Impact of Measurement Error on Residual Extropy Estimation

Abstract

In scientific analyses, measurement errors in data can significantly impact statistical inferences, and ignoring them may lead to biased and invalid results. This study focuses on the estimation of the residual extropy function, in the presence of measurement errors. We developed an estimator for the extropy function and established its asymptotic properties. A comprehensive simulation study evaluates the performance of the proposed estimators under various error scenarios, while their practical utility and precision are demonstrated through an application to a real-world data set.

1. Introduction

Entropy, as introduced by [1], is frequently used to measure the uncertainty in the probability distribution of a random variable (rv). The concept of entropy plays a fundamental role in statistical theory and its applications (see [2,3]). A well-known application of entropy in statistics is the test for normality, based on the property that the normal distribution has the highest entropy among all continuous distributions with a given variance (see [4]). A recent review of Shannon entropy and related measures like Rényi and Tsallis entropy can be found in [5]. The differential entropy for a non-negative continuous rv X, with a probability density function (pdf) $f_X\left(x\right)$, can be defined as:

$$\mathcal{D}\left(X\right)=-{\int }_0^{+\infty }{f}_X\left(x\right)log{f}_X\left(x\right)dx,$$

(1)

When a unit has survived up to a specific age t, understanding the distribution of its remaining lifetime is especially important in reliability and survival analysis. To deal with this, Ref. [6] introduced the notion of residual entropy. Later, Ref. [7] suggested a way to describe the lifetime distribution by using conditional Shannon’s entropy. Based on these, Refs. [6,7] studied certain ordering and aging characteristics of lifetime distributions. Ref. [8] expanded some results presented by [9]. Ref. [10] described a distribution using the functional connections between residual entropy and the hazard rate function. Refs. [11,12,13] studied different generalizations of Ebrahimi‘s measure. For a non-negative $rv$ X, which represents the lifetime of a component, the residual entropy function can be defined as:

$${\mathcal{D}}_t\left(X\right)=-{\int }_t^{\infty }{\displaystyle \frac{f_X\left(x\right)}{{\overline{F}}_X\left(t\right)}}ln\left({\displaystyle \frac{f_X\left(x\right)}{{\overline{F}}_X\left(t\right)}}\right)dx.$$

where ${\overline{F}}_X\left(t\right)$ is the survival function ($sf$).

The measure called extropy is considered as the complementary dual of entropy. For a non-negative and absolutely continuous rv X, which represents the lifetime of a component with $pdf$ $f_X\left(x\right)$, the differential extropy of X, as introduced by [14], is defined as:

$$M\left(X\right)=-{\displaystyle \frac{1}{2}}{\int }_0^{+\infty }{f}_X^2\left(x\right)dx.$$

(2)

This measure provides an alternative way to assess the concentration or spread of the pdf over the domain of X, offering a different perspective on the uncertainty or distribution associated with the rv’s lifetime. By utilizing extropy, researchers can quantify the relative uncertainty of one variable compared to another, which is particularly beneficial in areas such as reliability engineering, information theory, and decision-making processes.

Extropy serves as a complementary measure to entropy for quantifying uncertainty in rv’s, and it is particularly useful for comparing the uncertainties of two rv’s. The uncertainty in a $rv$ X can be quantified by considering the difference between the outcomes of two independent repetitions of the same experiment. Let $X_1$ and $X_2$ denote such outcomes. Then, the difference $X_1-X_2$ reflects the uncertainty associated with X. Then, the $pdf$ of $X_1-X_2$ is given by,

$$d\left(u\right)={\int }_{-\infty }^{+\infty }{f}_X\left(x\right)f_X(x-u)dx.$$

(3)

It follows that the probability of $X_1=X_2$ can be approximated as $d\left(0\right)=-2M\left(X\right)$. Therefore, if the extropy of $X_1$ is smaller than that of another $rv$ $X_2$, i.e., $M\left(X_1\right)<M\left(X_2\right)$, then $X_1$ possesses greater uncertainty than $X_2$ (see [15]); further foundational studies on extropy can be found in [16]. As the concept of $M\left(X\right)$ is not suitable for a rv that has already persisted for a certain period, Ref. [17] proposed the concept of residual extropy, defined as:

$${\mathcal{M}}_t\left(X\right)=-{\displaystyle \frac{1}{2}}{\int }_t^{+\infty }{\left({\displaystyle \frac{f_X\left(x\right)}{\overline{F_X}\left(x\right)}}\right)}^{2}dx,$$

(4)

where ${\overline{F}}_X(.)$ is the $sf$.

Residual extropy was introduced as a tool to assess the uncertainty associated with a rv. This measure has gained growing importance in areas such as survival analysis, reliability analysis, and actuarial science, as it provides crucial insights into the behavior and dynamics of systems and processes over time. In commercial or scientific fields such as astronomical measurements of heat distribution in galaxies, extropic analysis offers valuable insights worthy of further exploration (see [18,19]). The concept of residual extropy has been studied in various contexts, including the work by [20], which focuses on k-record values. Ref. [21] proposed a kernel-based estimator for the residual extropy function under the $\alpha$-mixing dependence condition.

Data gathered for statistical analysis in various scientific disciplines often include errors. In meteorology, weather data often contain inaccuracies caused by instrument limitations or unpredictable atmospheric changes. For example, in manufacturing industry, production data can contain errors due to problems with machines or mistakes made during the inspection process. Similarly, in agriculture, the measurement of crop yields can be unreliable because of differences in how samples are collected or changes in weather and other environmental conditions. These kinds of mistakes are known as measurement errors, and they happen when the data we observe is different from the actual or true values. This can occur for many reasons, such as faulty equipment, poor measuring tools, changes in the environment, or even simple human mistakes during data collection. To deal with these issues, measurement error models are used. These models help us understand and correct the inaccuracies in the data, making it possible to get more accurate and meaningful results. They are especially important in fields like manufacturing, agriculture, medicine, and social sciences, where reliable data is crucial for making decisions and drawing conclusions.

The classical error model is used when we want to determine X, but cannot do so directly because of different types of errors in measurement. For example, measuring systolic blood pressure (SBP) can be affected by daily and seasonal changes. Errors may come from machine recording issues, how the measurement is taken, etc. In such cases, it is often reasonable to assume an additive error model. For additional details and examples of classical error models, refer to [22]. Additive measurement error models have been widely studied due to their significance in handling contaminated data. In cases where the variable of interest X, which cannot be directly observed. Instead, an independent sample $Y_1,Y_2,\dots ,Y_n$ are drawn according to the model:

$$Y_i=X_i+{\omega }_{i}i=1,2,\dots ,n,$$

(5)

where the measurement error $\omega$ is independent of X. The primary goal is to estimate $f_X\left(x\right)$, the unknown pdf of X. In this model, the distribution of $\omega$ is exactly known; however, this may not possible always. In such cases, $\omega$ can be estimated from replicated measurements, as discussed by [23]. In cases where replicates are unavailable, estimating the measurement error distribution becomes more challenging. One approach is to estimate the error distribution from an independent validation data set, where a subset of the data includes both the contaminated observations and their corresponding error-free measurements. From this, the distribution of the measurement error can be directly estimated by [22]. This strategy is particularly useful in industrial applications. Additionally, simulation-based deconvolution techniques have been proposed to estimate the error distribution even when it is unknown, such as the SIMEX (Simulation-Extrapolation)-based approach introduced by [24].

In the context of extropy, Ref. [25] extended the theory of extropy estimation to handle data contaminated by additive measurement error. Building on this, Ref. [26] developed the theory for past extropy estimation under measurement error, offering an estimation framework supported by asymptotic theory and simulation evidence. However, residual extropy, despite its practical relevance, has not yet been studied under the presence of measurement error.

This lack of literature presents a critical gap: while uncertainty about the future is often the focus in predictive tasks, we currently lack statistical tools to estimate residual extropy when the observed data are contaminated. Given the pervasiveness of measurement error in lifetime and reliability data, addressing this issue is essential to ensure accurate and reliable inference.

Therefore, this study is motivated by the need to develop a non-parametric estimation method for residual extropy in the presence of additive measurement error. Our approach leverages kernel-based deconvolution techniques to correct for the error and accurately estimate the residual extropy function. By doing so, we extend the existing body of work on extropy and contribute a novel and practical solution to uncertainty quantification under imperfect data conditions.

The structure of the paper is as follows: Section 2 introduces the estimator for residual extropy in the presence of measurement error in the data. In Section 3, we examine the asymptotic properties of the proposed estimator. Section 4 and Section 5 provide a comparison between the proposed estimator and the empirical estimator based on contaminated data, through simulation and data analysis, respectively. The paper concludes with Section 6.

2. Estimation of Residual Extropy in the Presence of Measurement Error

This section addresses the estimation of the residual extropy function when measurement error is present.

2.1. Deconvolution Estimator

Assume that the rv X has a continuous pdf $f_X\left(x\right)$ and the characteristic function (cf) of $\omega$, ${\varphi }_{\omega }$ exhibits the property of being non-vanishing, which ensures the identifiability of the deconvolution procedure,

$$|{\varphi }_{\omega }\left(t\right)|>0,\forall t\in \mathbb{R}.$$

(6)

This holds true in many relevant cases, particularly in the normal model, where the $cf$ of the normal distribution satisfies the non-vanishing condition (see [27]). Furthermore, the assumption fails in case of uniform distribution (see [28]). Assume the kernel K is a symmetric, bounded $pdf$ used in the construction of the deconvolution estimator, with a $cf$${\varphi }_K$, satisfying the following conditions for any fixed positive $\lambda >0$:

$$\underset{t\in \mathbb{R}}{\mathrm{Sup}}\left|{\displaystyle \frac{{\varphi }_K\left(t\right)}{{\varphi }_{\omega }(t/\lambda )}}\right|<+\infty ,{\int }_{-\infty }^{+\infty }\left|{\displaystyle \frac{{\varphi }_K\left(t\right)}{{\varphi }_{\omega }(t/\lambda )}}\right|dt<+\infty .$$

(7)

This condition ensures that ${\varphi }_K^2\left(t\right)/|{\varphi }_{\omega }\left({\displaystyle \frac{t}{\lambda }}\right){|}^2,\left|{\varphi }_K\right|$ and ${\varphi }_K^2$ are integrable and ${\varphi }_K$ is invertible, which often requires ${\varphi }_K$ to have a compact support. Ref. [29] suggested kernels such as Gaussian or Laplacian have $cf$s that are not compactly supported but exhibit rapid decay and these kernels are used in practice due to their smoothness and computational advantages. That is

$$K\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }{e}^{-itx}{\varphi }_K\left(t\right)dt.$$

(8)

The deconvolution kernel estimator of $f_X\left(x\right)$, proposed by [30] and later extended by [27] is:

$$\begin{array}{cc}\hfill \widehat{f_X}\left(x\right)& ={\displaystyle \frac{1}{n}}\sum _{j=1}^n{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }{e}^{it(Y_j-x)}{\displaystyle \frac{{\varphi }_K\left(t\lambda \right)}{{\varphi }_{\omega }\left(t\right)}}dt,\hfill \\ \hfill & ={\displaystyle \frac{1}{n\lambda }}\sum _{j=1}^{n}L\left({\displaystyle \frac{x-Y_j}{\lambda }}\right),\hfill \end{array}$$

(9)

where K: $\mathbb{R}\to {\mathbb{R}}^{+}$, $\lambda >0$, ${\varphi }_{\omega }$ and ${\varphi }_K$ are the cf s of $\omega$ and K, respectively, and

$$\mathcal{L}\left(z\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }{e}^{-itz}{\displaystyle \frac{{\varphi }_K\left(t\right)}{{\varphi }_{\omega }\left(\frac{t}{\lambda }\right)}}dt.$$

(10)

The estimator is valid for any non-zero ${\varphi }_{\omega }$, provided ${\varphi }_K$ has compact support.

2.2. Estimation of Residual Extropy

The estimator for residual extropy function is developed based on the standard deconvolution estimator to handle with the contaminated data. The proposed estimator is expressed as:

$${\widehat{\mathcal{M}}}_{nt}\left(X\right)=-{\displaystyle \frac{1}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}{\int }_t^{+\infty }{\widehat{f}}_X^2\left(x\right)dx.$$

(11)

The function ${\widehat{f}}_X\left(x\right)$ refers to the standard deconvolution estimator, as specified in Equation (9) and,

$${\widehat{\overline{F}}}_X\left(t\right)={\int }_t^{{\tau }_{\mathcal{F}}}{\widehat{f}}_X\left(x\right)dx,$$

(12)

where ${\tau }_{\mathcal{F}}=inf\{x:F\left(x\right)=1\}$

Based on the tail behavior of the characteristic function ${\varphi }_{\omega }$, the following cases are commonly identified.

• Ordinary Smooth Errors: The tails of characteristic function ${\varphi }_{\omega }$ are polynomial, which can be described as:

  $$c_0{\left|t\right|}^{-\beta }\le |{\varphi }_{\omega }\left(t\right)|\le c_1{\left|t\right|}^{-\beta },\forall t\in \mathbb{R},$$

  for some $c_1>0,c_0>0,$ and $\beta >0$.
• Super Smooth Errors: The tails of characteristic function ${\varphi }_{\omega }$ are exponential, which follow the following form:

  $$c_{0}exp\left(-{\gamma \left|t\right|}^{-\beta }\right)\le \left|{\varphi }_{\omega }\left(t\right)\right|\le c_{1}exp\left(-{\gamma \left|t\right|}^{-\beta }\right),\forall t\in \mathbb{R},$$

  for some $c_1>0,c_0>0,\gamma >0$ and $\beta >0$.

Distributions such as the Laplace, gamma, and double exponential are considered examples of ordinary smooth errors, while the normal, mixture normal, and Cauchy distributions are examples of super-smooth errors.

3. Asymptotic Results

This section focuses on investigating the asymptotic distribution and consistency of the proposed estimator.

Theorem 1. 

Let ${\varphi }_{\omega }$ and ${\varphi }_K$ satisfy the conditions given in Equations (6) and (7). Assume that $\lambda \to 0$ and ${\left(n\lambda \right)}^{-1}{\int }_{-\infty }^{+\infty }{\varphi }_K^2\left(t\right){\left|{\varphi }_{\omega }\left({\displaystyle \frac{t}{\lambda }}\right)\right|}^{-2}dt\to 0$ as $n\to +\infty$. Under these assumptions, as $n\to +\infty$, the estimator ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ converges in probability to ${\mathcal{M}}_t\left(X\right)$, expressed as:

$${\widehat{\mathcal{M}}}_{nt}\left(X\right)\stackrel{p}{\to }{\mathcal{M}}_t\left(X\right),$$

where $\stackrel{p}{\to }$ denotes convergence in probability.

Proof. 

The corresponding Theorem’s proof is outlined in Appendix A. □

Corollary 1. 

Under the condition that error measurements follow a normal distribution with mean zero and standard deviation ${\sigma }_{\omega }$, and λ approaches 0 as n tends to infinity, Theorem 1 applies in this case.

Corollary 2. 

If the error measurements follows a Laplace distribution with mean zero, and as λ approaches zero, when $n\to +\infty$, then Theorem 1 holds in this context.

Theorem 2. 

Assume that ${\int }_{-\infty }^{+\infty }({\varphi }_K\left(t\lambda \right)/{\varphi }_{\omega }\left(t\right))dt\to casn\to +\infty ,$ for some constant $c\ge 0$, and the conditions outlined in Theorem 1 are satisfied. Then, as $n\to +\infty$, for a fixed t, the following expression holds:

$$\sqrt{n}\left({\widehat{\mathcal{M}}}_{nt}\left(X\right)-{\mathcal{M}}_t\left(X\right)\right)\stackrel{d}{\to }\mathcal{N}(0,{\sigma }^2).$$

Assuming that ${\varphi }_{\omega }$ and ${\varphi }_K$ meet the conditions outlined in Equations (6) and (7), ${\sigma }^2$ is defined as:

$${\sigma }^2={\displaystyle \frac{1}{{\widehat{\overline{F}}}_X^4\left(t\right)}}\tau {\int }_t^{+\infty }{f}_X^2\left(x\right)dx,$$

where $\tau =Var\left(L_1(x-Y_1)\right)$, and

$${\mathcal{L}}_1\left(z\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }{e}^{-itz}({\varphi }_K\left(t\lambda \right)/{\varphi }_{\omega }\left(t\right))dt,$$

with ${\widehat{\overline{F}}}_X\left(t\right)$ is defined in Equation (12).

Proof. 

We observe that,

$$\begin{array}{cc}\hfill \sqrt{n}\left({\widehat{\mathcal{M}}}_{nt}\left(X\right)-{\mathcal{M}}_t\left(X\right)\right)& =\sqrt{n}\left[-{\displaystyle \frac{1}{2}}{\int }_t^{+\infty }{\displaystyle \frac{{\widehat{f}}_X^2\left(x\right)}{{\widehat{\overline{F}}}_X^2\left(t\right)}}dx-{\displaystyle \frac{1}{2}}{\int }_t^{+\infty }{\displaystyle \frac{f_X^2\left(x\right)}{{\overline{F}}_X^2\left(t\right)}}dx\right]\hfill \\ & =-{\displaystyle \frac{\sqrt{n}}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\int }_t^{+\infty }{\widehat{f}}_X^2\left(x\right)dx-{\displaystyle \frac{{\widehat{\overline{F}}}_X^2\left(t\right)}{{\overline{F}}_X^2\left(t\right)}}{\int }_t^{+\infty }{f}_X^2\left(x\right)dx\right]\hfill \\ & =-{\displaystyle \frac{\sqrt{n}}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\int }_t^{+\infty }{\widehat{f}}_X^2\left(x\right)dx-{\int }_t^{+\infty }{f}_X^2\left(x\right)dx+{\int }_t^{+\infty }{f}_X^2\left(x\right)dx\right]\hfill \\ & -{\displaystyle \frac{\sqrt{n}}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\displaystyle \frac{{\widehat{\overline{F}}}_X^2\left(t\right)}{{\overline{F}}_X^2\left(t\right)}}{\int }_t^{+\infty }{f}_X^2\left(x\right)dx\right]\hfill \\ & =-{\displaystyle \frac{\sqrt{n}}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\int }_t^{+\infty }{\widehat{f}}_X^2\left(x\right)dx-{\int }_t^{+\infty }{f}_X^2\left(x\right)dx\right]\hfill \\ & -{\displaystyle \frac{\sqrt{n}}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\int }_t^{+\infty }{f}_X^2\left(x\right)dx-{\displaystyle \frac{{\widehat{\overline{F}}}_X^2\left(t\right)}{{\overline{F}}_X^2\left(t\right)}}{\int }_t^{+\infty }{f}_X^2\left(x\right)dx\right]\hfill \\ & =-{\displaystyle \frac{\sqrt{n}}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\int }_t^{+\infty }{\widehat{f}}_X^2\left(x\right)dx-{\int }_t^{+\infty }{f}_X^2\left(x\right)dx\right]\hfill \\ & -{\displaystyle \frac{\sqrt{n}}{2{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\displaystyle \frac{{\overline{F}}_X^2\left(t\right)-{\widehat{\overline{F}}}_X^2\left(t\right)}{{\overline{F}}_X^2\left(t\right)}}{\int }_t^{+\infty }{f}_X^2\left(x\right)dx\right]\hfill \\ & \approx -{\displaystyle \frac{\sqrt{n}}{{\widehat{\overline{F}}}_X^2\left(t\right)}}\left({\int }_t^{+\infty }\left({\widehat{f}}_X\left(x\right)-f_X\left(x\right)\right)f_X\left(x\right)dx\right)\hfill \\ & -{\displaystyle \frac{2\sqrt{n}}{{\widehat{\overline{F}}}_X^2\left(t\right)}}\left({\mathcal{M}}_t\left(X\right)\left({\widehat{\overline{F}}}_X\left(t\right)-{\overline{F}}_X\left(t\right)\right){\overline{F}}_X\left(t\right)\right)\hfill \\ & \approx -{\displaystyle \frac{\sqrt{n}}{{\widehat{\overline{F}}}_X^2\left(t\right)}}\left[{\int }_t^{+\infty }\left({\widehat{f}}_X\left(x\right)-f_X\left(x\right)\right)\left(f_X\left(x\right)+2{\mathcal{M}}_t\left(X\right){\overline{F}}_X\left(t\right)\right)dx\right].\hfill \end{array}$$

By utilizing the asymptotic normality of the standard deconvolution estimator, as described in [25],

$${\displaystyle \frac{\sqrt{n}}{\sqrt{\tau }}}\left({\widehat{f}}_X\left(x\right)-f_X\left(x\right)\right)\stackrel{d}{\to }N(0,1),n\to +\infty .$$

Consequently, as $n\to +\infty$ for a fixed t,

$$\sqrt{n}\left({\widehat{\mathcal{M}}}_{nt}\left(X\right)-{\mathcal{M}}_t\left(X\right)\right)\stackrel{d}{\to }\mathcal{N}(0,{\displaystyle \frac{1}{{\widehat{\overline{F}}}_X^4\left(t\right)}}\tau {\int }_t^{+\infty }{f}_X^2\left(x\right)dx),$$

where $\tau =\mathrm{Var}\left(L_1(x-Y_1)\right)$, and

$${\mathcal{L}}_1\left(z\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }{e}^{-itz}({\varphi }_K\left(t\lambda \right)/{\varphi }_{\omega }\left(t\right))dt,$$

□

Selecting bandwidth in deconvolution problems has been extensively explored in the literature. A theoretical study of the cross-validation (CV) bandwidth selection procedure was conducted by [31]. A bootstrap procedure for estimating the optimal bandwidth was introduced by [32], who also demonstrated its consistency. In their subsequent work, Ref. [29] compared several plug-in bandwidth selectors with the CV and bootstrap methods. Furthermore, the plug-in and bootstrap methods were extended to cases with heteroscedastic errors by [33]. Then, we use the rule-of-thumb and plug-in methods for bandwidth selection from [34].

4. Simulation

A simulation study was conducted to assess the performance of the proposed estimator, comparing it with that of the empirical estimator of residual extropy, using the contaminated sample $Y_1,\dots ,Y_n$. The estimator is defined as:

$$\begin{array}{c}\hfill M_{t\ast }\left(Y\right)=-{\displaystyle \frac{\frac{1}{2n}{\sum }_{i=1}^n{\widehat{f}}_Y\left(Y_i\right)I(Y_i\ge t)}{{\left(G_n\left(t\right)\right)}^2}}.\end{array}$$

(13)

The empirical estimator of the sf ${\overline{F}}_Y\left(t\right)$, denoted by $G_n\left(t\right)$, is based on the contaminated sample $Y_i$,

$$I(Y_i\ge t)=\left\{\begin{array}{cc}1:\hfill & \mathrm{if}{Y}_i\ge t\hfill \\ 0:\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

The simulation framework included the following scenarios and conditions, with the unobserved distribution assumed to be as follows:

• Gamma distribution with shape parameter 2 and scale parameter $\sqrt{3}$.
• Weibull distribution with shape parameter 2 and scale parameter 3.
• Log-normal distribution with shape parameter 0 and scale parameter 1.

The measurement errors are assumed to follow the distributions outlined below:

• Normal distribution with mean zero and standard deviation ${\sigma }_{\omega }$.
• Laplace distribution with mean zero and standard deviation ${\sigma }_{\omega }$.

The parameter ${\sigma }_{\omega }$ was chosen to achieve a specific level of contamination, as determined by the signal-to-noise ratio $V\left(X\right)/{\sigma }_{\omega }$. We examined contamination levels of 15% and 30% in measurement error distributions with a mean of 0 in both cases. Convergence rates shows how quickly an estimator approaches the true parameter or function as the sample size increases. They are important to understand the efficiency and accuracy of the estimator under different error distributions and smoothness conditions. For normal and Laplace error distributions, we applied the rule-of-thumb (ROT) bandwidth selection method, which sets the bandwidth proportional to the noise standard deviation and sample size. Regarding the error distribution,

$$b_{\mathrm{ROT},N}=\sqrt{2}{\sigma }_{\omega }{(logn)}^{-1/2},$$

and

$$b_{\mathrm{ROT},L}={\left({\displaystyle \frac{5{\sigma }_{\omega }^4}{n}}\right)}^{1/9}.$$

are optimum bandwidth values for normal and Laplace errors, respectively (see [34]).

In kernel density estimation with error-free data, it is generally accepted that the choice of the kernel function $\mathit{K}$ has little influence on the estimator’s accuracy. On the other hand, when performing deconvolution kernel estimation with contaminated data, certain conditions concerning the structure of deconvolution estimators must be fulfilled under specific circumstances. For normal errors, we applied a second-order kernel with a characteristic function that exhibits symmetric and compact support (see, [28,32]),

$$K\left(x\right)={\displaystyle \frac{48cosx}{\pi x^4}}\left(1-{\displaystyle \frac{15}{x^2}}\right)-{\displaystyle \frac{144sinx}{\pi x^5}}\left(2-{\displaystyle \frac{5}{x^2}}\right)$$

(14)

with characteristic function,

$${\varphi }_K\left(t\right)={(1-t^2)}^3{I}_{[-1,1]}\left(t\right).$$

In case of Laplacian errors, the standard normal kernel function was considered. The estimators defined in Equations (9) and (13) were calculated for different sample sizes n, while keeping t = 1 fixed for each sample. Repeating this procedure for 1000 samples, the bias and mean squared error $\left(MSE\right)$ were evaluated for various sample sizes n = 50, 100, 200, 400, and 500.

Examining

Table 1. The estimated value (J), absolute bias ($\left|\mathrm{bias}\right|$), and the MSE of ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ and $M_{t\ast }\left(Y\right)$ are derived from a Weibull(2,3) distribution with normal contamination, where ${\mathcal{M}}_t\left(X\right)=-0.17385$.

n	Estimator	Error with 15% Contamination			Error with 30% Contamination
		$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE	$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE
50	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.26047$	$0.08662$	$0.00750$	$-0.22099$	$0.04714$	$0.00222$
	$M_{t\ast }\left(Y\right)$	$-0.34356$	$0.17151$	$0.02941$	$-0.32238$	$0.06542$	$0.00428$
100	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.22634$	$0.05249$	$0.00276$	$-0.20145$	$0.02760$	$0.00076$
	$M_{t\ast }\left(Y\right)$	$-0.32047$	$0.14662$	$0.02150$	$-0.30550$	$0.04853$	$0.00236$
200	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.20694$	$0.03309$	$0.00109$	$-0.18915$	$0.01530$	$0.00023$
	$M_{t\ast }\left(Y\right)$	$-0.31912$	$0.14527$	$0.02110$	$-0.29507$	$0.03810$	$0.00145$
400	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.18502$	$0.01117$	$0.00012$	$-0.18534$	$0.01149$	$0.00013$
	$M_{t\ast }\left(Y\right)$	$-0.31171$	$0.13786$	$0.01901$	$-0.27502$	$0.01805$	$0.00033$
500	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.18424$	$0.01039$	$0.00011$	$-0.17771$	$0.01214$	$0.00000$
	$M_{t\ast }\left(Y\right)$	$-0.28717$	$0.11332$	$0.01284$	$-0.27746$	$0.01449$	$0.00021$

,

Table 2. The estimated value (J), absolute bias ($\left|\mathrm{bias}\right|$), and the MSE of ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ and $M_{t\ast }\left(Y\right)$ are derived from a Weibull(2,3) distribution with Laplace contamination, where ${\mathcal{M}}_t\left(X\right)=-0.17385$.

n	Estimator	Error with 15% Contamination			Error with 30% Contamination
		$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE	$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE
50	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.21742$	$0.04357$	$0.00190$	$-0.20427$	$0.03042$	$0.00093$
	$M_{t\ast }\left(Y\right)$	$-0.09855$	$0.04686$	$0.00220$	$-0.11949$	$0.06780$	$0.00460$
100	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.21693$	$0.04308$	$0.00186$	$-0.19377$	$0.01992$	$0.00040$
	$M_{t\ast }\left(Y\right)$	$-0.09609$	$0.04440$	$0.00197$	$-0.10673$	$0.05504$	$0.00303$
200	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.20263$	$0.02878$	$0.00083$	$-0.15843$	$0.01542$	$0.00024$
	$M_{t\ast }\left(Y\right)$	$-0.09185$	$0.04016$	$0.00161$	$-0.09606$	$0.04437$	$0.00197$
400	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.19958$	$0.02573$	$0.00066$	$-0.16784$	$0.00601$	$0.00004$
	$M_{t\ast }\left(Y\right)$	$-0.09045$	$0.03876$	$0.00150$	$-0.09355$	$0.04186$	$0.00175$
500	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.18966$	$0.01581$	$0.00025$	$-0.17007$	$0.00378$	$0.00001$
	$M_{t\ast }\left(Y\right)$	$-0.08380$	$0.03211$	$0.00103$	$-0.09218$	$0.04049$	$0.00164$

,

Table 3. The estimated value (J), absolute bias ($\left|\mathrm{bias}\right|$), and the MSE of ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ and $M_{t\ast }\left(Y\right)$ derived from a Gamma(2,$\sqrt{3}$) distribution with normal contamination, where ${\mathcal{M}}_t\left(X\right)=-0.05169$.

n	Estimator	Error with 15% Contamination			Error with 30% Contamination
		$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE	$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE
50	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.07385$	$0.02215$	$0.00049$	$-0.03169$	$0.02000$	$0.00040$
	$M_{t\ast }\left(Y\right)$	$-0.10962$	$0.05793$	$0.00336$	$-0.1072$	$0.05551$	$0.00308$
100	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.07217$	$0.02048$	$0.00042$	$-0.06163$	$0.00994$	$0.00010$
	$M_{t\ast }\left(Y\right)$	$-0.10208$	$0.05039$	$0.00254$	$-0.10549$	$0.05380$	$0.00289$
200	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.07375$	$0.01434$	$0.00021$	$-0.04600$	$0.00569$	$0.00003$
	$M_{t\ast }\left(Y\right)$	$-0.09745$	$0.04579$	$0.00210$	$-0.10499$	$0.05330$	$0.00284$
400	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.06508$	$0.01338$	$0.00018$	$-0.04355$	$0.00491$	$0.00003$
	$M_{t\ast }\left(Y\right)$	$-0.09311$	$0.04142$	$0.00172$	$-0.09729$	$0.04560$	$0.00208$
500	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.06516$	$0.01347$	$0.00018$	$-0.05036$	$0.00133$	$0.00000$
	$M_{t\ast }\left(Y\right)$	$-0.09295$	$0.04126$	$0.00170$	$-0.09355$	$0.04186$	$0.00175$

,

Table 4. The estimated value (J), absolute bias ($\left|\mathrm{bias}\right|$), and the MSE of ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ and $M_{t\ast }\left(Y\right)$ derived from a Gamma(2,$\sqrt{3}$) distribution with Laplace contamination, where ${\mathcal{M}}_t\left(X\right)=-0.05169$.

n	Estimator	Error with 15% Contamination			Error with 30% Contamination
		$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE	$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE
50	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.06955$	$0.01786$	$0.00022$	$-0.06922$	$0.01753$	$0.00031$
	$M_{t\ast }\left(Y\right)$	$-0.09555$	$0.04686$	$0.0032$	$-0.11949$	$0.06780$	$0.00460$
100	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.06701$	$0.01532$	$0.00023$	$-0.06493$	$0.01324$	$0.00018$
	$M_{t\ast }\left(Y\right)$	$-0.09609$	$0.04440$	$0.00197$	$-0.10673$	$0.05504$	$0.00303$
200	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.03748$	$0.01421$	$0.0002$	$-0.04826$	$0.00343$	$0.00001$
	$M_{t\ast }\left(Y\right)$	$-0.09185$	$0.04016$	$0.00161$	$-0.09606$	$0.04437$	$0.00197$
400	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.06279$	$0.01110$	$0.00012$	$-0.05159$	$0.01110$	$0.0001$
	$M_{t\ast }\left(Y\right)$	$-0.09045$	$0.03876$	$0.00150$	$-0.09355$	$0.04186$	$0.00175$
500	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.05913$	$0.00744$	$0.00006$	$-0.0509$	$0.0002$	$0.0000$
	$M_{t\ast }\left(Y\right)$	$-0.0838$	$0.03211$	$0.00103$	$-0.09218$	$0.04049$	$0.00164$

,

Table 5. The estimated value (J), absolute bias ($\left|\mathrm{bias}\right|$), and the MSE of ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ and $M_{t\ast }\left(Y\right)$ derived from a Log-normal (0,1) distribution with normal contamination, where ${\mathcal{M}}_t\left(X\right)=-0.14053$.

n	Estimator	Error with 15% Contamination			Error with 30% Contamination
		$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE	$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE
50	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.23882$	$0.08266$	$0.00683$	$-0.08875$	$0.05178$	$0.00268$
	$M_{t\ast }\left(Y\right)$	$-0.05787$	$0.09829$	$0.00966$	$-0.03233$	$0.10820$	$0.01171$
100	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.18850$	$0.04798$	$0.00230$	$-0.09773$	$0.04280$	$0.00183$
	$M_{t\ast }\left(Y\right)$	$-0.07095$	$0.06958$	$0.00484$	$-0.03422$	$0.10631$	$0.01130$
200	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.18014$	$0.03961$	$0.00157$	$-0.10520$	$0.03533$	$0.00125$
	$M_{t\ast }\left(Y\right)$	$-0.07266$	$0.06787$	$0.00461$	$-0.03651$	$0.10402$	$0.01082$
400	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.16971$	$0.02918$	$0.00085$	$-0.10662$	$0.03391$	$0.00115$
	$M_{t\ast }\left(Y\right)$	$-0.07340$	$0.06712$	$0.00451$	$-0.04351$	$0.09702$	$0.00941$
500	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.15635$	$0.01583$	$0.00025$	$-0.12425$	$0.01628$	$0.00026$
	$M_{t\ast }\left(Y\right)$	$-0.07455$	$0.06597$	$0.00435$	$-0.05718$	$0.08334$	$0.00695$

and

Table 6. The estimated value (J), absolute bias ($\left|\mathrm{bias}\right|$), and the MSE of ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ and $M_{t\ast }\left(Y\right)$ derived from a Log-normal (0,1) distribution with Laplace contamination, where ${\mathcal{M}}_t\left(X\right)=-0.14053$.

n	Estimator	Error with 15% Contamination			Error with 30% Contamination
		$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE	$\mathit{J}$	$\left|\mathbf{bias}\right|$	MSE
50	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.21534$	$0.07482$	$0.00560$	$-0.07974$	$0.06079$	$0.00370$
	$M_{t\ast }\left(Y\right)$	$-0.05107$	$0.08946$	$0.00800$	$-0.03124$	$0.10928$	$0.01194$
100	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.08929$	$0.05124$	$0.00263$	$-0.09416$	$0.04637$	$0.00215$
	$M_{t\ast }\left(Y\right)$	$-0.05623$	$0.08430$	$0.00711$	$-0.03458$	$0.10595$	$0.01122$
200	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.12196$	$0.01856$	$0.00034$	$-0.10055$	$0.03998$	$0.00160$
	$M_{t\ast }\left(Y\right)$	$-0.06527$	$0.07526$	$0.00566$	$-0.03978$	$0.10074$	$0.01015$
400	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.15525$	$0.01472$	$0.00022$	$-0.10584$	$0.03469$	$0.00120$
	$M_{t\ast }\left(Y\right)$	$-0.06827$	$0.07226$	$0.00522$	$-0.04121$	$0.09932$	$0.00987$
500	${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.14123$	$0.00071$	$0.00000$	$-0.11417$	$0.02636$	$0.00069$
	$M_{t\ast }\left(Y\right)$	$-0.07066$	$0.06987$	$0.00488$	$-0.04418$	$0.09635$	$0.00928$

, we notice a consistent pattern where both bias and MSE decrease with increasing sample size. When we compare the performance of the estimators based on the MSE, it can be concluded that ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ performs better than $M_{t\ast }\left(Y\right)$.

5. Data Analysis

We evaluated the performance of the proposed estimator ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$ and the empirical estimator $M_{t\ast }\left(Y\right)$ using an actual testing data set of piston pumps, which is taken from [35]. The data set used in this study originates from an accelerated testing experiment conducted on a hydraulic piston pump. It consists of two parts: a degradation test and a lifetime test. In the degradation test, the return oil flow (measured in L/min) is used as a performance degradation indicator. Three pumps were tested under a constant pressure condition of 27.6 MPa and a rotation speed of 6000 r/min. The return oil flow was recorded every 10 h for a total duration of 220 h, producing a time series of degradation values for each pump. This data helps monitor the gradual performance decay of the pumps. In the lifetime test, eight pumps were tested under the same conditions. The failure threshold was defined as a drop in the return oil flow to 2.15 L/min, at which point the pump was considered to have failed. The time taken by each pump to reach this failure threshold is recorded as the lifetime. Both the degradation and lifetime data are used to assess the reliability of the system, where degradation readings inform the progression of wear, fatigue, and ageing and lifetime measurements represent the total usable life until failure occurs.

The primary function of a hydraulic piston pump is to produce hydraulic energy for the hydraulic system. Powered by an engine, the drive shaft rotates the cylinder block, causing the pistons to move reciprocally within the block. Meanwhile, the valve plate and swash plate remain stationary. With each revolution, the pistons perform a back-and-forth motion within their chambers, transforming mechanical energy into hydraulic energy. In this study, the standard deviation is assumed to be 0.01. The context of degradation testing error is proposed in [35,36].

The proposed estimators have been assessed by determining their estimated values and MSE. A comparative study was also carried out to examine the effect of the kernel estimator of extropy on contaminated data. Let $M_n\left(t\right)$ be the kernel estimator of residual extropy, which is given by

$$M_n\left(t\right)=-{\displaystyle \frac{1}{2}}{\int }_t^{+\infty }{\left({\displaystyle \frac{f_n\left(x\right)}{{\overline{F}}_n\left(t\right)}}\right)}^{2}dt,$$

(15)

where $f_n\left(x\right)$ represents the standard kernel density estimator introduced by [37] and

$$F_n\left(t\right)={\int }_0^t{f}_n\left(x\right)dx,$$

(16)

is the kernel density estimator.

As shown in

Table 7. The estimated value (J) and MSE of ${\widehat{\mathcal{M}}}_{nt}\left(X\right)$, $M_{t\ast }\left(Y\right)$, and $M_n\left(t\right)$ for the hydraulic piston pump study.

Estimator	J	MSE
${\widehat{\mathcal{M}}}_{nt}\left(X\right)$	$-0.000002$	33,452.53
$M_{t\ast }\left(Y\right)$	$-0.003205$	33,453.64
$M_n\left(t\right)$	$-0.004779$	33,454.18

, our proposed estimator generally exhibits better performance than the empirical estimator in terms of MSE. The primary objective here is to show that the performance of our proposed estimator is either comparable to or better than that of the empirical estimator when dealing with contaminated data. A comparison with the kernel density estimator was also conducted, demonstrating that the proposed estimator performs better. The results confirm that this is indeed the case. These findings confirm the effectiveness and reliability of the proposed estimator in challenging data environments.

6. Conclusions

We have estimated the residual extropy function under the presence of measurement error and demonstrated the asymptotic properties of the estimator. We assessed the performance of the estimator through a simulation study and data analysis, comparing it with the empirical estimator derived from contaminated samples. We also compared our estimator with the kernel density estimator, and the results indicate that our proposed estimator outperforms it. The simulation study, which shows decreasing MSE and bias, confirms the effectiveness of the estimator. Additionally, data analysis confirms that the proposed estimator performs better in the presence of contaminated data. Overall, these findings confirm that the proposed estimator is effective in the presence of contaminated or noisy data.