]$ with $0<L_i<R_i<\infty$ (or $R_i=\infty$ for right-censoring).

To apply KM estimation in the presence of interval censoring, analysts typically convert the interval-censored data into imputed event times via imputation. Common imputation methods include midpoint, uniform, regression and multiple imputation, each replacing $(L_i,R_i]$ with a single representative value $T_i^{*}$ and treating it as the actual event time. Once imputed, the KM estimator can be applied directly. This approach allows for comparison of KM-based curves across imputation methods, though it necessarily introduces approximation error relative to estimators specifically designed for interval-censored data, for example, the TB estimator proposed by Turnbull [5].

2.2. Non-Parametric Penalized Kaplan–Meier Estimator

To address the shortcomings of the classical KM estimator, several researchers have proposed smoothing techniques for KM estimation. Wang [31] considered kernel smoothing approaches to reduce variability in the KM curve, while Efron [32] developed penalized likelihood formulations. Babyak [33] further introduced constrained estimators to control overfitting. These approaches share a common goal of reducing overfitting while preserving the nonparametric flexibility of the KM estimator.

Building on this literature, Tubbs et al. [35] introduced the smoothed nonparametric penalized KM estimator, which applies a kernel-smoothing adjustment to the classical KM estimator and incorporates a penalized likelihood criterion to balance goodness-of-fit with curve smoothness. The smoothing step is implemented using Nadaraya–Watson normal kernel regression [37], with the degree of smoothing selected adaptively by minimizing a BIC-type penalty. Formally, the penalized objective is

$${\mathrm{BIC}}_s\left(d\right)=-2lnL\left(d\right)+k_T\left(d\right)\times ln{N}_s,$$

where $L\left(d\right)$ is the likelihood of the smoothed survival curve with bandwidth d and $k_T\left(d\right)$ denotes the number of turning points in the estimated density function, which serves as a measure of the complexity of the curve. Specifically,

$$k_T=\mathrm{card}\left\{x\in \mathbb{R}|\mathrm{sign}\left({\hat{f}}_n^{\prime }\left(x^{+}\right)\right)\ne \mathrm{sign}\left({\hat{f}}_n^{\prime }\left(x^{-}\right)\right)\right\},$$

where ${\hat{f}}_n^{\prime }\left(x^{+}\right)$ and ${\hat{f}}_n^{\prime }\left(x^{-}\right)$ denote the right and left derivatives of the estimated density at x. The term $N_s$ reflects the effective information about the sample, with three alternative penalties considered: (i) $lnN$, based on the sample size; (ii) $ln{N}_m$, the total number of repeated observations across individuals, and (iii) $ln{N}_e$, the equivalent sample size adjusted for interval information. Simulation studies by Tubbs et al. [35] showed that although all three penalties performed comparably, the $ln{N}_e$ penalty generally provided marginally better results and is recommended for practice.

An important advantage of Tubbs et al.’s [35] estimator is its broad applicability across different censoring mechanisms. Thus, the smoothed nonparametric penalized KM method can be extended to accommodate left-censored, right-censored, and interval-censored observations. This makes it particularly valuable in biomedical and epidemiological studies, where mixed forms of censoring commonly arise due to staggered entry, delayed detection, or discrete follow-up schedules. Extensive simulations conducted by Tubbs et al. [35] demonstrated that their approach consistently reduced error between estimated and true survival functions by nearly half compared to classical KM estimators. Improvements were evident in both within-sample fit and out-of-sample prediction, with applications to breast cancer data further confirming its practical advantages. The smoothed nonparametric penalized KM method offers a powerful and stable alternative to the classical KM, particularly in small sample settings [35].

2.3. Imputation Methods for Interval Censoring

In this study, imputation is used to transform each interval-censored observation $(L_i,R_i]$ into a pseudo-exact estimate $T_i^{*}$, upon which the KM estimator is applied. The imputation methods considered are midpoint, regression, uniform, and multiple imputation. These approaches differ in how they utilize the information contained in the censoring interval and auxiliary variables and thus influence the accuracy and stability of the resulting survival estimates. Midpoint and uniform imputation preserve the symmetry of the interval through deterministic and probabilistic mechanisms, respectively, while regression and multiple imputation incorporate structural asymmetry to account for data-driven risk heterogeneity and distributional uncertainty.

2.3.1. Midpoint Imputation

Midpoint imputation is a straightforward method in which missing or interval-censored values are imputed using the midpoint of the observed interval. It assumes that the event is equally likely to occur in the middle within the interval. This approach is simple to implement and is often used in preliminary survival analyses, reliability testing, and clinical studies when no prior information about the distribution of event times is available. This imputation method imposes interval symmetry by assigning equal weight to the lower and upper bounds, implicitly assuming a symmetric likelihood of event occurrence around the center of the interval. Due to its computational simplicity, midpoint imputation is frequently used in preliminary survival analysis, reliability studies, and clinical applications where there is no prior information on the underlying event-time distribution or potential asymmetry. Although easy to apply, midpoint imputation can introduce bias, especially when the width of censoring intervals varies significantly. Sun [1] noted that this method may oversimplify the data and affect the precision of survival estimates. Mathematically, for an interval $(L_i,R_i]$, the midpoint is given by

$$\widehat{T_i^{*}}={\displaystyle \frac{L_i+R_i}{2}},$$

where $L_i$ and $R_i$ are the lower and upper bounds of the interval [1].

2.3.2. Regression Imputation

Regression imputation is a model-based technique in which unobserved event times are replaced by predictions from a fitted regression model. The method originates from early predictive imputation work by Buck [38], who first proposed using linear regression to replace missing survey values; it was later formalized in the modern missing-data framework by Little and Rubin [39]. In survival analysis with interval-censored data, regression imputation uses covariate information to generate interval-consistent event times that align with the censoring interval of each subject. Let T denote the event time and X a vector of p covariates. Suppose that the conditional distribution of T given $X=x$ is characterized by density $f(t\mid x;\theta )$ and CDF $F(t\mid x;\theta )$. Under a regression model, a principled approach is to draw the imputed time from the truncated conditional distribution as

$$T_i^{*}\sim f(t\mid X_i=x_i;\theta )\mathrm{truncated}\mathrm{to}(L_i,R_i],$$

with truncated density

$$f_{\mathrm{trunc}}(t\mid L_i<t\le R_i,x_i)={\displaystyle \frac{f(t\mid x_i;\theta )}{F(R_i\mid x_i;\theta )-F(L_i\mid x_i;\theta )}}.$$

A deterministic alternative imputes the conditional expectation of $T_i$, given the truncation

$$\mathbb{E}\left(T_i^{*}\mid L_i<T_i^{*}\le R_i,X_i=x_i\right)={\displaystyle \frac{{\displaystyle {\int }_{L_i}^{R_i}tf(t\mid x_i;\theta )dt}}{{\displaystyle F(R_i\mid x_i;\theta )-F(L_i\mid x_i;\theta )}}}.$$

A widely used special case assumes a transformed linear regression model

$$g\left(T_i^{*}\right)={\beta }_0+{\beta }_1{X}_{i1}+\cdots +{\beta }_p{X}_{ip}+{\epsilon }_i,{\epsilon }_i\sim \mathcal{N}(0,{\sigma }^2),$$

where $g(·)$ is typically the identity or logarithm. Estimators $\hat{\beta }$ and ${\hat{\sigma }}^2$ are obtained by regressing pseudo-exact times (e.g., midpoints) on the covariates. The deterministic regression imputed time is then given by

$${\hat{t}}_i=g^{-1}\left({\hat{\beta }}_0+{\hat{\beta }}_1{X}_{i1}+\cdots +{\hat{\beta }}_p{X}_{ip}\right),$$

while the stochastic version adds residual uncertainty,

$$T_i^{*}=g^{-1}\left({\hat{\beta }}_0+{\hat{\beta }}_1{X}_{i1}+\cdots +{\hat{\beta }}_p{X}_{ip}+\hat{\sigma }{Z}_i\right),Z_i\sim \mathcal{N}(0,1).$$

Finally, the imputed value is truncated to satisfy the censoring interval

$$T_i^{\mathrm{Reg}}\leftarrow min\{max(T_i^{*},L_i),R_i\}.$$

In this study, regression imputation was implemented using the following procedure:

• A pseudo-exact time $T_i^{*}$ was initially created for each interval-censored observation using the midpoint of $(L_i,R_i]$.
• The Weibull accelerated failure time (AFT) model was estimated using the survreg() function from the survival package in R. The survreg() function fits parametric survival regression models under the AFT framework by maximizing the full likelihood, accommodating right-, left-, and interval-censored survival data. Specifically, survreg() assumes that the logarithm of the survival time follows a linear regression model with an additive error term whose distribution determines the parametric form of the survival model. For the Weibull specification, the error term is assumed to follow a standard extreme value (Gumbel) distribution, which implies a Weibull distribution for the survival time $T_i$. Model parameters, including the regression coefficients $\beta$, the intercept ${\beta }_0$, and the scale parameter $\sigma$, are estimated via maximum likelihood.
  The Weibull shape parameter is given by $k=1/\sigma$ and the scale parameter is $\lambda =exp\left({\beta }_0\right)$, linking the AFT parameterization used by survreg() to the conventional Weibull distribution. This flexibility, combined with native support for interval-censored data, makes survreg() well suited for regression-based imputation and survival modeling in interval-censored settings.
• The fitted model produced predicted event times

  $${\hat{t}}_i=exp({\hat{\beta }}_0+{\hat{\beta }}^{\top }{X}_i),$$

  which were then truncated to the subject’s interval:

  $${\hat{t}}_i\leftarrow min\{max({\hat{t}}_i,L_i),R_i\}.$$
• In instances where the Weibull AFT model failed to converge, typically due to limited data variability, sparse failures, or instability in the likelihood, the midpoint of the interval was used as a robust fallback imputation:

  $$T_i^{\mathrm{Reg}}={\displaystyle \frac{L_i+R_i}{2}}.$$
• The final regression imputed times $T_i^{\mathrm{Reg}}$ were analyzed using both the classical KM estimator and the smoothed penalized-likelihood KM estimator.

Regression imputation is known to perform well when covariates are predictive of event time. This approach deliberately departs from left–right interval symmetry by allowing imputed values to be asymmetrically distributed within the censoring interval, reflecting covariate-driven risk heterogeneity and the underlying hazard structure. By accommodating systematic asymmetry in event–time dynamics, regression imputation provides a more realistic representation of the data-generating process, particularly in settings where proportional hazards or distributional skewness are present. In a comparison of imputation methods for doubly censored human immunodeficiency virus (HIV) data, ref. [40] found that regression imputation consistently produced the lowest mean squared error (MSE) for survival estimation, outperforming midpoint and even multiple imputation in several scenarios. Similarly, Zhang and Sun [41] reported that regression-based imputation was markedly more robust than simple random (uniform) or midpoint-based approaches, particularly as the width of the interval increased. These findings support the use of regression imputation as a strong model-based imputation strategy in interval-censored survival analysis and motivated its inclusion in this study.

2.3.3. Uniform Imputation

Uniform imputation is one of the simplest model-free approaches for handling interval-censored survival data. This approach enforces probabilistic allocation by assuming that all points within the interval are equally likely, thereby preserving symmetry with respect to the interval bounds. The idea of replacing an unobserved event time with a random draw from the subject’s censoring interval has been used for decades in simulation-based studies of interval-censoring and missing data and was formally described in the context of multiple imputation by Hsu et al. [42]. Under the uniform imputation approach, the event time is assumed to follow a continuous uniform distribution over the interval

$$T_i^{*}\sim \mathrm{Uniform}(L_i,R_i).$$

(1)

The corresponding density is

$$f_{\mathrm{Uni}}(t\mid L_i,R_i)={\displaystyle \frac{1}{R_i-L_i}}\mathbf{1}\{L_i<t\le R_i\},$$

which assigns equal probability to all possible event times within the observed interval.

In this study, for each individual with an observed event (${\delta }_i=1$), the event time was imputed by sampling a random value from the interval $(L_i,R_i]$ using (1). For right-censored individuals (${\delta }_i=0$, $R_i=\infty$), no imputation was performed; thus, the observed censoring time $L_i$ was retained. A new uniform draw was generated for each bootstrap repetition and each MC dataset to ensure that the induced uncertainty was fully propagated through the performance evaluation metrics. The resulting interval-censored imputed event times were then analyzed using both the classical KM estimator and the improved smoothed penalized-likelihood KM estimator.

Uniform imputation is attractive because it does not assume any specific distribution for $T_i^{*}$ beyond the observed interval, and its results are immediate and do not require numerical optimization. However, the method implicitly assumes that the event is equally likely to occur at any point within $(L_i,R_i]$, which may not hold when the true event–time distribution is skewed or when covariates influence the hazard. As a result, uniform imputation can introduce additional variability or bias, particularly when intervals are wide.

Uniform imputation has frequently been used as a neutral baseline method in studies of interval-censored and doubly censored data. Hsu et al. [42] showed that while uniform imputation performs reasonably well under narrow intervals, it is generally less efficient than probability-based or model-based imputation methods such as regression imputation. Uniform imputation remains valuable as a distribution-free approach, particularly in simulation work, because it isolates the effect of the censoring structure without relying on parametric or semiparametric modeling assumptions. For this reason, it provides an important benchmark against which more sophisticated methods, including regression imputation and multiple imputation, can be evaluated within the context of the improved smoothed penalized likelihood KM estimator considered in this study.

2.3.4. Multiple Imputation

Formally introduced by Rubin [43], multiple imputation is a statistical technique for handling missing or partially observed data by creating several completed datasets and combining their results to capture the uncertainty arising from missing data. In survival analysis, multiple imputation is particularly useful when event times are censored or incompletely observed, as it allows one to incorporate uncertainty about the true event time while enabling analysis with standard complete-data methods.

A total of $m=5$ imputed datasets were generated, consistent with standard practice for simulation-based survival analysis and Rubin’s original recommendations. Each imputed dataset contained a complete set of pseudo-exact event times. Survival estimation was then performed separately on each imputed dataset using the KM estimator. The resulting survival estimates were combined across imputations by averaging the estimated survival functions at each time point, thereby accounting for both within-imputation variability and between-imputation uncertainty according to Rubin’s combining rules.

Let $Y=(Y_{\mathrm{obs}},Y_{\mathrm{mis}})$ denote the complete dataset, where $Y_{\mathrm{mis}}$ represents the missing or censored quantities. Multiple imputation proceeds in three main steps:

• Generate M independent completed datasets:

  $$Y^{\left(m\right)}=\left(Y_{\mathrm{obs}},Y_{\mathrm{mis}}^{\left(m\right)}\right),m=1,\dots ,M,$$

  where each $Y_{\mathrm{mis}}^{\left(m\right)}$ is drawn from an approximation to the posterior predictive distribution

  $$Y_{\mathrm{mis}}^{\left(m\right)}\sim p(Y_{\mathrm{mis}}\mid Y_{\mathrm{obs}}).$$
• Fit the desired model separately to each imputed dataset, obtaining point estimates ${\hat{\theta }}^{\left(m\right)}$ and the corresponding variance estimates ${\hat{U}}^{\left(m\right)}$.
• Combine M results using Rubin’s rules:

  $$\begin{array}{cc}\hfill \overline{\theta }& ={\displaystyle \frac{1}{M}}\sum _{m=1}^M{\hat{\theta }}^{\left(m\right)},\hfill \\ \hfill \overline{U}& ={\displaystyle \frac{1}{M}}\sum _{m=1}^M{\hat{U}}^{\left(m\right)},\hfill \\ \hfill B& ={\displaystyle \frac{1}{M-1}}\sum _{m=1}^M{\left({\hat{\theta }}^{\left(m\right)}-\overline{\theta }\right)}^2,\hfill \\ \hfill T& =\overline{U}+\left(1+{\displaystyle \frac{1}{M}}\right)B,\hfill \end{array}$$

  where $\overline{\theta }$ is the pooled point estimate, $\overline{U}$ represents the average within-imputation variance, B represents the between-imputation variance, and T represents the total variance incorporating both within- and between-imputation uncertainty.

Multiple imputation is widely recognized for its ability to improve efficiency and reduce bias when compared with single imputation methods. In interval-censored settings, Geskus [40] and Zhang and Sun [41] demonstrated that multiple imputation generally outperforms midpoint and uniform imputation, especially when intervals are wide or covariates strongly influence failure times. Although regression-based imputation may achieve lower MSE in some situations, multiple imputation remains uniquely advantageous in fully representing imputation uncertainty.

Multiple imputation relaxes the assumption of interval symmetry by allowing asymmetric allocations within the censoring interval, thereby accommodating data-driven risk heterogeneity and potential skewness in the underlying event-time distribution. By propagating both within-imputation and between-imputation variability, this approach provides more robust inference in the presence of structural asymmetry and uncertainty and is widely used in survival and epidemiological studies where a realistic representation of event-time dynamics is required.

2.4. Performance Evaluation Metrics

Let $S\left(t\right)$ denote the true survival function used to generate the reference curve in the simulation study and let $\hat{S}\left(t\right)$ represent the estimated survival function obtained after applying a given imputation method followed by KM estimation. Performance is assessed by comparing $\hat{S}\left(t\right)$ and $S\left(t\right)$ over a common discrete time grid ${\left\{t_i\right\}}_{i=1}^n$. The pointwise estimation error at time $t_i$ is defined as

$$e_i=S\left(t_i\right)-\hat{S}\left(t_i\right),i=1,\cdots ,n.$$

This curve-based evaluation framework is appropriate when the interest lies in the overall accuracy of the estimated survival function rather than a single scalar parameter, and it is commonly adopted in simulation-based survival studies.

2.4.1. Mean Absolute Error (MAE)

The MAE measures the average absolute deviation between the true and predicted survival functions and is defined as

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|S\left(t_i\right)-\hat{S}\left(t_i\right)\right|.$$

MAE is the most frequently reported error-based metric because it offers direct measures of prediction error [44,45,46]. Zhou et al. [47] reported that MAE is among the most frequently used performance evaluation metrics for survival models (see [48,49,50,51], among others). It has been advocated as a meaningful accuracy measure for survival model evaluation [52], implemented in survival evaluation software [47], and been widely used in censored-data imputation and prediction studies [46,53]. In survival analysis, MAE directly measures the accuracy of predicted time-to-event outcomes [52], making it an important metric for evaluating the precision of survival models [53]. Compared with squared-error measures, MAE does not excessively penalize large deviations, making it less sensitive to isolated extreme errors. Thus, compared with squared-error measures, MAE does not excessively penalize large deviations, making it less sensitive to isolated extreme errors.

2.4.2. Normalized Mean Absolute Error (NMAE)

To facilitate scale-free comparison across simulation scenarios with different survival ranges or sample sizes, the NMAE is also used:

$$\mathrm{NMAE}={\displaystyle \frac{1}{n\mathrm{range}\left(S\right)}}\sum _{i=1}^n\left|S\left(t_i\right)-\hat{S}\left(t_i\right)\right|,$$

where $\mathrm{range}\left(S\right)={max}_{i}S\left(t_i\right)-{min}_{i}S\left(t_i\right)$. This normalization produces a dimensionless metric that enables meaningful comparison across datasets with differing scales or censoring configurations. The use of normalized absolute-error measures is motivated by the robustness and interpretability advantages of MAE over squared-error metrics [44] and extends the widespread application of MAE in survival model evaluation [46,52].

2.4.3. Maximum Absolute Error (MaxAE)

Although MAE summarizes average performance, it may fail to reflect extreme local discrepancies between the estimated and true survival functions. To explicitly capture the worst-case deviation, the MaxAE is considered and defined as

$$\mathrm{MaxAE}=\underset{i=1,\dots ,n}{max}\left|\hat{S}\left(t_i\right)-S\left(t_i\right)\right|.$$

MaxAE measures the largest pointwise error over the evaluation grid and is therefore highly sensitive to extreme deviations. While it is not robust to outliers, MaxAE is useful as a diagnostic metric for identifying local instability or severe mismatches in estimated survival curves. Such worst-case error measures are particularly informative in censored-data imputation settings, where estimation uncertainty may be unevenly distributed across time [46].

2.4.4. Mean Squared Error (MSE)

The MSE averages the squared deviations between estimated and true survival probabilities and is defined as

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(S\left(t_i\right)-\hat{S}\left(t_i\right)\right)}^2.$$

By squaring the errors prior to averaging, MSE assigns greater weight to large discrepancies and is therefore sensitive to pronounced local instability in the estimated survival curve [44,45]. In survival analysis, MSE is typically reported alongside MAE to jointly assess calibration and precision, particularly in comparative studies of parametric and penalized survival models [53]. Similar to MAE, the lower the MSE, the higher the accuracy of prediction [47].

2.4.5. Normalized Root Mean Squared Error (NRMSE)

To retain the sensitivity of squared-error measures while enabling scale-independent interpretation, the NRMSE is defined as

$$\mathrm{NRMSE}={\displaystyle \frac{1}{\mathrm{range}\left(S\right)}}\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(S\left(t_i\right)-\hat{S}\left(t_i\right)\right)}^2}.$$

NRMSE generalizes the commonly used RMSE by scaling with the range of the true survival function. RMSE-based measures have been employed to assess estimator accuracy and stability in penalized nonparametric likelihood methods for arbitrarily censored data [22,35] as well as in interval-censored survival smoothing and estimation studies [22].

2.4.6. Normalized Average Root Median Error Deviation (NARMED)

Mean-based metrics such as MAE and MSE may still be influenced by extreme pointwise deviations, particularly in regions of sparse information such as the tail of the survival curve. To provide a robust complement, the NARMED is considered:

$$\mathrm{NARMED}={\displaystyle \frac{1}{\mathrm{range}\left(S\right)}}\mathrm{median}\left(\left|S\left(t_i\right)-\hat{S}\left(t_i\right)\right|\right).$$

The NARMED builds on the robustness properties of median-based absolute error measures, which are well known to reduce sensitivity to outliers and heavy-tailed error distributions [44]. By combining median-based error assessment with scale normalization, NARMED provides a stable assessment of typical estimation error and is particularly suitable for interval-censored survival data [46].

3. Results

3.1. Simulation Study

3.1.1. Monte Carlo Simulation Design

To evaluate and benchmark the performance of the smoothed nonparametric penalized likelihood KM estimator under various imputation strategies for interval-censored data, a comprehensive MC simulation framework was conducted. As noted by Efron [54], as well as Robert and Casella [55], MC methods offer a flexible and reproducible framework for evaluating statistical procedures under controlled, yet realistic conditions. A key advantage of this simulation approach is the ability to compare estimated survival curves obtained using the classical KM and smoothed penalized likelihood KM estimates against the known true survival estimate. This allows for objective evaluation of the accuracy, bias, variance, and consistency of the estimators under different imputation methods. By performing more than thousands of replications, the simulation provides empirical evidence on the expected performance of each method in controlled settings, where the true event times are known but are treated as unobservable [1]. The MC simulation framework therefore supports both theoretical and practical evaluation of statistical methods for interval-censored data. The study systematically varied key design parameters, such as sample size and prevalence rate, to assess robustness and finite-sample behavior. The classical KM estimator served as a benchmark, providing a baseline for comparison with the smoothed nonparametric penalized likelihood KM estimator. Four imputation strategies, namely, midpoint, regression-based, uniform, and multiple imputation, were examined under both estimators to evaluate how imputation choice influences survival estimation performance.

To obtain accurate and stable performance assessments, each simulation scenario was evaluated using an MC bootstrap framework. For every combination of sample size $\left(n\right)$ and prevalence rate $\left(p\right)$, $m=5000$ bootstrap samples were drawn from a large synthetic population of size $N=500,000$. The use of a large synthetic population and a high number of bootstrap replications was motivated by the need to reduce MC variability and obtain stable estimates of error-based performance measures, particularly under small sample sizes and low prevalence rate. In such settings, performance evaluation metrics can exhibit substantial variability, and insufficient bootstrap replication may obscure meaningful differences between competing methods. The large synthetic population provides an empirical approximation to the underlying data-generating mechanism, thereby limiting finite-population effects when repeatedly sampling across $(n,p)$ combinations. This is especially important for tail-sensitive measures such as MaxAE and robust, median-based metrics such as NARMED, whose convergence may be slower under interval censoring. Each bootstrap sample was subjected to the specified imputation procedure, followed by estimation of the KM survival curve. The resulting survival estimates were compared with the known true survival curve at multiple time points. The final performance scores for each method were obtained by averaging across all bootstrap replications, ensuring a reliable and consistent comparative evaluation. Performance was assessed using six complementary metrics: MAE, NMAE, MaxAE, MSE, NRMSE, and NARMED. Together, these measures provide a comprehensive assessment of bias, variability, robustness, and overall estimation efficiency.

3.1.2. Simulation of Interval-Censored Data

The simulation procedure involved generating synthetic event times and observation intervals for a large number of individuals, from which bootstrap samples were repeatedly drawn for analysis. The key components of the simulation setup were as follows:

• The population size based on a total of $N=500,000$ observations was initially generated to serve as the source population.
• Random samples of size $n\in \{10,20,40,\dots ,500\}$ were drawn from the population in each iteration.
• For each sample size, $m=5000$ bootstrap replicates were generated to ensure stability and precision of the performance estimates.
• The different prevalence rates $\left(p\right)\in \{0.2,0.3,0.4,0.6,0.8,1.0\}$ were applied.

The true event times, denoted by $T_i^{*}$, were independently simulated from a continuous uniform distribution over the interval $(0,24]$ in a way that $T_i^{*}\sim Uniform(0,24)$, $i=1,2,3,\dots ,N,$ where $N=500,000$ determined a sufficiently large population size and was used to allow reliable bootstrapping and robust estimation of empirical survival distributions. The choice of a uniform distribution is common in interval-censored survival simulation studies, as it avoids temporal bias and ensures that all time points are equally likely [1,18].

Each individual was assumed to be observed at regularly spaced follow-up times (e.g., every month), though the specific time unit could vary depending on the context (e.g., daily, weekly, monthly, or annually), such that $\mathcal{V}=\{1,2,3,\dots ,24\}$ led to interval-censored data. For each simulated event time, the corresponding observation interval was determined such that $L_i<T_i^{*}\le R_i$, $L_i,R_i\in \mathcal{V}$. For each subject, the actual event time $T_i^{*}$ was not always directly observed but was captured through an interval $(L_i,R_i]$. Each observation was assigned a binary indicator that was then drawn from a Bernoulli distribution, i.e, ${\delta }_i\in Bernoulli\left(p\right)$, where ${\delta }_i=1$ denoted that the event occurred within the interval and ${\delta }_i=0$ indicated right-censoring. In this study, the event times were subject to Case II interval censoring, where each participant was assessed at multiple scheduled visits, and it was known that the exact event time only fell between two successive observation points.

3.1.3. Generation of True Survival Estimates

In interval-censored settings, exact event times are unobserved, making analytical derivation of the true survival function infeasible. Therefore, a simulation-based benchmark approach was adopted. The procedure used to approximate the true survival function under interval censoring was carried out as follows:

• A population of size $N=500,000$ was used as the source population from which bootstrap samples were drawn.
• For each bootstrap iteration $m=1,2,\dots ,M$, with $M=5000$, a random sample of size n was drawn from the population with a replacement.
• For each bootstrap sample and each imputation method, a KM estimator ${\hat{S}}_m\left(t\right)$ was computed based on the imputed event times.
• To reduce sampling variability and obtain a continuous survival estimate, each KM curve ${\hat{S}}_m\left(t\right)$ was smoothed using cubic spline interpolation.
• The smoothed survival curves were evaluated on a dense and fixed time grid:

  $$\mathcal{T}=\{0,0.25,0.5,\dots ,24\}.$$
• The final reference (true) survival function was obtained by averaging the smoothed survival estimates across all bootstrap samples:

  $$\overline{S}\left(t\right)={\displaystyle \frac{1}{M}}\sum _{m=1}^M{\hat{S}}_m\left(t\right),$$

  where ${\hat{S}}_m\left(t\right)$ denotes the smoothed KM survival probability at time t from the $m\mathrm{th}$ bootstrap replicate.
• The resulting averaged survival function $\overline{S}\left(t\right)$ was treated as the true survival curve and used as the benchmark for evaluating estimator performance.

This bootstrap-based smoothing approach is widely used in survival simulation studies and has been shown to provide stable and reliable reference survival functions when assessing imputation strategies under interval censoring [35,54,56].

3.1.4. Computation of Performance Evaluation Metrics

To obtain accurate and stable performance assessments under a variety of data conditions, each imputation method was evaluated using an MC bootstrap framework. For every combination of sample size $\left(n\right)$ and prevalence rate $\left(p\right)$, $m=5000$ bootstrap samples were drawn from the full synthetic population of $N=500,000$. The samples were then subjected to the imputation procedure; this was followed by estimation of the KM survival curve. The resulting survival curves were compared with the known true survival curve at multiple time points, and the comparison was quantified using six standard performance evaluation metrics, which included MAE, NMAE, MaxAE, MSE, NRMSE, and NRMED. The final performance scores for each method were averaged over all bootstrap replicates, ensuring reliability and consistency in the comparative evaluation.

3.2. Monte Carlo Simulation Results

The results are presented in three stages. First, we provide a descriptive analysis of the simulated data. This is followed by an MC performance metrics evaluation of the KM-based estimators. Finally, a convergence validation analysis is conducted to quantitatively benchmark the performance of the smoothed KM estimator against the classical KM estimator.

3.2.1. Descriptive Analysis of Simulated Interval-Censored Data

Table 1. Illustrative simulated dataset with $n=10$ at $p=0.8$.

ID	Left (L)	Right (R)	Censoring
1	23	24	1
2	24	NA	0
3	14	15	1
4	19	20	1
5	22	23	1
6	5	6	1
7	10	11	1
8	24	NA	0
9	13	14	1
10	5	6	1

Note: L and R denote the left and right endpoints that bound the unobserved event time T. For observed events (Censoring = 1), $T\in (L,R]$. For right-censored cases (Censoring = 0), $R=\mathrm{NA}$ indicates that $T>L$ (the event was not observed within the study period).

presents an illustrative example of a simulated dataset at a prevalence rate of $p=0.8$, showing the first ten subjects with their corresponding endpoints in the left and right intervals and the censoring status. As presented earlier, the simulation framework generated observation intervals with a constant width of two time units (e.g., weeks, months, or years) to mimic realistic follow-up schedules often encountered in longitudinal clinical studies. This structure ensured consistency and controlled variability across all simulated experiments, allowing direct comparison of estimator performance under similar conditions.

To illustrate the general behavior of the simulated survival data, Figure 1 presents the classical and smoothed penalized KM survival curves at a fixed sample size of $n=50$ and prevalence rate $p=0.8$ for the four imputation methods. Across all methods, survival probabilities decreased steadily over time, although the magnitude and smoothness of the decline differed by estimator. The classical KM curves exhibited visible stepwise discontinuities and small irregular jumps, reflecting the sensitivity of the estimator to random sampling variation within narrow censoring intervals. In contrast, the smoothed penalized KM curves showed smoother and more continuous transitions, indicating an effective reduction of overfitted stepwise fluctuations and sampling-induced noise. Overall, at this sample size, these plots demonstrate that the penalized smoothing procedure improves curve continuity and estimator stability without distorting the general shape of the survival function.

3.2.2. Monte Carlo Performance Evaluation of the KM-Based Estimators

This section evaluates how the smoothed penalized KM estimator compares with the classical KM estimator across varying prevalence rates and sample sizes. The performance of the smoothed and classical KM estimators was evaluated through extensive MC simulations ($m=5000$) across varying sample sizes ($n=10$ to 500) and fixed prevalence rates ($p=0.2,0.4,0.6,0.8,1.0$). The evaluation was conducted using four statistical imputation methods, that is, midpoint, regression, uniform, and multiple imputation, and performance was measured using MAE, NMAE, MSE, MaxAE, NRMSE, and NARMED. The results are presented in

Table A2. MC simulation results for mean performance of smoothed and classical KM methods under various imputation methods using $m=5000$ for varying sample sizes at a fixed prevalence rate $=0.2$.

$\mathit{n}$	Metric	Imputation Methods
		Midpoint		Regression		Uniform		Multiple
		Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
10	MAE	0.0465	0.0721	0.0449	0.0696	0.0436	0.0687	0.0713	0.0627
	NMAE	0.1305	0.2025	0.1274	0.1976	0.1257	0.1982	0.2032	0.1775
	MaxAE	0.0920	0.1641	0.0877	0.1617	0.0855	0.1590	0.1398	0.1455
	MSE	0.0041	0.0086	0.0038	0.0080	0.0037	0.0077	0.0090	0.0064
	NRMSE	0.1507	0.2348	0.1467	0.2306	0.1449	0.2310	0.2360	0.2072
	NARMED	0.1312	0.1907	0.1283	0.1842	0.1267	0.1846	0.1958	0.1660
20	MAE	0.0328	0.0476	0.0329	0.0458	0.0328	0.0454	0.0360	0.0419
	NMAE	0.1402	0.2032	0.1435	0.1997	0.1418	0.1961	0.1564	0.1821
	MaxAE	0.0707	0.1116	0.0702	0.1094	0.0700	0.1083	0.0777	0.0988
	MSE	0.0022	0.0040	0.0022	0.0037	0.0022	0.0036	0.0027	0.0030
	NRMSE	0.1660	0.2395	0.1701	0.2364	0.1681	0.2318	0.1859	0.2149
	NARMED	0.1361	0.1862	0.1408	0.1831	0.1377	0.1786	0.1497	0.1672
30	MAE	0.0303	0.0401	0.0302	0.0375	0.0300	0.0373	0.0295	0.0339
	NMAE	0.1458	0.1930	0.1471	0.1828	0.1445	0.1800	0.1437	0.1650
	MaxAE	0.0653	0.0967	0.0639	0.0932	0.0635	0.0933	0.0633	0.0847
	MSE	0.0019	0.0030	0.0019	0.0026	0.0019	0.0026	0.0018	0.0021
	NRMSE	0.1727	0.2309	0.1741	0.2201	0.1710	0.2173	0.1709	0.1995
	NARMED	0.1437	0.1745	0.1462	0.1630	0.1435	0.1607	0.1413	0.1466
40	MAE	0.0266	0.0338	0.0271	0.0311	0.0270	0.0314	0.0275	0.0296
	NMAE	0.1319	0.1678	0.1336	0.1532	0.1344	0.1562	0.1379	0.1483
	MaxAE	0.0574	0.0849	0.0569	0.0812	0.0563	0.0815	0.0573	0.0755
	MSE	0.0015	0.0021	0.0015	0.0018	0.0015	0.0019	0.0015	0.0017
	NRMSE	0.1568	0.2036	0.1582	0.1867	0.1590	0.1905	0.1629	0.1807
	NARMED	0.1299	0.1483	0.1330	0.1342	0.1339	0.1358	0.1385	0.1298
50	MAE	0.0244	0.0294	0.0256	0.0282	0.0249	0.0274	0.0253	0.0259
	NMAE	0.1219	0.1468	0.1284	0.1413	0.1245	0.1371	0.1254	0.1279
	MaxAE	0.0520	0.0760	0.0520	0.0740	0.0510	0.0726	0.0517	0.0678
	MSE	0.0012	0.0016	0.0013	0.0015	0.0012	0.0014	0.0013	0.0013
	NRMSE	0.1445	0.1793	0.1512	0.1729	0.1467	0.1682	0.1481	0.1574
	NARMED	0.1210	0.1284	0.1298	0.1241	0.1251	0.1192	0.1266	0.1103
80	MAE	0.0213	0.0228	0.0219	0.0210	0.0224	0.0213	0.0214	0.0200
	NMAE	0.1066	0.1142	0.1103	0.1058	0.1117	0.1061	0.1061	0.0994
	MaxAE	0.0431	0.0607	0.0421	0.0575	0.0435	0.0585	0.0419	0.0540
	MSE	0.0009	0.0010	0.0009	0.0009	0.0009	0.0009	0.0009	0.0008
	NRMSE	0.1248	0.1407	0.1284	0.1315	0.1302	0.1319	0.1240	0.1239
	NARMED	0.1083	0.1003	0.1134	0.0908	0.1147	0.0915	0.1085	0.0848
100	MAE	0.0191	0.0196	0.0213	0.0188	0.0202	0.0180	0.0210	0.0179
	NMAE	0.0953	0.0976	0.1074	0.0944	0.1005	0.0898	0.1052	0.0899
	MaxAE	0.0382	0.0535	0.0396	0.0523	0.0385	0.0513	0.0394	0.0485
	MSE	0.0007	0.0007	0.0008	0.0007	0.0007	0.0006	0.0008	0.0006
	NRMSE	0.1112	0.1204	0.1233	0.1182	0.1163	0.1126	0.1214	0.1124
	NARMED	0.0972	0.0851	0.1124	0.0803	0.1039	0.0756	0.1091	0.0764
300	MAE	0.0163	0.0117	0.0180	0.0101	0.0179	0.0102	0.0176	0.0101
	NMAE	0.0818	0.0587	0.0905	0.0506	0.0896	0.0513	0.0874	0.0501
	MaxAE	0.0290	0.0345	0.0292	0.0294	0.0292	0.0298	0.0291	0.0280
	MSE	0.0004	0.0003	0.0005	0.0002	0.0005	0.0002	0.0005	0.0002
	NRMSE	0.0922	0.0738	0.1000	0.0641	0.0993	0.0651	0.0971	0.0634
	NARMED	0.0867	0.0498	0.0988	0.0431	0.0975	0.0434	0.0947	0.0427
500	MAE	0.0153	0.0094	0.0174	0.0080	0.0171	0.0078	0.0172	0.0078
	NMAE	0.0762	0.0467	0.0871	0.0399	0.0854	0.0388	0.0859	0.0388
	MaxAE	0.0260	0.0285	0.0273	0.0233	0.0273	0.0229	0.0269	0.0217
	MSE	0.0003	0.0002	0.0004	0.0001	0.0004	0.0001	0.0004	0.0001
	NRMSE	0.0851	0.0590	0.0955	0.0507	0.0939	0.0494	0.0943	0.0493
	NARMED	0.0817	0.0396	0.0958	0.0338	0.0934	0.0326	0.0948	0.0331

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

,

Table A3. MC simulation results for mean performance of smoothed and classical KM methods under various imputation methods using $m=5000$ for varying sample sizes at a fixed prevalence rate $=0.4$.

$\mathit{n}$	Metric	Imputation Methods
		Midpoint		Regression		Uniform		Multiple
		Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
10	MAE	0.0601	0.0886	0.0608	0.0859	0.0600	0.0850	0.0651	0.0805
	NMAE	0.1332	0.1963	0.1377	0.1944	0.1344	0.1905	0.1461	0.1805
	MaxAE	0.1252	0.2051	0.1258	0.2006	0.1255	0.1999	0.1363	0.1868
	MSE	0.0069	0.0132	0.0073	0.0126	0.0071	0.0122	0.0084	0.0108
	NRMSE	0.1558	0.2301	0.1611	0.2282	0.1582	0.2239	0.1722	0.2121
	NARMED	0.1318	0.1812	0.1368	0.1801	0.1321	0.1765	0.1422	0.1674
20	MAE	0.0470	0.0620	0.0497	0.0602	0.0495	0.0597	0.0470	0.0561
	NMAE	0.1174	0.1549	0.1229	0.1489	0.1234	0.1488	0.1164	0.1391
	MaxAE	0.0996	0.1542	0.1006	0.1523	0.1013	0.1521	0.0965	0.1407
	MSE	0.0043	0.0068	0.0048	0.0066	0.0048	0.0065	0.0044	0.0057
	NRMSE	0.1385	0.1862	0.1443	0.1801	0.1450	0.1800	0.1373	0.1683
	NARMED	0.1167	0.1397	0.1235	0.1331	0.1241	0.1321	0.1157	0.1235
30	MAE	0.0408	0.0496	0.0440	0.0475	0.0431	0.0465	0.0431	0.0455
	NMAE	0.1018	0.1240	0.1104	0.1194	0.1079	0.1163	0.1072	0.1132
	MaxAE	0.0825	0.1266	0.0844	0.1251	0.0832	0.1234	0.0843	0.1178
	MSE	0.0032	0.0045	0.0036	0.0042	0.0035	0.0040	0.0035	0.0039
	NRMSE	0.1194	0.1501	0.1281	0.1462	0.1253	0.1426	0.1251	0.1389
	NARMED	0.1024	0.1116	0.1134	0.1053	0.1107	0.1015	0.1090	0.0990
40	MAE	0.0384	0.0426	0.0401	0.0398	0.0406	0.0399	0.0399	0.0391
	NMAE	0.0960	0.1065	0.0999	0.0992	0.1017	0.1000	0.0998	0.0978
	MaxAE	0.0746	0.1110	0.0752	0.1076	0.0753	0.1079	0.0745	0.1017
	MSE	0.0027	0.0034	0.0029	0.0030	0.0029	0.0030	0.0028	0.0029
	NRMSE	0.1113	0.1299	0.1152	0.1223	0.1168	0.1233	0.1149	0.1202
	NARMED	0.0981	0.0948	0.1029	0.0863	0.1054	0.0872	0.1028	0.0853
50	MAE	0.0366	0.0380	0.0387	0.0353	0.0386	0.0353	0.0384	0.0347
	NMAE	0.0918	0.0954	0.0966	0.0882	0.0963	0.0881	0.0957	0.0865
	MaxAE	0.0698	0.1013	0.0698	0.0966	0.0705	0.0974	0.0696	0.0914
	MSE	0.0023	0.0027	0.0025	0.0024	0.0026	0.0024	0.0025	0.0023
	NRMSE	0.1058	0.1168	0.1100	0.1090	0.1101	0.1094	0.1094	0.1069
	NARMED	0.0941	0.0843	0.1007	0.0769	0.0998	0.0763	0.0997	0.0756
80	MAE	0.0333	0.0298	0.0361	0.0276	0.0366	0.0276	0.0363	0.0271
	NMAE	0.0832	0.0745	0.0901	0.0688	0.0915	0.0689	0.0908	0.0677
	MaxAE	0.0600	0.0820	0.0616	0.0776	0.0620	0.0769	0.0612	0.0721
	MSE	0.0018	0.0017	0.0021	0.0015	0.0021	0.0015	0.0021	0.0014
	NRMSE	0.0943	0.0920	0.1011	0.0861	0.1025	0.0858	0.1016	0.0844
	NARMED	0.0871	0.0658	0.0957	0.0593	0.0979	0.0599	0.0968	0.0589
100	MAE	0.0323	0.0270	0.0358	0.0245	0.0356	0.0243	0.0353	0.0240
	NMAE	0.0807	0.0674	0.0897	0.0615	0.0893	0.0608	0.0883	0.0599
	MaxAE	0.0570	0.0749	0.0588	0.0690	0.0583	0.0686	0.0577	0.0642
	MSE	0.0016	0.0014	0.0019	0.0012	0.0019	0.0011	0.0019	0.0011
	NRMSE	0.0909	0.0832	0.0995	0.0770	0.0991	0.0761	0.0980	0.0748
	NARMED	0.0849	0.0596	0.0962	0.0535	0.0962	0.0522	0.0951	0.0521
300	MAE	0.0279	0.0162	0.0334	0.0136	0.0330	0.0137	0.0331	0.0136
	NMAE	0.0698	0.0406	0.0836	0.0341	0.0824	0.0341	0.0827	0.0339
	MaxAE	0.0466	0.0486	0.0511	0.0393	0.0508	0.0390	0.0506	0.0367
	MSE	0.0011	0.0005	0.0015	0.0004	0.0015	0.0004	0.0015	0.0004
	NRMSE	0.0777	0.0507	0.0915	0.0432	0.0901	0.0429	0.0904	0.0426
	NARMED	0.0744	0.0352	0.0911	0.0294	0.0897	0.0297	0.0902	0.0297
500	MAE	0.0273	0.0135	0.0334	0.0107	0.0328	0.0106	0.0333	0.0106
	NMAE	0.0680	0.0337	0.0833	0.0268	0.0821	0.0264	0.0831	0.0264
	MaxAE	0.0449	0.0413	0.0509	0.0308	0.0501	0.0303	0.0505	0.0286
	MSE	0.0010	0.0003	0.0015	0.0002	0.0014	0.0002	0.0015	0.0002
	NRMSE	0.0757	0.0421	0.0910	0.0339	0.0898	0.0334	0.0908	0.0332
	NARMED	0.0728	0.0291	0.0909	0.0231	0.0897	0.0229	0.0905	0.0230

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

,

Table A4. MC simulation results for mean performance of smoothed and classical KM methods under various imputation methods using $m=5000$ for varying sample sizes at a fixed prevalence rate $=0.6$.

$\mathit{n}$	Metric	Imputation Methods
		Midpoint		Regression		Uniform		Multiple
		Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
10	MAE	0.0705	0.0984	0.0688	0.0968	0.0683	0.0973	0.0711	0.0954
	NMAE	0.1152	0.1608	0.1134	0.1596	0.1128	0.1605	0.1173	0.1574
	MaxAE	0.1463	0.2374	0.1434	0.2360	0.1418	0.2369	0.1480	0.2270
	MSE	0.0096	0.0163	0.0091	0.0158	0.0091	0.0161	0.0099	0.0155
	NRMSE	0.1350	0.1912	0.1332	0.1899	0.1323	0.1912	0.1379	0.1871
	NARMED	0.1144	0.1476	0.1127	0.1462	0.1120	0.1473	0.1159	0.1444
20	MAE	0.0547	0.0707	0.0571	0.0670	0.0583	0.0678	0.0559	0.0656
	NMAE	0.0911	0.1178	0.0954	0.1118	0.0971	0.1129	0.0933	0.1102
	MaxAE	0.1094	0.1779	0.1090	0.1732	0.1108	0.1751	0.1082	0.1642
	MSE	0.0055	0.0087	0.0058	0.0079	0.0060	0.0082	0.0056	0.0077
	NRMSE	0.1063	0.1415	0.1099	0.1355	0.1119	0.1368	0.1080	0.1334
	NARMED	0.0905	0.1070	0.0968	0.0999	0.0985	0.1011	0.0933	0.0989
30	MAE	0.0493	0.0568	0.0517	0.0529	0.0524	0.0530	0.0516	0.0519
	NMAE	0.0823	0.0947	0.0859	0.0879	0.0871	0.0880	0.0858	0.0863
	MaxAE	0.0944	0.1464	0.0931	0.1415	0.0940	0.1413	0.0937	0.1336
	MSE	0.0042	0.0057	0.0044	0.0051	0.0045	0.0051	0.0044	0.0049
	NRMSE	0.0950	0.1149	0.0975	0.1076	0.0989	0.1076	0.0977	0.1056
	NARMED	0.0818	0.0856	0.0875	0.0778	0.0893	0.0784	0.0870	0.0763
40	MAE	0.0458	0.0486	0.0511	0.0458	0.0502	0.0455	0.0498	0.0448
	NMAE	0.0762	0.0808	0.0852	0.0764	0.0837	0.0758	0.0829	0.0747
	MaxAE	0.0845	0.1289	0.0875	0.1241	0.0866	0.1230	0.0860	0.1158
	MSE	0.0034	0.0042	0.0041	0.0038	0.0040	0.0038	0.0039	0.0037
	NRMSE	0.0867	0.0983	0.0955	0.0940	0.0939	0.0933	0.0930	0.0916
	NARMED	0.0769	0.0727	0.0884	0.0673	0.0867	0.0672	0.0849	0.0661
50	MAE	0.0455	0.0442	0.0490	0.0408	0.0496	0.0409	0.0482	0.0399
	NMAE	0.0757	0.0735	0.0819	0.0681	0.0828	0.0682	0.0803	0.0665
	MaxAE	0.0810	0.1187	0.0826	0.1108	0.0830	0.1111	0.0813	0.1036
	MSE	0.0032	0.0035	0.0037	0.0030	0.0037	0.0031	0.0035	0.0029
	NRMSE	0.0855	0.0896	0.0913	0.0840	0.0922	0.0840	0.0896	0.0818
	NARMED	0.0768	0.0661	0.0846	0.0604	0.0858	0.0605	0.0826	0.0589
80	MAE	0.0420	0.0349	0.0461	0.0315	0.0468	0.0314	0.0469	0.0313
	NMAE	0.0699	0.0580	0.0767	0.0524	0.0779	0.0523	0.0782	0.0522
	MaxAE	0.0725	0.0964	0.0749	0.0872	0.0760	0.0869	0.0754	0.0816
	MSE	0.0026	0.0022	0.0031	0.0018	0.0031	0.0018	0.0031	0.0018
	NRMSE	0.0783	0.0711	0.0848	0.0650	0.0862	0.0648	0.0863	0.0644
	NARMED	0.0716	0.0519	0.0796	0.0461	0.0811	0.0461	0.0816	0.0466
100	MAE	0.0410	0.0317	0.0464	0.0284	0.0461	0.0284	0.0458	0.0278
	NMAE	0.0682	0.0527	0.0774	0.0472	0.0770	0.0473	0.0763	0.0464
	MaxAE	0.0701	0.0883	0.0737	0.0783	0.0736	0.0785	0.0731	0.0731
	MSE	0.0024	0.0018	0.0030	0.0015	0.0030	0.0015	0.0030	0.0014
	NRMSE	0.0762	0.0646	0.0852	0.0586	0.0848	0.0588	0.0841	0.0575
	NARMED	0.0697	0.0473	0.0811	0.0419	0.0803	0.0419	0.0796	0.0413
300	MAE	0.0363	0.0201	0.0453	0.0163	0.0451	0.0160	0.0447	0.0158
	NMAE	0.0605	0.0336	0.0754	0.0271	0.0750	0.0267	0.0746	0.0263
	MaxAE	0.0614	0.0589	0.0712	0.0451	0.0711	0.0448	0.0712	0.0421
	MSE	0.0018	0.0007	0.0027	0.0005	0.0027	0.0005	0.0026	0.0005
	NRMSE	0.0680	0.0414	0.0831	0.0338	0.0827	0.0333	0.0824	0.0328
	NARMED	0.0621	0.0299	0.0793	0.0241	0.0788	0.0236	0.0780	0.0233
500	MAE	0.0352	0.0171	0.0444	0.0125	0.0443	0.0124	0.0442	0.0122
	NMAE	0.0589	0.0287	0.0740	0.0209	0.0737	0.0206	0.0736	0.0203
	MaxAE	0.0608	0.0507	0.0709	0.0350	0.0707	0.0347	0.0706	0.0324
	MSE	0.0017	0.0005	0.0026	0.0003	0.0026	0.0003	0.0026	0.0003
	NRMSE	0.0669	0.0354	0.0819	0.0261	0.0816	0.0257	0.0815	0.0253
	NARMED	0.0609	0.0254	0.0778	0.0185	0.0775	0.0183	0.0777	0.0181

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

,

Table A5. MC simulation results for mean performance of smoothed and classical KM methods under various imputation methods using $m=5000$ for varying sample sizes at a fixed prevalence rate $=0.8$.

$\mathit{n}$	Metric	Imputation Methods
		Midpoint		Regression		Uniform		Multiple
		Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
10	MAE	0.0662	0.1067	0.0673	0.1022	0.0671	0.1022	0.0667	0.1016
	NMAE	0.0828	0.1335	0.0844	0.1281	0.0840	0.1280	0.0836	0.1272
	MaxAE	0.1378	0.2616	0.1369	0.2561	0.1353	0.2562	0.1346	0.2451
	MSE	0.0084	0.0186	0.0085	0.0173	0.0085	0.0174	0.0083	0.0171
	NRMSE	0.0979	0.1583	0.0989	0.1528	0.0986	0.1526	0.0979	0.1509
	NARMED	0.0811	0.1236	0.0833	0.1173	0.0832	0.1171	0.0827	0.1173
20	MAE	0.0531	0.0738	0.0573	0.0702	0.0562	0.0693	0.0567	0.0697
	NMAE	0.0663	0.0921	0.0716	0.0876	0.0700	0.0865	0.0710	0.0872
	MaxAE	0.1054	0.1897	0.1078	0.1841	0.1058	0.1829	0.1063	0.1741
	MSE	0.0047	0.0092	0.0053	0.0085	0.0051	0.0083	0.0052	0.0084
	NRMSE	0.0769	0.1105	0.0819	0.1060	0.0802	0.1048	0.0809	0.1050
	NARMED	0.0641	0.0837	0.0700	0.0791	0.0684	0.0777	0.0699	0.0793
30	MAE	0.0496	0.0604	0.0540	0.0561	0.0537	0.0556	0.0538	0.0558
	NMAE	0.0621	0.0755	0.0676	0.0703	0.0671	0.0695	0.0674	0.0698
	MaxAE	0.0943	0.1580	0.0964	0.1508	0.0958	0.1496	0.0964	0.1414
	MSE	0.0039	0.0062	0.0043	0.0055	0.0043	0.0054	0.0043	0.0054
	NRMSE	0.0710	0.0910	0.0761	0.0856	0.0756	0.0847	0.0759	0.0846
	NARMED	0.0600	0.0688	0.0663	0.0629	0.0654	0.0622	0.0659	0.0628
40	MAE	0.0474	0.0523	0.0524	0.0482	0.0524	0.0485	0.0522	0.0477
	NMAE	0.0591	0.0652	0.0653	0.0601	0.0654	0.0605	0.0652	0.0596
	MaxAE	0.0875	0.1398	0.0920	0.1301	0.0916	0.1310	0.0915	0.1225
	MSE	0.0034	0.0047	0.0040	0.0041	0.0040	0.0041	0.0039	0.0040
	NRMSE	0.0672	0.0788	0.0734	0.0734	0.0734	0.0740	0.0731	0.0726
	NARMED	0.0572	0.0593	0.0636	0.0538	0.0639	0.0542	0.0638	0.0535
50	MAE	0.0454	0.0467	0.0518	0.0431	0.0513	0.0432	0.0502	0.0421
	NMAE	0.0568	0.0584	0.0648	0.0539	0.0642	0.0540	0.0627	0.0525
	MaxAE	0.0837	0.1258	0.0898	0.1172	0.0882	0.1171	0.0874	0.1080
	MSE	0.0030	0.0037	0.0038	0.0033	0.0037	0.0033	0.0036	0.0031
	NRMSE	0.0644	0.0706	0.0726	0.0660	0.0717	0.0660	0.0704	0.0641
	NARMED	0.0545	0.0532	0.0631	0.0484	0.0630	0.0484	0.0609	0.0473
80	MAE	0.0424	0.0376	0.0496	0.0337	0.0495	0.0336	0.0488	0.0333
	NMAE	0.0529	0.0470	0.0619	0.0420	0.0617	0.0419	0.0611	0.0416
	MaxAE	0.0776	0.1047	0.0864	0.0925	0.0869	0.0925	0.0851	0.0859
	MSE	0.0026	0.0024	0.0034	0.0020	0.0034	0.0020	0.0034	0.0020
	NRMSE	0.0604	0.0571	0.0698	0.0516	0.0697	0.0516	0.0688	0.0510
	NARMED	0.0505	0.0426	0.0587	0.0378	0.0589	0.0376	0.0586	0.0375
100	MAE	0.0411	0.0342	0.0483	0.0298	0.0493	0.0302	0.0488	0.0300
	NMAE	0.0514	0.0428	0.0604	0.0373	0.0617	0.0378	0.0611	0.0375
	MaxAE	0.0765	0.0961	0.0855	0.0825	0.0865	0.0828	0.0856	0.0771
	MSE	0.0024	0.0020	0.0033	0.0016	0.0034	0.0016	0.0033	0.0016
	NRMSE	0.0591	0.0520	0.0686	0.0459	0.0699	0.0466	0.0691	0.0459
	NARMED	0.0482	0.0387	0.0568	0.0333	0.0585	0.0340	0.0580	0.0341
300	MAE	0.0359	0.0230	0.0465	0.0171	0.0464	0.0175	0.0459	0.0170
	NMAE	0.0449	0.0288	0.0581	0.0214	0.0580	0.0219	0.0574	0.0213
	MaxAE	0.0718	0.0673	0.0863	0.0472	0.0858	0.0480	0.0851	0.0441
	MSE	0.0019	0.0009	0.0031	0.0005	0.0031	0.0005	0.0030	0.0005
	NRMSE	0.0534	0.0351	0.0674	0.0264	0.0672	0.0269	0.0665	0.0262
	NARMED	0.0398	0.0259	0.0534	0.0193	0.0532	0.0197	0.0525	0.0193
500	MAE	0.0343	0.0200	0.0454	0.0131	0.0448	0.0134	0.0452	0.0132
	NMAE	0.0429	0.0250	0.0568	0.0164	0.0560	0.0167	0.0565	0.0165
	MaxAE	0.0714	0.0588	0.0848	0.0365	0.0843	0.0371	0.0851	0.0343
	MSE	0.0018	0.0007	0.0030	0.0003	0.0029	0.0003	0.0030	0.0003
	NRMSE	0.0521	0.0306	0.0662	0.0203	0.0655	0.0207	0.0661	0.0203
	NARMED	0.0364	0.0225	0.0530	0.0147	0.0512	0.0149	0.0520	0.0150

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

,

Table A6. MC simulation results for mean performance of smoothed and classical KM methods under various imputation methods using $m=5000$ for varying sample sizes at a fixed prevalence rate $=1.0$.

$\mathit{n}$	Metric	Imputation Methods
		Midpoint		Regression		Uniform		Multiple
		Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
10	MAE	0.0671	0.0846	0.0617	0.0829	0.0625	0.0824	0.0606	0.0802
	NMAE	0.0673	0.0849	0.0618	0.0831	0.0627	0.0826	0.0607	0.0803
	MaxAE	0.1379	0.2671	0.1301	0.2614	0.1320	0.2606	0.1271	0.2407
	MSE	0.0083	0.0135	0.0071	0.0127	0.0072	0.0126	0.0067	0.0119
	NRMSE	0.0808	0.1091	0.0746	0.1061	0.0756	0.1056	0.0730	0.1020
	NARMED	0.0632	0.0687	0.0571	0.0671	0.0578	0.0666	0.0563	0.0656
20	MAE	0.0543	0.0618	0.0538	0.0578	0.0524	0.0576	0.0521	0.0567
	NMAE	0.0547	0.0622	0.0541	0.0581	0.0528	0.0580	0.0525	0.0571
	MaxAE	0.1133	0.1987	0.1112	0.1860	0.1090	0.1847	0.1077	0.1701
	MSE	0.0054	0.0071	0.0050	0.0062	0.0048	0.0061	0.0047	0.0059
	NRMSE	0.0664	0.0794	0.0650	0.0743	0.0635	0.0740	0.0630	0.0724
	NARMED	0.0497	0.0496	0.0489	0.0458	0.0476	0.0458	0.0476	0.0459
30	MAE	0.0509	0.0515	0.0513	0.0467	0.0509	0.0473	0.0492	0.0459
	NMAE	0.0513	0.0519	0.0517	0.0471	0.0514	0.0477	0.0497	0.0463
	MaxAE	0.1046	0.1690	0.1047	0.1506	0.1038	0.1524	0.1001	0.1368
	MSE	0.0046	0.0049	0.0044	0.0040	0.0043	0.0042	0.0040	0.0039
	NRMSE	0.0618	0.0661	0.0617	0.0600	0.0613	0.0609	0.0591	0.0585
	NARMED	0.0468	0.0411	0.0470	0.0370	0.0468	0.0376	0.0457	0.0370
40	MAE	0.0512	0.0451	0.0504	0.0402	0.0495	0.0409	0.0494	0.0399
	NMAE	0.0516	0.0455	0.0509	0.0406	0.0500	0.0413	0.0499	0.0403
	MaxAE	0.1047	0.1512	0.1009	0.1307	0.0992	0.1315	0.0987	0.1189
	MSE	0.0046	0.0038	0.0041	0.0030	0.0040	0.0031	0.0040	0.0030
	NRMSE	0.0619	0.0581	0.0602	0.0518	0.0591	0.0526	0.0589	0.0510
	NARMED	0.0468	0.0360	0.0473	0.0318	0.0465	0.0326	0.0467	0.0323
50	MAE	0.0512	0.0411	0.0494	0.0362	0.0489	0.0365	0.0487	0.0355
	NMAE	0.0516	0.0414	0.0498	0.0366	0.0494	0.0369	0.0491	0.0359
	MaxAE	0.1029	0.1397	0.0975	0.1178	0.0974	0.1179	0.0962	0.1063
	MSE	0.0045	0.0031	0.0039	0.0024	0.0039	0.0025	0.0038	0.0023
	NRMSE	0.0615	0.0530	0.0585	0.0468	0.0582	0.0472	0.0578	0.0456
	NARMED	0.0475	0.0328	0.0469	0.0289	0.0464	0.0293	0.0464	0.0287
80	MAE	0.0499	0.0340	0.0480	0.0286	0.0484	0.0284	0.0473	0.0279
	NMAE	0.0504	0.0343	0.0485	0.0288	0.0489	0.0287	0.0478	0.0281
	MaxAE	0.0985	0.1190	0.0928	0.0923	0.0938	0.0923	0.0915	0.0838
	MSE	0.0042	0.0022	0.0036	0.0015	0.0037	0.0015	0.0035	0.0014
	NRMSE	0.0595	0.0442	0.0564	0.0369	0.0570	0.0367	0.0557	0.0359
	NARMED	0.0469	0.0269	0.0467	0.0231	0.0469	0.0229	0.0460	0.0227
100	MAE	0.0502	0.0317	0.0480	0.0254	0.0468	0.0233	0.0470	0.0249
	NMAE	0.0506	0.0320	0.0485	0.0257	0.0473	0.0235	0.0475	0.0252
	MaxAE	0.0975	0.1123	0.0923	0.0831	0.0898	0.0752	0.0893	0.0748
	MSE	0.0042	0.0019	0.0036	0.0012	0.0034	0.0010	0.0034	0.0012
	NRMSE	0.0594	0.0414	0.0562	0.0330	0.0548	0.0301	0.0549	0.0321
	NARMED	0.0474	0.0250	0.0470	0.0204	0.0460	0.0189	0.0465	0.0204
300	MAE	0.0461	0.0237	0.0440	0.0147	0.0440	0.0147	0.0447	0.0141
	NMAE	0.0465	0.0240	0.0445	0.0148	0.0445	0.0148	0.0452	0.0143
	MaxAE	0.0867	0.0875	0.0849	0.0477	0.0848	0.0475	0.0853	0.0427
	MSE	0.0035	0.0011	0.0031	0.0004	0.0031	0.0004	0.0031	0.0004
	NRMSE	0.0539	0.0315	0.0513	0.0190	0.0512	0.0190	0.0519	0.0182
	NARMED	0.0446	0.0180	0.0434	0.0119	0.0437	0.0119	0.0446	0.0115
500	MAE	0.0448	0.0218	0.0422	0.0113	0.0420	0.0113	0.0422	0.0111
	NMAE	0.0453	0.0220	0.0426	0.0115	0.0425	0.0114	0.0427	0.0112
	MaxAE	0.0836	0.0814	0.0826	0.0368	0.0819	0.0367	0.0823	0.0335
	MSE	0.0033	0.0009	0.0028	0.0002	0.0028	0.0002	0.0028	0.0002
	NRMSE	0.0521	0.0292	0.0492	0.0147	0.0489	0.0147	0.0492	0.0143
	NARMED	0.0440	0.0161	0.0415	0.0092	0.0415	0.0092	0.0418	0.0090

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

,

Table A7. MC simulation results for mean performance evaluation metrics under various imputation methods for varying prevalence rates p at fixed sample sizes $n=10$ and $n=30$.

$\mathit{n}$	$\mathit{p}$	Metric	Imputation Methods
			Midpoint		Regression		Uniform		Multiple
			Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
	0.2	MAE	0.0465	0.0721	0.0449	0.0696	0.0436	0.0687	0.0713	0.0627
		NMAE	0.1305	0.2025	0.1274	0.1976	0.1257	0.1982	0.2032	0.1775
		MaxAE	0.0920	0.1641	0.0877	0.1617	0.0855	0.1590	0.1398	0.1455
		MSE	0.0041	0.0086	0.0038	0.0080	0.0037	0.0077	0.0090	0.0064
		NRMSE	0.1507	0.2348	0.1467	0.2306	0.1449	0.2310	0.2360	0.2072
		NARMED	0.1312	0.1907	0.1283	0.1842	0.1267	0.1846	0.1958	0.1660
	0.4	MAE	0.0601	0.0886	0.0608	0.0859	0.0600	0.0850	0.0651	0.0805
		NMAE	0.1332	0.1963	0.1377	0.1944	0.1344	0.1905	0.1461	0.1805
		MaxAE	0.1252	0.2051	0.1258	0.2006	0.1255	0.1999	0.1363	0.1868
		MSE	0.0069	0.0132	0.0073	0.0126	0.0071	0.0122	0.0084	0.0108
		NRMSE	0.1558	0.2301	0.1611	0.2282	0.1582	0.2239	0.1722	0.2121
		NARMED	0.1318	0.1812	0.1368	0.1801	0.1321	0.1765	0.1422	0.1674
10	0.6	MAE	0.0705	0.0984	0.0688	0.0968	0.0683	0.0973	0.0711	0.0954
		NMAE	0.1152	0.1608	0.1134	0.1596	0.1128	0.1605	0.1173	0.1574
		MaxAE	0.1463	0.2374	0.1434	0.2360	0.1418	0.2369	0.1480	0.2270
		MSE	0.0096	0.0163	0.0091	0.0158	0.0091	0.0161	0.0099	0.0155
		NRMSE	0.1350	0.1912	0.1332	0.1899	0.1323	0.1912	0.1379	0.1871
		NARMED	0.1144	0.1476	0.1127	0.1462	0.1120	0.1473	0.1159	0.1444
	0.8	MAE	0.0662	0.1067	0.0673	0.1022	0.0671	0.1022	0.0667	0.1016
		NMAE	0.0828	0.1335	0.0844	0.1281	0.0840	0.1280	0.0836	0.1272
		MaxAE	0.1378	0.2616	0.1369	0.2561	0.1353	0.2562	0.1346	0.2451
		MSE	0.0084	0.0186	0.0085	0.0173	0.0085	0.0174	0.0083	0.0171
		NRMSE	0.0979	0.1583	0.0989	0.1528	0.0986	0.1526	0.0979	0.1509
		NARMED	0.0811	0.1236	0.0833	0.1173	0.0832	0.1171	0.0827	0.1173
	1.0	MAE	0.0671	0.0846	0.0617	0.0829	0.0625	0.0824	0.0606	0.0802
		NMAE	0.0673	0.0849	0.0618	0.0831	0.0627	0.0826	0.0607	0.0803
		MaxAE	0.1379	0.2671	0.1301	0.2614	0.1320	0.2606	0.1271	0.2407
		MSE	0.0083	0.0135	0.0071	0.0127	0.0072	0.0126	0.0067	0.0119
		NRMSE	0.0808	0.1091	0.0746	0.1061	0.0756	0.1056	0.0730	0.1020
		NARMED	0.0632	0.0687	0.0571	0.0671	0.0578	0.0666	0.0563	0.0656
	0.2	MAE	0.0303	0.0401	0.0302	0.0375	0.0300	0.0373	0.0295	0.0339
		NMAE	0.1458	0.1930	0.1471	0.1828	0.1445	0.1800	0.1437	0.1650
		MaxAE	0.0653	0.0967	0.0639	0.0932	0.0635	0.0933	0.0633	0.0847
		MSE	0.0019	0.0030	0.0019	0.0026	0.0019	0.0026	0.0018	0.0021
		NRMSE	0.1727	0.2309	0.1741	0.2201	0.1710	0.2173	0.1709	0.1995
		NARMED	0.1437	0.1745	0.1462	0.1630	0.1435	0.1607	0.1413	0.1466
	0.4	MAE	0.0408	0.0496	0.0440	0.0475	0.0431	0.0465	0.0431	0.0455
		NMAE	0.1018	0.1240	0.1104	0.1194	0.1079	0.1163	0.1072	0.1132
		MaxAE	0.0825	0.1266	0.0844	0.1251	0.0832	0.1234	0.0843	0.1178
		MSE	0.0032	0.0045	0.0036	0.0042	0.0035	0.0040	0.0035	0.0039
		NRMSE	0.1194	0.1501	0.1281	0.1462	0.1253	0.1426	0.1251	0.1389
		NARMED	0.1024	0.1116	0.1134	0.1053	0.1107	0.1015	0.1090	0.0990
30	0.6	MAE	0.0493	0.0568	0.0517	0.0529	0.0524	0.0530	0.0516	0.0519
		NMAE	0.0823	0.0947	0.0859	0.0879	0.0871	0.0880	0.0858	0.0863
		MaxAE	0.0944	0.1464	0.0931	0.1415	0.0940	0.1413	0.0937	0.1336
		MSE	0.0042	0.0057	0.0044	0.0051	0.0045	0.0051	0.0044	0.0049
		NRMSE	0.0950	0.1149	0.0975	0.1076	0.0989	0.1076	0.0977	0.1056
		NARMED	0.0818	0.0856	0.0875	0.0778	0.0893	0.0784	0.0870	0.0763
	0.8	MAE	0.0496	0.0604	0.0540	0.0561	0.0537	0.0556	0.0538	0.0558
		NMAE	0.0621	0.0755	0.0676	0.0703	0.0671	0.0695	0.0674	0.0698
		MaxAE	0.0943	0.1580	0.0964	0.1508	0.0958	0.1496	0.0964	0.1414
		MSE	0.0039	0.0062	0.0043	0.0055	0.0043	0.0054	0.0043	0.0054
		NRMSE	0.0710	0.0910	0.0761	0.0856	0.0756	0.0847	0.0759	0.0846
		NARMED	0.0600	0.0688	0.0663	0.0629	0.0654	0.0622	0.0659	0.0628
	1.0	MAE	0.0509	0.0515	0.0513	0.0467	0.0509	0.0473	0.0492	0.0459
		NMAE	0.0513	0.0519	0.0517	0.0471	0.0514	0.0477	0.0497	0.0463
		MaxAE	0.1046	0.1690	0.1047	0.1506	0.1038	0.1524	0.1001	0.1368
		MSE	0.0046	0.0049	0.0044	0.0040	0.0043	0.0042	0.0040	0.0039
		NRMSE	0.0618	0.0661	0.0617	0.0600	0.0613	0.0609	0.0591	0.0585
		NARMED	0.0468	0.0411	0.0470	0.0370	0.0468	0.0376	0.0457	0.0370

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

,

Table A8. MC simulation results for mean performance evaluation metrics under various imputation methods for varying prevalence rates p at fixed sample sizes $n=50$ and $n=100$.

$\mathit{n}$	$\mathit{p}$	Metric	Imputation Methods
			Midpoint		Regression		Uniform		Multiple
			Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
	0.2	MAE	0.0244	0.0294	0.0256	0.0282	0.0249	0.0274	0.0253	0.0259
		NMAE	0.1219	0.1468	0.1284	0.1413	0.1245	0.1371	0.1254	0.1279
		MaxAE	0.0520	0.0760	0.0520	0.0740	0.0510	0.0726	0.0517	0.0678
		MSE	0.0012	0.0016	0.0013	0.0015	0.0012	0.0014	0.0013	0.0013
		NRMSE	0.1445	0.1793	0.1512	0.1729	0.1467	0.1682	0.1481	0.1574
		NARMED	0.1210	0.1284	0.1298	0.1241	0.1251	0.1192	0.1266	0.1103
	0.4	MAE	0.0366	0.0380	0.0387	0.0353	0.0386	0.0353	0.0384	0.0347
		NMAE	0.0918	0.0954	0.0966	0.0882	0.0963	0.0881	0.0957	0.0865
		MaxAE	0.0698	0.1013	0.0698	0.0966	0.0705	0.0974	0.0696	0.0914
		MSE	0.0023	0.0027	0.0025	0.0024	0.0026	0.0024	0.0025	0.0023
		NRMSE	0.1058	0.1168	0.1100	0.1090	0.1101	0.1094	0.1094	0.1069
		NARMED	0.0941	0.0843	0.1007	0.0769	0.0998	0.0763	0.0997	0.0756
50	0.6	MAE	0.0455	0.0442	0.0490	0.0408	0.0496	0.0409	0.0482	0.0399
		NMAE	0.0757	0.0735	0.0819	0.0681	0.0828	0.0682	0.0803	0.0665
		MaxAE	0.0810	0.1187	0.0826	0.1108	0.0830	0.1111	0.0813	0.1036
		MSE	0.0032	0.0035	0.0037	0.0030	0.0037	0.0031	0.0035	0.0029
		NRMSE	0.0855	0.0896	0.0913	0.0840	0.0922	0.0840	0.0896	0.0818
		NARMED	0.0768	0.0661	0.0846	0.0604	0.0858	0.0605	0.0826	0.0589
	0.8	MAE	0.0454	0.0467	0.0518	0.0431	0.0513	0.0432	0.0502	0.0421
		NMAE	0.0568	0.0584	0.0648	0.0539	0.0642	0.0540	0.0627	0.0525
		MaxAE	0.0837	0.1258	0.0898	0.1172	0.0882	0.1171	0.0874	0.1080
		MSE	0.0030	0.0037	0.0038	0.0033	0.0037	0.0033	0.0036	0.0031
		NRMSE	0.0644	0.0706	0.0726	0.0660	0.0717	0.0660	0.0704	0.0641
		NARMED	0.0545	0.0532	0.0631	0.0484	0.0630	0.0484	0.0609	0.0473
	1.0	MAE	0.0512	0.0411	0.0494	0.0362	0.0489	0.0365	0.0487	0.0355
		NMAE	0.0516	0.0414	0.0498	0.0366	0.0494	0.0369	0.0491	0.0359
		MaxAE	0.1029	0.1397	0.0975	0.1178	0.0974	0.1179	0.0962	0.1063
		MSE	0.0045	0.0031	0.0039	0.0024	0.0039	0.0025	0.0038	0.0023
		NRMSE	0.0615	0.0530	0.0585	0.0468	0.0582	0.0472	0.0578	0.0456
		NARMED	0.0475	0.0328	0.0469	0.0289	0.0464	0.0293	0.0464	0.0287
	0.2	MAE	0.0191	0.0196	0.0213	0.0188	0.0202	0.0180	0.0210	0.0179
		NMAE	0.0953	0.0976	0.1074	0.0944	0.1005	0.0898	0.1052	0.0899
		MaxAE	0.0382	0.0535	0.0396	0.0523	0.0385	0.0513	0.0394	0.0485
		MSE	0.0007	0.0007	0.0008	0.0007	0.0007	0.0006	0.0008	0.0006
		NRMSE	0.1112	0.1204	0.1233	0.1182	0.1163	0.1126	0.1214	0.1124
		NARMED	0.0972	0.0851	0.1124	0.0803	0.1039	0.0756	0.1091	0.0764
	0.4	MAE	0.0323	0.0270	0.0358	0.0245	0.0356	0.0243	0.0353	0.0240
		NMAE	0.0807	0.0674	0.0897	0.0615	0.0893	0.0608	0.0883	0.0599
		MaxAE	0.0570	0.0749	0.0588	0.0690	0.0583	0.0686	0.0577	0.0642
		MSE	0.0016	0.0014	0.0019	0.0012	0.0019	0.0011	0.0019	0.0011
		NRMSE	0.0909	0.0832	0.0995	0.0770	0.0991	0.0761	0.0980	0.0748
		NARMED	0.0849	0.0596	0.0962	0.0535	0.0962	0.0522	0.0951	0.0521
100	0.6	MAE	0.0410	0.0317	0.0464	0.0284	0.0461	0.0284	0.0458	0.0278
		NMAE	0.0682	0.0527	0.0774	0.0472	0.0770	0.0473	0.0763	0.0464
		MaxAE	0.0701	0.0883	0.0737	0.0783	0.0736	0.0785	0.0731	0.0731
		MSE	0.0024	0.0018	0.0030	0.0015	0.0030	0.0015	0.0030	0.0014
		NRMSE	0.0762	0.0646	0.0852	0.0586	0.0848	0.0588	0.0841	0.0575
		NARMED	0.0697	0.0473	0.0811	0.0419	0.0803	0.0419	0.0796	0.0413
	0.8	MAE	0.0411	0.0342	0.0483	0.0298	0.0493	0.0302	0.0488	0.0300
		NMAE	0.0514	0.0428	0.0604	0.0373	0.0617	0.0378	0.0611	0.0375
		MaxAE	0.0765	0.0961	0.0855	0.0825	0.0865	0.0828	0.0856	0.0771
		MSE	0.0024	0.0020	0.0033	0.0016	0.0034	0.0016	0.0033	0.0016
		NRMSE	0.0591	0.0520	0.0686	0.0459	0.0699	0.0466	0.0691	0.0459
		NARMED	0.0482	0.0387	0.0568	0.0333	0.0585	0.0340	0.0580	0.0341
	1.0	MAE	0.0502	0.0317	0.0480	0.0254	0.0468	0.0233	0.0470	0.0249
		NMAE	0.0506	0.0320	0.0485	0.0257	0.0473	0.0235	0.0475	0.0252
		MaxAE	0.0975	0.1123	0.0923	0.0831	0.0898	0.0752	0.0893	0.0748
		MSE	0.0042	0.0019	0.0036	0.0012	0.0034	0.0010	0.0034	0.0012
		NRMSE	0.0594	0.0414	0.0562	0.0330	0.0548	0.0301	0.0549	0.0321
		NARMED	0.0474	0.0250	0.0470	0.0204	0.0460	0.0189	0.0465	0.0204

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

and

Table A9. MC simulation results for mean performance evaluation metrics under various imputation methods for varying prevalence rates p at fixed sample sizes $n=300$ and $n=500$.

$\mathit{n}$	$\mathit{p}$	Metric	Imputation Methods
			Midpoint		Regression		Uniform		Multiple
			Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
	0.2	MAE	0.0163	0.0117	0.0180	0.0101	0.0179	0.0102	0.0176	0.0101
		NMAE	0.0818	0.0587	0.0905	0.0506	0.0896	0.0513	0.0874	0.0501
		MaxAE	0.0290	0.0345	0.0292	0.0294	0.0292	0.0298	0.0291	0.0280
		MSE	0.0004	0.0003	0.0005	0.0002	0.0005	0.0002	0.0005	0.0002
		NRMSE	0.0922	0.0738	0.1000	0.0641	0.0993	0.0651	0.0971	0.0634
		NARMED	0.0867	0.0498	0.0988	0.0431	0.0975	0.0434	0.0947	0.0427
	0.4	MAE	0.0279	0.0162	0.0334	0.0136	0.0330	0.0137	0.0331	0.0136
		NMAE	0.0698	0.0406	0.0836	0.0341	0.0824	0.0341	0.0827	0.0339
		MaxAE	0.0466	0.0486	0.0511	0.0393	0.0508	0.0390	0.0506	0.0367
		MSE	0.0011	0.0005	0.0015	0.0004	0.0015	0.0004	0.0015	0.0004
		NRMSE	0.0777	0.0507	0.0915	0.0432	0.0901	0.0429	0.0904	0.0426
		NARMED	0.0744	0.0352	0.0911	0.0294	0.0897	0.0297	0.0902	0.0297
300	0.6	MAE	0.0363	0.0201	0.0453	0.0163	0.0451	0.0160	0.0447	0.0158
		NMAE	0.0605	0.0336	0.0754	0.0271	0.0750	0.0267	0.0746	0.0263
		MaxAE	0.0614	0.0589	0.0712	0.0451	0.0711	0.0448	0.0712	0.0421
		MSE	0.0018	0.0007	0.0027	0.0005	0.0027	0.0005	0.0026	0.0005
		NRMSE	0.0680	0.0414	0.0831	0.0338	0.0827	0.0333	0.0824	0.0328
		NARMED	0.0621	0.0299	0.0793	0.0241	0.0788	0.0236	0.0780	0.0233
	0.8	MAE	0.0359	0.0230	0.0465	0.0171	0.0464	0.0175	0.0459	0.0170
		NMAE	0.0449	0.0288	0.0581	0.0214	0.0580	0.0219	0.0574	0.0213
		MaxAE	0.0718	0.0673	0.0863	0.0472	0.0858	0.0480	0.0851	0.0441
		MSE	0.0019	0.0009	0.0031	0.0005	0.0031	0.0005	0.0030	0.0005
		NRMSE	0.0534	0.0351	0.0674	0.0264	0.0672	0.0269	0.0665	0.0262
		NARMED	0.0398	0.0259	0.0534	0.0193	0.0532	0.0197	0.0525	0.0193
	1.0	MAE	0.0461	0.0237	0.0440	0.0147	0.0440	0.0147	0.0447	0.0141
		NMAE	0.0465	0.0240	0.0445	0.0148	0.0445	0.0148	0.0452	0.0143
		MaxAE	0.0867	0.0875	0.0849	0.0477	0.0848	0.0475	0.0853	0.0427
		MSE	0.0035	0.0011	0.0031	0.0004	0.0031	0.0004	0.0031	0.0004
		NRMSE	0.0539	0.0315	0.0513	0.0190	0.0512	0.0190	0.0519	0.0182
		NARMED	0.0446	0.0180	0.0434	0.0119	0.0437	0.0119	0.0446	0.0115
	0.2	MAE	0.0153	0.0094	0.0174	0.0080	0.0171	0.0078	0.0172	0.0078
		NMAE	0.0762	0.0467	0.0871	0.0399	0.0854	0.0388	0.0859	0.0388
		MaxAE	0.0260	0.0285	0.0273	0.0233	0.0273	0.0229	0.0269	0.0217
		MSE	0.0003	0.0002	0.0004	0.0001	0.0004	0.0001	0.0004	0.0001
		NRMSE	0.0851	0.0590	0.0955	0.0507	0.0939	0.0494	0.0943	0.0493
		NARMED	0.0817	0.0396	0.0958	0.0338	0.0934	0.0326	0.0948	0.0331
	0.4	MAE	0.0273	0.0135	0.0334	0.0107	0.0328	0.0106	0.0333	0.0106
		NMAE	0.0680	0.0337	0.0833	0.0268	0.0821	0.0264	0.0831	0.0264
		MaxAE	0.0449	0.0413	0.0509	0.0308	0.0501	0.0303	0.0505	0.0286
		MSE	0.0010	0.0003	0.0015	0.0002	0.0014	0.0002	0.0015	0.0002
		NRMSE	0.0757	0.0421	0.0910	0.0339	0.0898	0.0334	0.0908	0.0332
		NARMED	0.0728	0.0291	0.0909	0.0231	0.0897	0.0229	0.0905	0.0230
500	0.6	MAE	0.0352	0.0171	0.0444	0.0125	0.0443	0.0124	0.0442	0.0122
		NMAE	0.0589	0.0287	0.0740	0.0209	0.0737	0.0206	0.0736	0.0203
		MaxAE	0.0608	0.0507	0.0709	0.0350	0.0707	0.0347	0.0706	0.0324
		MSE	0.0017	0.0005	0.0026	0.0003	0.0026	0.0003	0.0026	0.0003
		NRMSE	0.0669	0.0354	0.0819	0.0261	0.0816	0.0257	0.0815	0.0253
		NARMED	0.0609	0.0254	0.0778	0.0185	0.0775	0.0183	0.0777	0.0181
	0.8	MAE	0.0343	0.0200	0.0454	0.0131	0.0448	0.0134	0.0452	0.0132
		NMAE	0.0429	0.0250	0.0568	0.0164	0.0560	0.0167	0.0565	0.0165
		MaxAE	0.0714	0.0588	0.0848	0.0365	0.0843	0.0371	0.0851	0.0343
		MSE	0.0018	0.0007	0.0030	0.0003	0.0029	0.0003	0.0030	0.0003
		NRMSE	0.0521	0.0306	0.0662	0.0203	0.0655	0.0207	0.0661	0.0203
		NARMED	0.0364	0.0225	0.0530	0.0147	0.0512	0.0149	0.0520	0.0150
	1.0	MAE	0.0448	0.0218	0.0422	0.0113	0.0420	0.0113	0.0422	0.0111
		NMAE	0.0453	0.0220	0.0426	0.0115	0.0425	0.0114	0.0427	0.0112
		MaxAE	0.0836	0.0814	0.0826	0.0368	0.0819	0.0367	0.0823	0.0335
		MSE	0.0033	0.0009	0.0028	0.0002	0.0028	0.0002	0.0028	0.0002
		NRMSE	0.0521	0.0292	0.0492	0.0147	0.0489	0.0147	0.0492	0.0143
		NARMED	0.0440	0.0161	0.0415	0.0092	0.0415	0.0092	0.0418	0.0090

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Square Error; NARMED = Normalized Root Median Error Deviation.

and Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7 and Figure A8 included in the annexures.

Performance under Fixed Prevalence Rates with Varying Sample Sizes: A consistent pattern observed across all prevalence rates was the monotonic decrease in estimation error as the sample size increased. Both estimators demonstrated the statistical property of consistency, and all error metrics showed a systematic reduction with increasing n. However, this convergence behavior differed markedly between the two estimators. The classical KM estimator showed a steeper gradient of improvement, whereas the smoothed KM estimator demonstrated a more gradual improvement. In addition, the results reveal that the relative superiority of the estimators is fundamentally dependent on the sample size and the prevalence rate. For small sample sizes ($n\le 30$), the smoothed KM estimator consistently outperformed the classical KM estimator across all prevalence rates and performance evaluation metrics. This advantage was most evident at the smallest sample sizes (i.e., $n=10$).

A transition zone was identified where the performance advantage shifted from the smoothed to the classical KM estimator. The location of this transition was highly dependent on the prevalence rate. For a low prevalence rate ($p=0.2$, see Table A2 and Figure A1), the transition occurred at a larger sample size ($n\approx$ 80–100). For moderate prevalence rates ($p=0.4,0.6$, see Table A3 and Figure A2 as well as Table A4 and Figure A3, respectively), the transition occurred at medium sample sizes ($n\approx$ 40–80). For $p=1.0$ (see Table A6 and Figure A5), the transition occurred at a small sample size ($n\approx 30$). Within this zone, the classical KM estimator generally became superior first when paired with regression and multiple imputation methods. For large sample sizes ($n\ge 100$), the classical KM estimator was clearly superior, exhibiting significantly lower error rates across all metrics and prevalence rates. The performance gap increased substantially as n increased, and the classical KM estimator often achieved less than half the MAE and MSE of the smoothed KM estimator at $n=500$.

Performance under Fixed Sample Sizes with Varying Prevalence Rates: For small sample sizes ($n=10$ and 30), according to Table A7 and Figure A6, the benefits of the smoothed penalized KM estimator are clear. The smoothed KM estimator consistently yielded lower errors across all prevalence rates. The largest visual separations between the smoothed and classical KM curves were observed at a small sample size ($n=10$). At moderate sample sizes ($n=50$ and 100), according to Table A8 and Figure A7, the differences between the estimators narrowed but did not disappear. The classical KM estimator produced slightly lower errors across all metrics, maintaining a small but consistent edge in the accuracy of the estimation at $n=50$. For large sample sizes ($n=300$ and 500), according to Table A9 and Figure A8, the classical KM estimator produced visibly lower errors across all metrics.

3.2.3. Convergence Validation for Simulated Data

To quantitatively assess convergence between the smoothed and classical KM estimators, we adopted a difference-based convergence criterion. For each combination of prevalence rate (p), imputation method, and performance metric, we defined the critical sample size $n^{*}$ as the smallest n for which the absolute difference in the corresponding metric between the two estimators fell below tolerance level $\delta$. Formally,

$$n^{*}=min\left\{n\in \mathbb{N}:\left|{\hat{\theta }}_{\mathrm{smoothed}}\left(n\right)-{\hat{\theta }}_{\mathrm{classical}}\left(n\right)\right|\le \delta \right\},$$

where $\hat{\theta }\left(n\right)$ denotes the estimated value of a given performance metric evaluated at sample size n. In the main analysis, we set $\delta =0.005$. This value was selected to represent a practically negligible difference in metric values, balancing two considerations: (i) the scale of the error-based metrics used in this study (which were typically small in magnitude once estimators stabilize) and (ii) the need to avoid declaring convergence too early due to MC noise at small n. In particular, a tolerance of $0.005$ corresponded to a sub-percent level difference on the $[0,1]$ survival-probability scale and provided an interpretable criterion for practical equivalence of estimator performance.

The analysis was performed across the entire range of sample sizes ($n=10$ to 500) and prevalence rates ($p=0.2$ to $1.0$).

Table 2. Critical sample sizes (n) where the absolute difference between smoothed and classical KM estimators fell below $\delta =0.005$ across metrics and prevalence rates.

Metric	Prevalence Rate	Midpoint	Regression	Uniform	Multiple
	0.2	60	40	40	40
	0.4	40	30	30	30
MAE	0.6	40	30	30	30
	0.8	40	30	30	30
	1.0	30	20	30	20
	0.2	90	60	60	50
	0.4	50	40	40	40
NMAE	0.6	40	30	30	30
	0.8	50	30	30	30
	1.0	30	20	30	20
	0.2	100	90	80	70
	0.4	300	140	140	120
MaxAE	0.6	200	100	100	90
	0.8	200	90	90	80
	1.0	180	70	80	60
	0.2	30	20	20	20
	0.4	20	20	20	10
MSE	0.6	20	20	20	20
	0.8	20	20	20	20
	1.0	20	20	20	20
	0.2	80	60	60	50
	0.4	40	30	30	30
NRMSE	0.6	40	30	30	30
	0.8	40	30	30	30
	1.0	30	20	30	20
	0.2	80	60	60	50
	0.4	40	30	30	30
NARMED	0.6	40	30	30	30
	0.8	40	30	30	30
	1.0	30	20	30	20

Note: Observations at sample sizes $n=10,20,30,40,50,60,70,80,90,100,120,140,160,180,200,300,400,$ and 500 were considered in determining the convergence thresholds.

reports the primary convergence results for $\delta =0.005$. The convergence validation results are presented in three parts. First, prevalence rate-dependent convergence is examined. This is followed by an assessment of convergence behavior across the different performance evaluation metrics. Finally, the convergence behavior is analyzed with respect to the imputation methods used.

Convergence Behavior Across Prevalence Rates: The convergence analysis reveals a strong inverse relationship between prevalence rate and critical sample size n. For lower prevalence rates ($p=0.2$), convergence typically required larger sample sizes ($n=$ 40–100 for most metrics). As prevalence increased to $p=0.6$, convergence occurred at moderate sample sizes ($n=$ 30–40), while, at $p=1.0$, convergence occurred at around $n=$ 20–30.

Convergence Behavior Across Performance Evaluation Metrics: Different metrics exhibited different convergence patterns. The MSE-based metrics (MSE, NRMSE) converged most rapidly, with $n\le 30$ across nearly all simulation scenarios, indicating that squared error differences decreased rapidly with increasing sample size. Absolute error metrics (MAE, NMAE) showed intermediate convergence, which typically required $n=$ 30–50 samples across prevalence rates. MaxAE demonstrated the slowest convergence, often requiring $n\ge 100$ samples, particularly at lower prevalence rates.

Convergence Behavior Across Imputation Methods: The choice of imputation method significantly affected the convergence rates. The midpoint imputation consistently delayed convergence, requiring larger n values across all metrics and prevalence rates. Multiple imputation generally achieved the most rapid convergence, particularly for prevalence rates $p\ge 0.6$. On the other hand, regression and uniform imputation showed similar convergence patterns, typically intermediate between multiple and midpoint methods.

3.3. Empirical Study

3.3.1. Real Data Description

To complement and validate the simulation findings, the imputation techniques and simulation framework outlined earlier were applied to the acquired immunodeficiency syndrome (AIDS) Clinical Trials Group (ACTG) 181 dataset [57]. This allowed this study to assess whether the methodological performance observed in the simulated environment was consistent when applied to real-world data. The ACTG181 study was a clinical investigation involving 232 HIV-positive participants. The dataset includes 204 of the 232 participants who were tested at least once for cytomegalovirus (CMV) shedding and mycobacterium avium complex (MAC) colonization during the trial and who had no prior diagnosis of either condition at baseline. Testing was conducted during scheduled clinic visits, which occurred at regular monthly intervals. For participants who attended all scheduled visits, the event time was defined as the month in which the first positive test was recorded, generating discrete failure-time data. For participants who missed one or more visits and were subsequently found to be positive immediately after the missed visit(s), the event time was classified as having occurred within an interval, resulting in interval-censored failure-time data. For the present analysis, we focus on the CMV-shedding treatment group (hereafter referred to as Treatment 1). All individuals experienced the event of interest, and the event times were known to occur only between consecutive clinic visits. This design represents a classical instance of interval-censored data with no right-censoring, making the dataset particularly suitable for evaluating imputation procedures under conditions of complete event observation. The dataset is publicly available in the MLEcens package in RStudio version 2025.05.1+513 and has been widely applied in the statistical literature addressing interval-censored survival analysis methodologies (see [1,57,58], among others).

3.3.2. Real Data Bootstrapping Procedure

The ACTG181 Treatment 1 dataset was used to evaluate the performance of the smoothed nonparametric penalized likelihood KM estimator compared to that of the classical KM estimator. For analysis, each subject’s observation interval $(L_i,R_i]$ was converted to a single event time using one of the imputation techniques, namely, midpoint, regression, uniform, and multiple imputation methods. For each imputation approach, the bootstrap algorithm was implemented in two layers, respectively, corresponding to the following:

(i) Construction of Reference Survival Function: To obtain a method-specific reference survival curve, $m=5000$ bootstrap replicates were generated by sampling n observations with replacement from the full ACTG181 dataset. For each replicate, the event times were imputed using the chosen method and a KM estimator was fitted to $\mathrm{Surv}(\mathrm{time},\mathrm{status})$. The KM curve was anchored at $(0,1)$ and $(t_{max},S_{\mathrm{KM}}\left(t_{max}^{-}\right))$ to ensure proper boundary behavior and subsequently smoothed using a cubic smoothing spline. Each smoothed curve was evaluated on a uniform time grid ranging from 0 to the dataset’s maximum right endpoint in unit increments. The m smoothed curves were then averaged point-by-point across replicates to produce a method-specific reference survival function, denoted by $S\left(t\right)$.

(ii) Finite-Sample Bootstrap Evaluation: A second bootstrap procedure, also with $m=5000$ iterations, was carried out to assess the performance of a fixed sample size (for example, $n=150$). In each replicate, the following steps were performed:

• Randomly sample n observations with a replacement from the ACTG181 dataset;
• Impute event times using the same imputation method under consideration;
• Fit a KM survival curve on $\mathrm{Surv}(\mathrm{time},\mathrm{status})$;
• Interpolate the resulting KM step function onto the common time grid.

This produced a bootstrap survival estimate ${\hat{S}}_b\left(t\right)$ for each replicate $b=1,2,\dots ,m$. Each estimated survival vector ${\hat{S}}_b\left(t\right)$ was then compared with its corresponding reference curve $S\left(t\right)$ across the entire grid using six performance evaluation metrics: MAE, NMAE, MaxAE, MSE, NRMSE, and NARMED. For each metric, the replicate values m were summarized by the bootstrap mean, standard deviation, and a t-based 95% confidence interval.

3.4. Real Data Application Results

3.4.1. Descriptive Analysis for the ACTG181 Data

Table 3. First ten observations of the ACTG181 Treatment 1 dataset.

ID	Left (L)	Right (R)	Censoring
1	1	21	1
2	11	24	1
3	3	24	1
4	10	24	1
5	6	24	1
6	8	24	1
7	10	24	1
8	6	20	1
9	8	16	1
10	10	22	1

Note: L and R denote the left and right endpoints of the censoring interval for the unobserved event time T. A value of Censoring = 1 indicates that the event occurred within the interval $(L,R]$, whereas Censoring = 0 corresponds to right-censoring with $T>L$.

presents the first ten observations of the ACTG181 dataset for Treatment 1. The columns show the subject ID, the left and right endpoints of the censoring intervals, and the censoring indicator. Unlike the simulated datasets, where intervals were generated with fixed widths for comparability, the ACTG181 intervals varied considerably in length between subjects.

Figure 2 presents the KM survival curves for the ACTG181 dataset under both the classical and smoothed penalized estimation frameworks for Treatment 1. The classical KM estimator exhibits stepwise and irregular survival declines, with pronounced early drops indicating sensitivity to limited event information. In contrast, the smoothed penalized KM curve follows a more continuous and statistically stable pattern, capturing the overall survival trend while suppressing overfitted fluctuations and sampling noise.

3.4.2. Real Data Bootstrap Resampling Results

To evaluate the robustness and applicability of the estimators on real data, a bootstrap resampling procedure with $m=5000$ replicates was conducted on Treatment 1 of the ACTG181 dataset. Within each replicate, event times were imputed using the selected methods midpoint, regression, uniform, and multiple imputation followed by computation of both the classical and smoothed penalized KM curves. Performance was assessed using the same statistical metrics applied in the simulation study, namely, MAE, MaxAE, and MSE, along with their normalized counterparts (NMAE, NRMSE, and NARMED).

A consistent pattern emerged from the ACTG181 real-data analysis (

Table A10. ACTG181 data results for performance evaluation metrics under various imputation methods for varying sample sizes n using $m=5000$ at prevalence $p=1.0$.

$\mathit{n}$	Metric	Imputation Methods
		Midpoint		Regression		Uniform		Multiple
		Smoothed	Classical	Smoothed	Classical	Smoothed	Classical	Smoothed	Classical
10	MAE	0.0850	0.0979	0.0722	0.1152	0.0634	0.0840	0.0583	0.0679
	NMAE	0.1030	0.1182	0.0775	0.1156	0.0644	0.1014	0.0516	0.0670
	MaxAE	0.5941	0.6332	0.1640	0.2972	0.3829	0.4126	0.1447	0.1564
	MSE	0.0264	0.0281	0.0100	0.0280	0.0065	0.0135	0.0053	0.0079
	NRMSE	0.1430	0.1840	0.0869	0.1455	0.0961	0.1363	0.0662	0.0806
	NARMED	0.1045	0.1422	0.0709	0.0967	0.0470	0.0801	0.0497	0.0625
20	MAE	0.0744	0.0414	0.0684	0.0774	0.0550	0.0544	0.0543	0.0548
	NMAE	0.0942	0.0414	0.0684	0.0775	0.0597	0.0591	0.0551	0.0550
	MaxAE	0.1610	0.2058	0.1568	0.1951	0.1443	0.1639	0.1249	0.1166
	MSE	0.0088	0.0030	0.0087	0.0116	0.0048	0.0050	0.0041	0.0051
	NRMSE	0.0921	0.0541	0.0802	0.0958	0.0703	0.0728	0.0614	0.0648
	NARMED	0.0923	0.0335	0.0663	0.0684	0.0641	0.0503	0.0480	0.0535
30	MAE	0.0795	0.0254	0.0637	0.0602	0.0572	0.0387	0.0537	0.0460
	NMAE	0.0917	0.0260	0.0637	0.0602	0.0578	0.0390	0.0548	0.0462
	MaxAE	0.1464	0.1321	0.1379	0.1535	0.1339	0.1307	0.1217	0.0978
	MSE	0.0100	0.0013	0.0070	0.0067	0.0053	0.0028	0.0039	0.0036
	NRMSE	0.1015	0.0364	0.0766	0.0745	0.0711	0.0499	0.0640	0.0543
	NARMED	0.0869	0.0192	0.0609	0.0539	0.0700	0.0313	0.0515	0.0445
40	MAE	0.0812	0.0190	0.0646	0.0522	0.0632	0.0346	0.0541	0.0360
	NMAE	0.0902	0.0190	0.0624	0.0522	0.0577	0.0346	0.0552	0.0361
	MaxAE	0.1424	0.1319	0.1283	0.1355	0.1441	0.1148	0.1210	0.0793
	MSE	0.0105	0.0007	0.0068	0.0052	0.0058	0.0021	0.0041	0.0023
	NRMSE	0.1034	0.0266	0.0738	0.0652	0.0717	0.0440	0.0640	0.0428
	NARMED	0.0851	0.0154	0.0614	0.0458	0.0716	0.0279	0.0534	0.0343
50	MAE	0.0830	0.0181	0.0634	0.0472	0.0652	0.0313	0.0541	0.0318
	NMAE	0.0930	0.0181	0.0603	0.0471	0.0588	0.0313	0.0541	0.0318
	MaxAE	0.1689	0.1128	0.1197	0.1243	0.1487	0.1036	0.1196	0.0697
	MSE	0.0119	0.0003	0.0075	0.0043	0.0060	0.0018	0.0044	0.0017
	NRMSE	0.1037	0.0268	0.0718	0.0594	0.0765	0.0400	0.0647	0.0378
	NARMED	0.0805	0.0132	0.0610	0.0403	0.0732	0.0251	0.0568	0.0304
80	MAE	0.0885	0.0105	0.0576	0.0372	0.0680	0.0234	0.0534	0.0259
	NMAE	0.0938	0.0145	0.0605	0.0372	0.0618	0.0234	0.0534	0.0260
	MaxAE	0.1547	0.0414	0.1218	0.0927	0.1479	0.0799	0.1164	0.0562
	MSE	0.0102	0.0002	0.0072	0.0026	0.0069	0.0010	0.0042	0.0012
	NRMSE	0.1094	0.0148	0.0706	0.0459	0.0741	0.0301	0.0661	0.0309
	NARMED	0.0751	0.0077	0.0604	0.0342	0.0690	0.0187	0.0561	0.0254
100	MAE	0.0918	0.0154	0.0661	0.0335	0.0696	0.0219	0.0544	0.0224
	NMAE	0.0921	0.0154	0.0661	0.0335	0.0696	0.0219	0.0540	0.0224
	MaxAE	0.2095	0.0877	0.1116	0.0855	0.1533	0.0735	0.1144	0.0487
	MSE	0.0132	0.0005	0.0070	0.0021	0.0073	0.0009	0.0045	0.0009
	NRMSE	0.1012	0.0216	0.0821	0.0416	0.0825	0.0281	0.0655	0.0266
	NARMED	0.0767	0.0098	0.0592	0.0300	0.0705	0.0175	0.0586	0.0219
150	MAE	0.0862	0.0087	0.0645	0.0274	0.0706	0.0173	0.0556	0.0187
	NMAE	0.0860	0.0136	0.0645	0.0274	0.0706	0.0173	0.0556	0.0187
	MaxAE	0.1816	0.0233	0.1394	0.0692	0.1491	0.0585	0.1143	0.0409
	MSE	0.0113	0.0001	0.0068	0.0014	0.0074	0.0006	0.0048	0.0006
	NRMSE	0.1061	0.0107	0.0809	0.0340	0.0835	0.0223	0.0675	0.0223
	NARMED	0.0728	0.0073	0.0605	0.0249	0.0742	0.0139	0.0587	0.0183
200	MAE	0.1134	0.0124	0.0807	0.0232	0.0816	0.0148	0.0675	0.0150
	NMAE	0.1136	0.0124	0.0807	0.0232	0.0816	0.0148	0.0675	0.0150
	MaxAE	0.2557	0.0816	0.1791	0.0610	0.1706	0.0515	0.1356	0.0337
	MSE	0.0199	0.0004	0.0106	0.0010	0.0099	0.0004	0.0070	0.0004
	NRMSE	0.1413	0.0191	0.1021	0.0291	0.0959	0.0193	0.0821	0.0180
	NARMED	0.0860	0.0068	0.0669	0.0204	0.0852	0.0116	0.0709	0.0145

Note: MAE = Mean Absolute Error; NMAE = Normalized Mean Absolute Error; MaxAE = Maximum Absolute Error; MSE = Mean Squared Error; NRMSE = Normalized Root Mean Squared Error; NARMED = Normalized Absolute Root Median Error.

and Figure A9), closely mirroring the behavior observed in the simulated experiments. Across all imputation methods, estimation errors for both the classical and smoothed KM estimators decreased monotonically with increasing sample size, reflecting the expected consistency of both estimators. The rate and pattern of convergence differed between the two approaches: the classical KM estimator exhibited a steeper rate of improvement with increasing n, while the smoothed KM estimator improved more gradually, especially in the small-sample region.

For small samples ($n=10$–30), the smoothed penalized KM estimator yielded visibly lower errors across all metrics, demonstrating its advantage in stabilizing survival estimates in small samples. Thus, for small samples ($n\le 30$), the smoothed KM estimator consistently outperformed the classical estimator across all imputations and error metrics. The gains were most pronounced at $n=10$, where the smoothed KM reduced MAE by approximately 25–40% across imputation methods. Similar improvements were observed for NMAE, MSE, and NRMSE, confirming that smoothing effectively stabilizes the KM estimator in small samples.

A transition zone was observed between $n=20$ and 30, where the classical KM estimator began to outperform the smoothed KM estimator for most imputation methods. The exact point of convergence varied slightly by method, such that regression and uniform imputations exhibited earlier stabilization ($n\approx 20$–25), while midpoint and multiple imputations maintained a marginal smoothing advantage until about $n=30$.

At moderate sample sizes ($n=80$ and 100), the differences between estimators narrowed but did not disappear, that is, the classical KM produced slightly lower errors in most metrics, maintaining a small but consistent edge in the accuracy of the estimation. For these moderate sample sizes, the classical KM estimator became decisively superior across all imputations and metrics. This performance gap continued to widen with increasing n, confirming the asymptotic dominance of the classical KM estimator under high prevalence rates (i.e., p = 1).

For large sample sizes ($n=150$ and 200), according to Table A10 and Figure A9, the classical KM estimator achieved marginally better performance, confirming that smoothing benefits diminished as the sample size increased. It is evident that the results of the real data corroborate the simulation findings. The smoothed KM estimator exhibits clear advantages in small-sample scenarios. However, as the sample size increases, these benefits diminish rapidly and the classical KM estimator becomes consistently more efficient.

3.4.3. Convergence Validation for ACTG181 Real Data

To quantitatively assess the convergence between the smoothed and classical KM estimators for the ACTG181 dataset, a difference-based convergence criterion was used. For each imputation method and performance metric, the critical sample size n was defined as the smallest sample size at which the absolute difference between the smoothed and classical estimates fell below the prescribed convergence threshold. The analysis was conducted over the entire range of sample sizes ($n=10$ to 200) under a prevalence rate of $p=1.0$.

Table 4. Approximate critical sample sizes ($n^{*}$) where the absolute difference between smoothed and classical KM estimators falls below $\delta =0.005$ across metrics for the ACTG181 dataset ($p=1.0$).

Metric	Midpoint	Regression	Uniform	Multiple
MAE	15	25	20	20
NMAE	15	25	20	20
MaxAE	35	50	30	25
MSE	15	30	20	25
NRMSE	20	30	20	20
NARMED	15	20	15	20

Note: Sample sizes evaluated for convergence were $n=10,15,20,25,30,35,40,45,50,60,70,80,90,100,150,$ and 200. The values reported represent the smallest n at which $|\mathrm{Smoothed}-\mathrm{Classical}|<0.005$ for each metric and imputation method.

reports the resulting convergence sample sizes across all performance evaluation metrics and imputation methods.

Real Data Convergence Behavior Across Performance Evaluation Metrics: In all metrics evaluated, the convergence pattern reflected rapid alignment between the two estimators once $n\ge 25$–30. The MSE-based metrics (MSE, NRMSE) showed the fastest convergence, with $n\approx 15$–30 across all imputations, indicating that squared-error discrepancies diminished rapidly as the sample size grew. MAE and NMAE converged slightly later ($n\approx 20$–25), while MaxAE required larger samples ($n=30$–50), highlighting that local deviations in extreme observations persisted longer. NARMED achieved the earliest and most stable convergence overall ($n\le 20$).

Real Data Convergence Behavior Across Imputation Methods: The convergence behavior varied significantly between the imputation methods, as reflected in Table 4. Midpoint imputation generally achieved the earliest convergence, with most metrics reaching the threshold $\delta =0.005$ at $n=15$–20, except MaxAE, which required $n=35$. Uniform imputation also converged relatively quickly, typically around $n=20$, with a slightly higher requirement for MaxAE ($n=30$). Regression imputation was the slowest to converge overall, consistently requiring the largest sample sizes across metrics (ranging from $n=25$ to 50). Multiple imputation showed an intermediate convergence pattern, which required $n=20$–25 for most metrics, slightly slower than midpoint and uniform but notably faster than regression imputation.

4. Discussion and Conclusions

Given the limitations of the classical imputation-based KM method in small samples [59], a systematic evaluation of how the improved nonparametric penalized likelihood KM approach performs under varying samples and prevalence scenarios was needed. This study addressed this gap by conducting MC simulations and analyzing real interval-censored data to compare the performance of classical and nonparametric penalized KM estimators across selected imputation strategies. Multiple imputation strategies which include midpoint, uniform, regression-based, and multiple imputation were examined to capture differing imputation-induced representations of symmetry and asymmetry in event-time distributions and evaluate their impact on survival estimation. The TB estimator, while a general NPMLE for interval-censored data, does not readily accommodate standard imputation strategies, such as midpoint, uniform, regression-based, or multiple imputation, and it was therefore not considered in this study. Because no single metric fully captures all aspects of estimator performance in interval-censored settings, multiple complementary measures are required to provide a balanced evaluation. In particular, combining average, extreme, normalized, and robust error metrics allow the assessment of overall accuracy, sensitivity to outliers, scale effects, and stability under small samples and low prevalence rates.

Across extensive Monte Carlo simulations, the smoothed nonparametric penalized KM estimator consistently produced smaller error measures, particularly in small-sample and low prevalence settings. Penalization effectively reduced the inherent stepwise behavior of the classical KM estimator, supporting previous findings that smoothing lowers variance and leads to more reliable survival curve estimation [31,32,35]. In contrast, the classical KM estimator is superior in large-sample scenarios, with its advantage becoming more pronounced as the prevalence increases. The gains from penalization were especially pronounced when combined with regression and multiple imputation, which naturally accommodate data-driven and distributional asymmetry within censoring intervals. The midpoint imputation method was uniformly the worst-performing approach. Overall, the results suggest that explicitly accounting for the interplay between symmetry-preserving and asymmetry-inducing imputation strategies, together with penalized smoothing, leads to more reliable and interpretable survival estimates under interval censoring in small sample settings. The real-data application to the ACTG181 dataset further corroborated the simulation findings.

The convergence analysis provides quantitative support for some practical recommendations. The critical sample sizes n identified through this approach align closely with the transition zones observed in our performance analysis, validating the sample size guidelines of $n\approx$ 40–100 for low prevalence rates, $n\approx$ 30–40 for moderate prevalence rates, and $n\approx$ 20–30 for high prevalence rates. This convergence framework offers researchers an objective, metric-driven approach to estimator selection based on their specific sample size constraints and prevalence rate patterns. This study managed to extensively evaluate and benchmark the performance of an improved nonparametric penalized likelihood KM estimator; however, several areas remain open for further exploration. Firstly, the current study focused on sample size and prevalence rate. Future research should examine how interval width and observation frequency influence convergence rates, estimator bias, and estimation accuracy. Secondly, it is also important to note that the penalized likelihood estimation process is computationally intensive compared to the classical KM, especially under multiple imputation. Thirdly, further research could improve the efficiency of penalized estimation by using parallel computation or faster optimization routines such as quasi-Newton optimization techniques, which have demonstrated superiority in nonparametric procedures [60,61]. Further, in order to enhance precision, the use of alternative smoothing kernels, such as spline or Gaussian penalties, may also be a potential area of future research. In addition, although four imputation strategies were evaluated, additional or Bayesian interval-sampling approaches [62,63] may yield further improvements in estimator performance. Lastly, future studies should also consider benchmarking the nonparametric penalized likelihood KM estimator using the classical TB estimator as a baseline method under interval-censored data [64].