Estimation of Parameters and Reliability Based on Unified Hybrid Censoring Schemes with an Application to COVID-19 Mortality Datasets

Abstract

This article presents maximum likelihood and Bayesian estimates for the parameters, reliability function, and hazard function of the Gumbel Type-II distribution using a unified hybrid censored sample. Bayesian estimates are derived under three loss functions: squared error, LINEX, and generalized entropy. The parameters are assumed to follow independent gamma prior distributions. Since closed-form solutions are not available, the MCMC approximation method is used to obtain the Bayesian estimates. The highest posterior density credible intervals for the model parameters are computed using importance sampling. Additionally, approximate confidence intervals are constructed based on the normal approximation to the maximum likelihood estimates. To derive asymptotic confidence intervals for the reliability and hazard functions, their variances are estimated using the delta method. A numerical study compares the proposed estimators in terms of their average values and mean squared error using Monte Carlo simulations. Finally, a real dataset is analyzed to illustrate the proposed estimation methods.

1. Introduction

In lifetime experiments, censoring is commonly used to address time and cost constraints, especially when testing modern, highly reliable items that require long testing periods in conventional studies. Various censoring techniques are applied in real-world scenarios with different objectives to improve efficiency and reduce unnecessary expenses. Censoring is widely used in fields such as clinical trials, biological studies, and industrial engineering. Two of the most commonly used methods are type-I and type-II censoring schemes, where testing ends at a predetermined time T or upon the occurrence of the r-th failure, respectively.

Epstein [1] introduced a hybrid Type-I censoring method in which the experiment ends at $T^{*}=min(T,y_{r:n})$, where $y_{r:n}$ represents the failure time of the r-th unit. In contrast, Childs et al. [2] proposed a Type-II hybrid censoring scheme that terminates at $T^{*}=max(T,y_{r:n})$. However, both Type-I and Type-II censoring methods share the same limitations as the Type-II hybrid censoring technique. To address these constraints, Chandrasekar et al. [3] introduced two generalized hybrid censoring methods. Although these approaches were designed to overcome the issues associated with Type-I and Type-II hybrid censoring schemes, they still have certain drawbacks. For instance, in the generalized Type-I hybrid censoring approach, the experimenter may fail to observe the r-th failure within the specified time frame.

To tackle these issues, Balakrishnan et al. [4] developed unified hybrid censoring methods (UHCS). The parameters of a life testing experiment with n objects are predetermined by the experimenter in the UHCS. k and $r\in \{0,\dots ,n\}$, where $k<r<n$, and $T_1<T_2$, are the preset values. However, if the k-th failure happens before $T_1$, the experiment stops at $min\{max\{y_{r:n},T_1\},T_2\}$. Should the k-th failure take place between $T_1$ and $T_2$, the experiment is ended at $min\left\{y_{r:n}\right\}$. If the k-th failure happens after $T_2$, the experiment is terminated at $y_k$. Due to the significance of this type of information, a number of authors have studied this topic, namely [5,6,7,8,9,10,11,12].

Consequently, we determine the six scenarios in

Table 1. Situations where UHCS analyzing will be finished.

Cases	UHCS	End of Time	Total Failures
I	$0<y_{k:n}<y_{r:n}<T_1<T_2$	$T_1$	$d_1$
II	$0<y_{k:n}<T_1<y_{r:n}<T_2$	$y_{r:n}$	r
III	$0<y_{k:n}<T_1<T_2<y_{r:n}$	$T_2$	$d_2$
IV	$0<T_1<y_{k:n}<y_{r:n}<T_2$	$y_{r:n}$	r
V	$0<T_1<y_{k:n}<T_2<y_{r:n}$	$T_2$	$d_2$
VI	$0<T_1<T_2<y_{k:n}<y_{r:n}$	$y_{k:n}$	k

in respect to the UHCS.

The likelihood function in accordance with the UHCS is

$$L(\mathbf{y},\theta )={\displaystyle \frac{n!}{(n-Q)!}}\left[\prod _{i=1}^{Q}f\left(y_i\right)\right]{[1-F(A\left)\right]}^{n-Q},$$

(1)

where

$$(Q,A)=\left\{\begin{array}{cc}(d_1,T_1),\hfill & \mathrm{for} \mathrm{case} \mathrm{I},\hfill \\ (r,y_{r:n}),\hfill & \mathrm{for} \mathrm{case} \mathrm{II},\hfill \\ (d_2,T_2),\hfill & \mathrm{for} \mathrm{case} \mathrm{III},\hfill \\ (r,y_{r:n}),\hfill & \mathrm{for} \mathrm{case} \mathrm{IV},\hfill \\ (d_2,T_2),\hfill & \mathrm{for} \mathrm{case} \mathrm{V},\hfill \\ (k,y_{k:n}),\hfill & \mathrm{for} \mathrm{case} \mathrm{VI}.\hfill \end{array}\right.$$

In this case, Q and A stand for the total number of test failures up until the end of time, respectively.

When analyzing real-world data on COVID-19 patient mortality rates, we often encounter information related to the time until a specific event occurs. Such data, commonly referred to as survival time data, typically exhibit a right-skewed distribution. The Gumbel Type-II distribution (G-IID), known for its positive skewness, is well-suited for modeling this type of data. Introduced by the German mathematician Gumbel [13], the G-IID has proven valuable for modeling extreme events, including floods, earthquakes, and other natural disasters. Its applications extend to life expectancy tables, hydrology, and rainfall studies. Gumbel himself demonstrated the effectiveness of this distribution in simulating expected lifespans during product lifetime comparisons. Additionally, the G-IID plays a crucial role in predicting the probabilities of natural hazards and meteorological phenomena.

Substantial progress has been made in statistical inference methods for the G-IID. For instance, Mousa et al. [14] investigated Bayesian estimation through simulations based on specific parameter values. Nadarajah and Kotz [15] enhanced the beta Gumbel distribution and proposed maximum likelihood algorithms, and Miladinovic and Tsokos [16] examined Bayesian reliability estimation for a modified Gumbel failure model using the square error loss function.

Focusing specifically on the G-IID, Feroze and Aslam [17] investigated Bayesian estimation methods for doubly censored samples under various loss functions. Abbas et al. [18] expanded the scope of inference by incorporating Bayesian estimation approaches for the G-IID. The Bayesian estimation of two competing units within this distribution was derived by Feroze and Aslam [19]. Reyad and Ahmed [20] estimated unknown shape parameters for joint Type-II censored data using Bayesian and E-Bayesian approaches. Sindhu et al. [21] examined Bayesian estimators and their associated risks under diverse informative and non-informative priors for left Type-II censored samples. Additionally, Abbas et al. [22] applied Lindley’s approximation to develop Bayesian estimators for Type-II censored data under a non-informative prior and multiple loss functions. More recently, Qiu and Gui [23] derived Bayesian estimates for two G-IID under joint Type-II censoring. For $\mu >0$ and $\upsilon >0$, the G-IID’s cumulative distribution function (CDF) is as described below:

$$F\left(y\right)=e^{-\upsilon y^{-\mu }},y>0,\upsilon >0,\mu >0,$$

(2)

additionally, the probability density function (PDF) that corresponds to it is given as follows:

$$f\left(y\right)=\mu \upsilon y^{-(\mu +1)}{e}^{-\upsilon y^{-\mu }},y>0,\upsilon >0,\mu >0.$$

(3)

The reliability function is defined as

$$S\left(y\right)=1-F\left(y\right)=1-e^{-\upsilon y^{-\mu }},$$

(4)

and the failure rate function $h\left(y\right)$ is expressed as follows:

$$h\left(y\right)={\displaystyle \frac{f\left(y\right)}{1-F\left(y\right)}}={\displaystyle \frac{\mu \upsilon y^{-(\mu +1)}{e}^{-\upsilon y^{-\mu }}}{1-e^{-\upsilon y^{-\mu }}}}.$$

(5)

In this context, the parameters $\mu$ and $\upsilon$ denote shape and scale, respectively.

Figure 1 shows the PDF, while Figure 2 illustrates the CDF of the G-IID for different values of the parameter $\mu$, with $\upsilon$ held constant. The G-IID was selected for this analysis due to both its theoretical properties and empirical performance in modeling positively skewed, heavy-tailed lifetime data. This type of distribution is well-suited for capturing the nature of COVID-19 mortality data, which often exhibits a long right tail reflecting delayed fatalities or rare high-mortality days during outbreak peaks.

1.1. Why G-IID over Weibull, Gompertz, or GEV?

• The Weibull distribution is widely used but often underestimates the probability of extreme outcomes due to its limited tail flexibility.
• The Gompertz distribution, while biologically interpretable, is more appropriate for modeling aging-related hazards rather than acute, outbreak-driven data.
• The Generalized extreme value (GEV) family is flexible but contains three parameters and may overfit small or moderately censored samples; in contrast, G-IID offers a parsimonious two-parameter alternative with similar right-tail behavior.

The Gumbel Type-II’s cumulative distribution function $F\left(y\right)=e^{-\nu y^{-\mu }}$ reflects a heavy right tail and positive skewness. The shape parameter $\mu$ controls the degree of skewness, while the scale parameter $\nu$ affects the spread. This flexibility allows the model to capture rare but significant increases in mortality; such as spikes due to emerging variants or delayed healthcare responses.

1.2. Interpretation of Parameters

• Higher values of $\mu$ correspond to sharper declines in the hazard rate, suggesting more concentrated mortality during epidemic peaks.
• Lower values of $\nu$ reflect long-tail behavior, i.e., mortality tapering off slowly rather than abruptly.

These properties make the G-IID suitable not only from a statistical fitting perspective but also for capturing the practical dynamics of COVID-19 mortality trends.

This research aims to develop and evaluate estimation methods for the parameters, reliability function, and hazard function of the G-IID under UHCS. It contributes by applying both classical and Bayesian approaches to estimate unknown parameters using real and simulated data. The study derives maximum likelihood estimators (MLEs), constructs asymptotic confidence intervals (ACIs) using the Fisher information matrix, and obtains Bayesian estimates under three distinct loss functions: squared error (SE), LINEX, and generalized entropy (GE). Since analytical Bayesian estimates are not available, the research utilizes Markov Chain Monte Carlo (MCMC) methods, including Metropolis–Hastings within Gibbs sampling, to generate posterior estimates and credible intervals. Through extensive simulation studies and real data application, the study highlights the efficiency and reliability of Bayesian methods, particularly under the GE loss function, in handling censored lifetime data. The work offers a practical and flexible framework for improving reliability analysis and life-testing experiments, providing new insights into the modeling of incomplete or censored data using the G-IID.

This study contributes to the Special Issue’s focus by addressing the challenge of making reliable inferences from censored and incomplete data. Using unified hybrid censoring schemes and real pandemic data, we demonstrate robust statistical estimation techniques that support uncertainty modeling and decision making in applied sciences.

The manuscript makes a substantial contribution to the field of reliability analysis and life-testing experiments by applying a UHCS alongside Bayesian estimation methods. To clarify its novelty, the work integrates both Type-I and Type-II censoring into a single unified framework, offering greater flexibility than previous studies that typically treated these schemes separately. While hybrid censoring has been applied to Gumbel-related distributions before, the unified approach presented here accommodates a wider range of experimental conditions. Additionally, the manuscript distinguishes itself through its application of Bayesian estimation under multiple loss functions—squared error, LINEX, and generalized entropy—unlike prior studies that often relied solely on maximum likelihood estimation (MLE) with a single loss criterion. The rationale for adopting Bayesian methods is well-founded; Bayesian inference allows for the incorporation of prior information and yields credible intervals, enhancing the reliability of estimates, especially under censored data scenarios, such as COVID-19 mortality datasets. These advantages are further highlighted by the limitations of MLE, including its potential bias in small samples and instability under heavy censoring, which can lead to large variances and unreliable estimates. In contrast, Bayesian methods maintain consistent performance under extreme censoring and offer robustness critical for practical applications. To reinforce this point, the manuscript would benefit from addressing the weaknesses of MLE explicitly in the Introduction. Doing so would provide a clearer rationale for the methodological choices made and prepare readers for the simulation-based comparisons discussed later. This framing would enhance the coherence of the argument and highlight the practical and theoretical strengths of the Bayesian approach within the context of the G-IID and UHCS.

This paper proceeds as follows. Section 2 provides the survival and hazard rate functions, as well as the MLEs for the unknown parameters. The Fisher information matrix (FIM) is used to create ACIs for interval estimation in Section 3. Bayesian parameter estimation using Gibbs sampling and importance sampling strategies under three distinct loss functions is shown in Section 4. A real dataset is analyzed in Section 5 to investigate parameter estimation. Section 6 presents a Monte Carlo simulation analysis to evaluate the proposed estimates based on average width, mean squared error (MSE), and average value. Finally, Section 7 provides the conclusions.

2. Maximum Likelihood Estimation

Assume that the random sample $y_1,y_2,\dots ,y_n$ is drawn from the G-IID. By incorporating Equations (2) and (3) into Equation (1) and omitting the constant term, the likelihood function is given by

$$L(\mathbf{y};\upsilon ,\mu )={\displaystyle \frac{n!}{(n-Q)!}}\left[\prod _{i=1}^Q\mu \upsilon y^{-(\mu +1)}{e}^{-\upsilon y^{-\mu }}\right]{\left[1-e^{-\upsilon A^{-\mu }}\right]}^{n-Q},$$

(6)

where ${\displaystyle \frac{n!}{(n-Q)!}}$ is constant.

$$L(\mathbf{y};\upsilon ,\mu )={\mu }^Q{\upsilon }^Q{e}^{-(\mu +1){\sum }_{i=1}^{Q}ln{y}_i}{e}^{-\upsilon {\sum }_{i=1}^Q{y}_i^{-\mu }}{e}^{(n-Q)ln\left(1-e^{-\upsilon A^{-\upsilon }}\right)}.$$

(7)

Using Equation (7), the log-likelihood function for $\mu$ and $\upsilon$ is expressed as

$$\begin{array}{cc}\hfill \mathsf{\ell }(\mathbf{y};\mu ,\upsilon )=& Qln\mu +Qln\upsilon -(\mu +1)\sum _{i=1}^Q{y}_i-\upsilon \sum _{i=1}^Q{y}_i^{-\mu }+(n-Q)ln\left(1-e^{-\upsilon A^{-\mu }}\right).\hfill \end{array}$$

(8)

Differentiating Equation (8) with respect to $\mu$ and $\upsilon$ gives the following partial derivatives of the likelihood function:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{Q}{\widehat{\mu }}}-\sum _{i=1}^{Q}ln{y}_i+\widehat{\upsilon }\sum _{i=1}^Q{y}_i^{-\widehat{\mu }}ln{y}_i-{\displaystyle \frac{(n-Q)\widehat{\upsilon }{A}^{-\widehat{\mu }}{e}^{-\widehat{\upsilon }{A}^{-\widehat{\mu }}}lnA}{1-e^{-\widehat{\upsilon }{A}^{-\widehat{\mu }}}}}=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{Q}{\widehat{\upsilon }}}-\sum _{i=1}^Q{y}_i^{-\widehat{\mu }}+{\displaystyle \frac{(n-Q)A^{-\widehat{\mu }}{e}^{-\widehat{\upsilon }{A}^{-\widehat{\mu }}}}{1-e^{-\widehat{\upsilon }{A}^{-\widehat{\mu }}}}}=0.\hfill \end{array}$$

(10)

Simultaneously solving the complex nonlinear Equations (9) and (10) enables the estimation of the MLEs for $\mu$ and $\upsilon$. However, deriving explicit closed-form solutions for these equations is difficult. Therefore, numerical methods, such as the Newton–Raphson algorithm, are used to compute the MLEs of the unknown parameters. Additionally, the consistency property of MLEs allows for the estimation of $S\left(t\right)$ and $h\left(t\right)$ at a given time t. To determine these MLEs, substitute the estimated values $\widehat{\mu }$ and $\widehat{\upsilon }$ into Equations (2) and (3), respectively.

$$\widehat{S}\left(y\right)=1-e^{-\widehat{\upsilon }{y}^{-\widehat{\mu }}},$$

(11)

and

$$\widehat{h}\left(y\right)={\displaystyle \frac{\widehat{\mu }\widehat{\upsilon }{y}^{-(\widehat{\mu }+1)}{e}^{-\widehat{\upsilon }{y}^{-\widehat{\mu }}}}{1-e^{-\widehat{\upsilon }{y}^{-\widehat{\mu }}}}}.$$

(12)

3. Fisher Information Matrix

To construct the ACIs in this subsection, the asymptotic variance–covariance matrix is required. This matrix is obtained by inverting the FIM. The inverse of the FIM $I^{-1}(\mu ,\upsilon )$ is given as follows:

$$\begin{array}{c}\hfill I^{-1}(\mu ,\upsilon )={\left(\begin{array}{cc}-{\displaystyle \frac{{\partial }^2\mathsf{\ell }}{\partial {\mu }^2}}& -{\displaystyle \frac{{\partial }^2\mathsf{\ell }}{\partial \mu \partial \upsilon }}\\ -{\displaystyle \frac{{\partial }^2\mathsf{\ell }}{\partial \upsilon \partial \mu }}& -{\displaystyle \frac{{\partial }^2\mathsf{\ell }}{\partial {\upsilon }^2}}\end{array}\right)}_{(\mu ,\upsilon )=(\widehat{\mu },\widehat{\upsilon })}^{-1}.\end{array}$$

However, deriving exact mathematical expressions for these assumptions is challenging. For further details, refer to Cohen [24]. As a result, the asymptotic variance–covariance matrix is given as follows:

$$\begin{array}{c}\hfill I^{-1}(\mu ,\upsilon )=\left(\begin{array}{cc}\widehat{var}\left(\widehat{\mu }\right)& cov(\widehat{\mu },\widehat{\upsilon })\\ cov(\widehat{\upsilon },\widehat{\mu })& \widehat{var}\left(\widehat{\upsilon }\right)\end{array}\right).\end{array}$$

Since the MLEs are asymptotically normal, ACIs can be constructed for the parameters $\upsilon$ and $\mu$. The $(1-\beta )100\%$ ACIs for these parameters are given by:

$$\begin{array}{c}\hfill \left(\widehat{\mu }\pm Z_{\beta /2}\sqrt{\widehat{var}\left(\widehat{\mu }\right)}\right)\mathrm{and}\left(\widehat{\upsilon }\pm Z_{\beta /2}\sqrt{\widehat{var}\left(\widehat{\upsilon }\right)}\right),\mathrm{respectively}.\end{array}$$

Greene [25] provides a detailed explanation of the delta method, a general approach for computing ACIs for functions of MLEs. This method is used to estimate the variances of $\widehat{S}\left(t\right)$ and $\widehat{h}\left(t\right)$, which depend on the parameters $\mu$ and $\upsilon$, to construct the corresponding ACIs. Consequently, the variances of $\widehat{S}\left(t\right)$ and $\widehat{h}\left(t\right)$ are provided as follows:

$${\widehat{\sigma }}_{S\left(t\right)}^2={\left[\nabla \widehat{S}\left(t\right)\right]}^T\left[\widehat{V}\right]\left[\nabla \widehat{S}\left(t\right)\right],$$

$${\widehat{\sigma }}_{h\left(t\right)}^2={\left[\nabla \widehat{h}\left(t\right)\right]}^T\left[\widehat{V}\right]\left[\nabla \widehat{h}\left(t\right)\right].$$

The gradients of $\widehat{S}\left(t\right)$ and $\widehat{h}\left(t\right)$ with regard to $\mu$ and $\upsilon$, respectively, are represented by $\widehat{V}=I^{-1}(\mu ,\upsilon )$ in these formulas with regard to $\mu$ and $\upsilon$, respectively. Consequently, the ACIs for $S\left(t\right)$ and $h\left(t\right)$ are calculated as follows:

$$\begin{array}{c}\hfill \left(\widehat{S}\left(t\right)\pm Z_{\beta /2}\sqrt{{\widehat{\sigma }}_{S\left(t\right)}^2}\right),\mathrm{and}\left(\widehat{h}\left(t\right)\pm Z_{\beta /2}\sqrt{{\widehat{\sigma }}_{h\left(t\right)}^2}\right),\mathrm{respectively}.\end{array}$$

4. Bayes Estimation

The fact that the Bayes estimate simultaneously considers the information from observed sample data and the prior information sets it apart from MLE. In a more reasonable and logical manner, it can describe the issues, considering the independent Gamma distributions of the parameters $\mu$ and $\upsilon$

$${\pi }_1\left(\upsilon \right)\propto {\upsilon }^{a_1-1}{e}^{-b_1\upsilon },\upsilon >0,$$

$${\pi }_2\left(\mu \right)\propto {\mu }^{a_2-1}{e}^{-b_2\mu },\mu >0,$$

where $a_1,b_1,a_2,$ and $b_2$ serve as hyperparameters. Consequently, the following is an expression for the joint prior distribution for $\mu$ and $\upsilon$:

$$\pi (\mu ,\upsilon )\propto {\mu }^{a_2-1}{\upsilon }^{a_1-1}{e}^{-b_2\mu -b_1\upsilon }.$$

(13)

Using Equations (7) and (13), the joint posterior density function for $\mu$ and $\upsilon$ is as follows:

$$\begin{array}{c}\hfill {\pi }^{*}(\mu ,\upsilon \mid \mathbf{y})\propto {\mu }^{Q+a_2-1}{\upsilon }^{Q+a_1-1}{e}^{-(\mu +1){\sum }_{i=1}^{Q}ln{y}_i-b_2\mu }{e}^{-\upsilon {\sum }_{i=1}^Q{y}_i^{-\mu }-b_1\upsilon }{e}^{(n-Q)ln\left(1-e^{-\upsilon A^{-\upsilon }}\right)}.\end{array}$$

(14)

It is not possible to simply calculate the Bayes estimator from Equation (14) using mathematical techniques. In order to generate crucial estimators for $\mu$ and $\upsilon$, as well as $S\left(t\right)$ and $h\left(t\right)$, and their respective credible intervals (CRIs), the MCMC technique should be utilized. Two practical MCMC techniques that have been applied extensively in statistics are Metropolis–Hastings (M-H) and Gibbs sampling.

Metropolis–Hastings with Gibbs Sampler Algorithm

The M-H with Gibbs sampler algorithm is used to generate samples from complex probability distributions when the conditional distribution of each variable is available, but the joint distribution is unknown or difficult to handle. This technique applies the Gibbs sampler when conditional distributions are easy to sample from, while incorporating M-H in steps where direct sampling is not feasible. Its importance lies in handling high-dimensional models or cases where conditional distributions are non-standard, making direct sampling impractical. This approach is widely used in Bayesian inference, providing approximate estimates of posterior distributions for unknown parameters, thereby facilitating the analysis of complex data and statistical estimation efficiently. For the parameters $\upsilon$ and $\mu$, the conditional posterior distributions are as follows:

$$\begin{array}{c}\hfill {\pi }^{*}\left(\upsilon \right|\mu ,\mathbf{y})\propto {\upsilon }^{Q+a_1-1}{e}^{-\upsilon {\sum }_{i=1}^Q{y}_i^{-\mu }-b_1\upsilon }{e}^{(n-Q)ln\left(1-e^{-\upsilon A^{-\upsilon }}\right)},\end{array}$$

(15)

$$\begin{array}{c}\hfill {\pi }^{*}\left(\mu \right|\upsilon ,\mathbf{y})\propto {\mu }^{Q+a_2-1}{e}^{-\mu (b_2-{\sum }_{i=1}^{Q}ln{y}_i)}.\end{array}$$

(16)

In this instance, the distribution of ${\pi }^{*}(\mu \mid \upsilon ,\mathbf{y})$ is $\mathrm{Gamma}\left[Q+a_2,b_2-{\sum }_{i=1}^{Q}ln{y}_i\right]$. To do this, the Gibbs sampler must be used to create random samples for $\mu$. Direct sampling using conventional methods is also not feasible since the conditional posterior distribution of $\upsilon$ cannot be analytically simplified into standard distributions. The M-H technique, which uses a normal proposal distribution, is utilized to generate samples. To evaluate the samples for $\mu$ and $\upsilon$, the following algorithm can be used.

• 1: Let $\left({\mu }^{\left(0\right)},{\upsilon }^{\left(0\right)}\right)$ be the starting values, and let B represent the burn-in time.
• 2: Put $j=1$.
• 3: Use $\mathrm{Gamma}\left[Q+a_2,b_2-{\sum }_{i=1}^{Q}ln{y}_i\right]$ to generate ${\mu }^{\left(j\right)}$.
• 4: By using Equations (16), ${\upsilon }^{\left(j\right)}$ can be generated using the M-H approach. A normal distribution such as $N({\upsilon }^{(j-1)},\mathrm{var}\left(\upsilon \right))$ is advised. The inverse FIM’s major diagonal can be used to compute $\mathrm{var}\left(\upsilon \right)$ in this case.
  (I) Calculate the rate of acceptance (r):

  $$r=\min \left[1,{\displaystyle \frac{{\pi }^{*}\left({\upsilon }^{*}\right|{\mu }^j,data)}{{\pi }^{*}\left({\upsilon }^{(j-1)}\right|{\mu }^j,data)}}\right].$$
  (II) Create a random value u from a uniform distribution throughout the range $(0,1)$.
  (III) Adopt the recommendation and, if $u\le r$, set ${\mu }^{\left(j\right)}$ to ${\mu }^{*}$; if not, maintain ${\mu }^{\left(j\right)}$ as ${\mu }^{(j-1)}$.
• 5: Construct $S^{\left(j\right)}\left(t\right)$ and $h^{\left(j\right)}\left(t\right)$ for a given number of $\left(t\right)$ in the manner described below:

  $$S^j\left(x\right)=1-e^{-{\upsilon }^j{y}^{-{\mu }^j}},$$

  (17)

  and

  $$h^j\left(x\right)={\displaystyle \frac{{\mu }^j{\upsilon }^j{y}^{-({\mu }^j+1)}{e}^{-{\upsilon }^j{y}^{-{\mu }^j}}}{1-e^{-{\upsilon }^j{y}^{-{\mu }^j}}}}$$

  (18)
• 6: Set $j=j+1$.
• 7: Repeat steps 2–7 Q times. Consequently, the posterior means of $(\mu ,\upsilon ,S(x),h(x\left)\right)$, denoted as $\lambda$ under the SE loss function, can be computed as follows:

  $${\widehat{\lambda }}_{BS}=E\left[\lambda \right|\underline{y}]={\displaystyle \frac{1}{Q-B}}\sum _{i=B+1}^Q{\lambda }^{\left(j\right)},$$

  (19)
  With the LINEX loss function, the Bayesian estimates of $\lambda$ are as follows:

  $${\widehat{\lambda }}_{BL}=-{\displaystyle \frac{1}{\epsilon }}ln\left[{\displaystyle \frac{1}{Q-B}}\sum _{i=B+1}^Q{e}^{-\epsilon {\lambda }^{\left(j\right)}}\right].$$

  (20)
  Lastly, applying the GE loss function, the Bayesian estimates for $\lambda$ are calculated as follows:

  $${\widehat{\lambda }}_{BG}={\left[{\displaystyle \frac{1}{Q-B}}\sum _{i=B+1}^Q{\left[{\lambda }^{\left(j\right)}\right]}^{-\epsilon }\right]}^{{\displaystyle \frac{-1}{\epsilon }}}.$$

  (21)

5. Application for COVID-19 Mortality Datasets

To illustrate and compare the different estimation methods examined in this study, this section presents real-world COVID-19 death rate data from France, as documented by Almetwally [26]. The dataset spans a 51-day period, from 1 January to 20 February 2021. The data are as follows in

Table 2. Daily COVID-19 death rate in France (1 January–20 February 2021).

Day	Rate	Day	Rate	Day	Rate
1	0.0995	18	0.0505	35	0.1131
2	0.0525	19	0.1434	36	0.1129
3	0.0615	20	0.2326	37	0.2054
4	0.0455	21	0.1089	38	0.0600
5	0.1474	22	0.1206	39	0.0534
6	0.3373	23	0.2242	40	0.1422
7	0.1087	24	0.0786	41	0.2235
8	0.1055	25	0.0587	42	0.0908
9	0.2235	26	0.1516	43	0.1092
10	0.0633	27	0.2070	44	0.1958
11	0.0565	28	0.1170	45	0.0580
12	0.2577	29	0.1141	46	0.0502
13	0.1345	30	0.2705	47	0.1229
14	0.0843	31	0.0793	48	0.1738
15	0.1023	32	0.0635	49	0.0917
16	0.2296	33	0.1474	50	0.0787
17	0.0691	34	0.2345	51	0.1654

. To assess the model’s effectiveness, we conducted several goodness-of-fit tests, including the Anderson–Darling, Kolmogorov–Smirnov (KS), and Cramér–von Mises tests.

To evaluate the model’s suitability for describing the data, we analyze both distance statistics and p-values. The null hypothesis ($H_0$) is rejected (and the alternative $H_1$ is accepted) at a significance level of $\beta =0.05$ if the p-value is below 0.05. The test statistics, presented in

Table 3. Statistical measures for goodness-of-fit analysis of observed data.

K-S		Cramer–von Mises		Anderson–Darling
Statistic	p-Value	Statistic	p-Value	Statistic	p-Value
0.1028	0.6168	0.1185	0.5012	0.7947	0.4838

, show relatively high p-values for all tests. Consequently, we fail to reject $H_0$, supporting the conclusion that the data follows the G-IID and confirming the model’s good fit to the real-world dataset.

For visual assessment, Figure 3 compares the fitted and empirical survival functions of the G-IID, demonstrating close agreement between the two. Further validation is provided by the observed vs. expected probability plots (Figure 4) and quantile plots (Figure 5), reinforcing the model’s appropriateness for analyzing the dataset.

Additionally, Figure 6, Figure 7 and Figure 8 depict the histogram, smoothed histogram, and box-and-whisker plot of the real data, respectively. Figure 9, Figure 10 and Figure 11 illustrate the profile log-likelihood function and contour plot for the parameters $\mu$ and $\upsilon$, confirming that it attains a unique maximum, further validating the model’s fit.

These tests capture different aspects of goodness-of-fit. The K–S test focuses on maximum deviation, the Anderson–Darling test gives more weight to tail discrepancies, and the Cramér–von Mises test assesses integrated squared error across the distribution. Using all three allows for a more comprehensive validation of the model fit.

We analyze the case where censoring is applied to the data. From the dataset, we generate six artificially constructed UHCD sets, as shown in

Table 4. Cases of UHCD.

Cases	k	r	${\mathit{T}}_1$	${\mathit{T}}_2$	The Investigation Ends at
(I)	20	25	30	35	$T_1=0.1206$ and Q = 30
(II)	20	32	30	35	$Q=y_{r:n}=32$
(III)	20	40	30	35	$T_2=0.1474$ and Q = 35
(IV)	22	25	20	28	$Q=y_{r:n}=25$
(V)	22	25	20	28	$T_2=0.1141$ and Q = 28
(VI)	40	45	20	28	$Q=y_{k:n}=40$

. We calculate and present frequentist and Bayesian estimates for the parameters $\mu$, $\upsilon$, the survival function $S\left(t\right)$, and the hazard function $h\left(t\right)$ at time $t=10$, using initial values $\mu =2$ and $\upsilon =0.005$. These results are displayed in

Table 5. The Classical estimate and Bayes estimate methods for the parameter $\mu$.

Cases	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(I)	1.8251	1.9080	2.0423	1.7961	1.9396	1.8124
(II)	2.1215	1.8811	2.0038	1.7772	1.9108	1.7915
(III)	1.7877	1.8977	2.0068	1.8018	1.9244	1.8163
(IV)	2.1565	1.8394	1.9913	1.7167	1.8757	1.7292
(V)	1.5915	1.8765	2.0100	1.7622	1.9091	1.7767
(VI)	2.2028	1.8862	1.9801	1.8024	1.9095	1.8151

,

Table 6. The Classical estimate and Bayes estimate methods for the parameter $\upsilon$.

Cases	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(I)	0.0198	0.0188	0.0184	0.0182	0.0180	0.0178
(II)	0.0063	0.0061	0.0059	0.0057	0.0055	0.0054
(III)	0.0177	0.0176	0.0177	0.0174	0.0172	0.0170
(IV)	0.0055	0.0054	0.0052	0.0051	0.0050	0.0048
(V)	0.0321	0.0319	0.0315	0.0313	0.0311	0.0309
(VI)	0.0054	0.0052	0.0050	0.0048	0.0045	0.0042

,

Table 7. The Classical estimate and Bayes estimate methods for the $S\left(t\right)$.

Cases	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(I)	0.00370	0.0034	0.0032	0.0031	0.0029	0.0025
(II)	0.00890	0.0063	0.0062	0.0058	0.0055	0.0052
(III)	0.00340	0.0032	0.0031	0.0033	0.0028	0.0029
(IV)	0.00077	0.0010	0.0011	0.0012	0.0014	0.0013
(V)	0.00074	0.0069	0.0065	0.0062	0.0060	0.0058
(VI)	0.00073	0.0015	0.0012	0.0011	0.0010	0.0010

and

Table 8. 95% CI and 95% CRI for the $h\left(t\right)$.

Cases	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(I)	0.7287	0.7619	0.7822	0.7431	0.7746	0.7235
(II)	0.8482	0.7520	0.7706	0.7345	0.7639	0.7161
(III)	0.7138	0.7579	0.7747	0.7419	0.7686	0.7252
(IV)	0.8622	0.7354	0.7580	0.7146	0.7499	0.6912
(V)	0.6342	0.7484	0.7689	0.7292	0.7616	0.7083
(VI)	0.8808	0.7541	0.7686	0.7402	0.7634	0.7256

, derived from the dataset in Table 4. Additionally,

Table 9. 95% CI and 95% CRI for $\mu$ for real data.

Cases	Classical Estimates			Bayes Estimates
	Lower	Upper	Length	Lower	Upper	Length
(I)	1.72109	1.9292	0.20811	1.28764	2.64947	1.36183
(II)	2.01474	2.22832	0.21358	1.28241	2.60194	1.31953
(III)	1.68244	1.89307	0.21063	1.32282	2.56606	1.24324
(IV)	2.0464	2.26661	0.22021	1.19765	2.6277	1.43005
(V)	1.48438	1.69873	0.21435	1.24412	2.62612	1.3820
(VI)	2.10122	2.30452	0.20331	1.33717	2.5039	1.16674

,

Table 10. 95% CI and 95% CRI for $\upsilon$ for real data.

Cases	Classical Estimates			Bayes Estimates
	Lower	Upper	Length	Lower	Upper	Length
(I)	0.0144	0.02531	0.01091	0.01872	0.01944	0.00072
(II)	0.00446	0.00814	0.00368	0.00631	0.00641	0.0001
(III)	0.01282	0.02265	0.00983	0.01744	0.01783	0.0004
(IV)	0.00384	0.00724	0.0034	0.00535	0.00556	0.00021
(V)	0.0232	0.04118	0.01798	0.03155	0.03217	0.00062
(VI)	0.00397	0.00697	0.00301	0.00543	0.00553	0.0001

,

Table 11. 95% CI and 95% CRI for $S\left(t\right)$ for real data.

Cases	Classical Estimates			Bayes Estimates
	Lower	Upper	Length	Lower	Upper	Length
(I)	0.00249	0.00495	0.00246	0.00168	0.00584	0.00416
(II)	0.00059	0.00122	0.00063	0.00059	0.00197	0.00138
(III)	0.00229	0.00459	0.0023	0.00168	0.00524	0.00357
(IV)	0.00049	0.00105	0.00056	0.00049	0.00182	0.00133
(V)	0.00495	0.00997	0.00502	0.00287	0.01017	0.0073
(VI)	0.00049	0.00097	0.00048	0.00055	0.00161	0.00106

and

Table 12. 95% CI and 95% CRI for $h\left(t\right)$ for real data.

Cases	Classical Estimates			Bayes Estimates
	Lower	Upper	Length	Lower	Upper	Length
(I)	0.68684	0.77056	0.08371	0.51356	1.05888	0.54532
(II)	0.80544	0.89102	0.08558	0.51246	1.04047	0.5280
(III)	0.67153	0.75621	0.08468	0.52775	1.02556	0.49781
(IV)	0.81816	0.90638	0.08822	0.47863	1.05083	0.5722
(V)	0.59097	0.67751	0.08654	0.49511	1.04893	0.55383
(VI)	0.84011	0.92155	0.08144	0.53444	1.00128	0.46685

provide the interval lengths for the 95% two-sided ACI and CRI estimates.

Due to the absence of prior data for the G-IID parameters ($\mu$, $\upsilon$, $S\left(t\right)$, and $h\left(t\right)$), the Bayesian analysis relies on non-informative priors under the SE, LINEX, and GE loss functions. To mitigate the influence of initial values, we apply a burn-in period of 3000 iterations from a total of 16,000 MCMC samples, following the algorithm detailed in Section 3. The initial guesses for $\mu$ and $\upsilon$ are based on their frequentist estimates.

6. Simulation Study Analysis

This section presents the results of a simulation study to support the theoretical conclusions discussed earlier. The study evaluates the performance of the proposed estimators obtained through MLE and Bayesian methods. We estimate the parameters, reliability function, and hazard rate function of G-IID under UHCS. A Monte Carlo simulation was conducted using Mathematica version 10. We compare the point estimators for the lifetime parameters based on MSEs. We also compare the average confidence lengths, where a smaller width indicates better interval estimate performance. We specifically focus on intervals derived from the distribution of the MLEs and the CRIs. The MSE is calculated by

$$\mathrm{MSE}\left(\widehat{\theta }\right)={\displaystyle \frac{{\sum }_{i=1}^{1000}{({\theta }_i-\widehat{\theta })}^2}{1000}}.$$

The simulation was conducted by setting different values for r, k, and T, while fixing $\mu =1.4$ and $\upsilon =1.2$ for all cases. A total of 18,000 UHC samples were generated from the G-IID, with the first 4000 samples discarded as burn-in. MLEs were used to estimate $\mu$, $\upsilon$, $S\left(t\right)$, and $h\left(t\right)$, with results presented in

Table 13. The estimated values of $\mu$ under various censoring schemes are presented, with their means and (MSEs) highlighted in bold.

$(\mathit{k},\mathit{r},\mathit{Q})$	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(15, 25, 30)	1.3674	1.2714	1.5528	1.1017	1.3933	1.1067
	0.5146	0.2807	0.2550	0.2502	0.2185	0.2260
(20, 25, 30)	1.4783	1.2575	1.5054	1.1022	1.37611	1.0984
	0.4792	0.2563	0.2647	0.2490	0.2040	0.2145
(15, 50, 50)	1.3954	1.7622	2.0379	1.5745	1.8646	1.6257
	0.3904	0.3042	0.2297	0.2444	0.2197	0.2286
(20, 50, 50)	1.3478	1.3538	1.5004	1.2419	1.4323	1.2474
	0.3044	0.2403	0.2221	0.2308	0.2101	0.2102
(15, 25, 24)	1.4247	1.2276	1.5026	1.0536	1.3673	1.0355
	0.2186	0.1756	0.1705	0.1623	0.1554	0.1558
(20, 25, 24)	1.2903	1.2714	1.6965	1.0542	1.4201	1.0651
	0.2147	0.1706	0.1610	0.1629	0.1579	0.1195
(15, 25, 25)	1.2431	1.5046	1.9194	1.2621	1.6708	1.2655
	0.1656	0.0926	0.0950	0.0877	0.0777	0.0264
(20, 25, 25)	1.4650	1.0822	1.3775	0.9139	1.2021	0.9138
	0.1088	0.0146	0.0101	0.0197	0.0126	0.0101
(15, 50, 40)	1.4261	1.3897	1.3897	1.5789	1.2486	1.4879
	0.1011	0.0143	0.0140	0.0145	0.0124	0.0100
(20, 50, 40)	1.6345	1.3428	1.52026	1.2093	1.4372	1.2110
	0.0950	0.0132	0.0132	0.0131	0.0095	0.0092
(15, 50, 15)	1.6584	1.1012	1.4945	0.8995	1.2980	0.8298
	0.0675	0.0124	0.0123	0.0114	0.0083	0.0080
(20, 50, 20)	1.3582	1.4297	1.9036	1.1775	1.6213	1.1639
	0.0526	0.0116	0.0115	0.0112	0.0078	0.0062

,

Table 14. The estimated values of $\upsilon$ under various censoring schemes are presented, with their means and MSEs highlighted in bold.

$(\mathit{k},\mathit{r},\mathit{Q})$	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(15, 25, 30)	1.0944	1.0942	1.0935	1.0939	1.0929	1.0925
	0.4160	0.2161	0.2152	0.2151	0.2150	0.2144
(20, 25, 30)	1.3229	1.3226	1.3222	1.3220	1.3215	1.3212
	0.4063	0.2063	0.2060	0.2055	0.2054	0.2051
(15, 50, 50)	1.2439	1.2442	1.2440	1.2438	1.2432	1.2428
	0.3178	0.1179	0.1179	0.1171	0.1159	0.1141
(20, 50, 50)	1.3633	1.3640	1.3629	1.3632	1.3625	1.3621
	0.2295	0.1098	0.1029	0.1021	0.1020	0.1011
(15, 25, 24)	1.1862	1.1860	1.1850	1.1852	1.1838	1.1830
	0.2004	0.0895	0.0875	0.0871	0.0865	0.0854
(20, 25, 24)	1.1750	1.1751	1.1748	1.1782	1.1735	1.1762
	0.1188	0.0688	0.0488	0.0480	0.0458	0.0392
(15, 25, 25)	1.2040	1.2035	1.2033	1.2036	1.2041	1.2032
	0.1036	0.0536	0.0432	0.0426	0.0424	0.0421
(20, 25, 25)	1.1801	1.1798	1.1795	1.1792	1.1784	1.1787
	0.1000	0.0290	0.0280	0.0292	0.0250	0.0244
(15, 50, 40)	1.2930	1.2931	1.2927	1.2930	1.2925	1.2922
	0.0988	0.0189	0.0185	0.0180	0.0183	0.0178
(20, 50, 40)	1.3134	1.3136	1.3132	1.3130	1.3126	1.3121
	0.0871	0.0171	0.0170	0.0161	0.0150	0.0152
(15, 50, 15)	1.0282	1.0283	1.0280	1.0278	1.0277	1.0275
	0.0554	0.0153	0.0150	0.0142	0.0140	0.0123
(20, 50, 20)	1.2009	1.2013	1.2011	1.2010	1.2009	1.2001
	0.0220	0.0118	0.0114	0.0112	0.0110	0.0101

,

Table 15. The estimated values of $S\left(t\right)$ under various censoring schemes are presented, with their means and MSEs highlighted in bold.

$(\mathit{r},\mathit{k})$	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(15, 25, 30)	0.3468	0.3727	0.3792	0.3662	0.3885	0.3445
	0.4019	0.0268	0.0260	0.0272	0.0257	0.0237
(20, 25, 30)	0.3834	0.4338	0.4409	0.4266	0.4487	0.4079
	0.4100	0.0229	0.0205	0.0293	0.0205	0.0201
(15, 50, 50)	0.37586	0.3199	0.3256	0.3143	0.3355	0.2944
	0.3507	0.0196	0.0191	0.0181	0.0156	0.0142
(20, 50, 50)	0.3576	0.4021	0.4106	0.3933	0.4198	0.3712
	0.3205	0.0141	0.0146	0.0138	0.0121	0.0114
(15, 25, 24)	0.4144	0.4153	0.4209	0.4095	0.4266	0.3977
	0.2324	0.0125	0.0140	0.0129	0.0136	0.0111
(20, 25, 24)	0.3799	0.3954	0.4040	0.3870	0.4158	0.3548
	0.2107	0.0109	0.0123	0.0116	0.0116	0.0108
(15, 25, 25)	0.40104	0.3487	0.3593	0.3381	0.3725	0.3627
	0.2046	0.0103	0.0110	0.0111	0.0105	0.0102
(20, 25, 25)	0.3457	0.4336	0.4410	0.4263	0.4500	0.4029
	0.2011	0.0100	0.0147	0.0152	0.0147	0.0180
(15, 50, 40)	0.3823	0.3918	0.3986	0.3849	0.4060	0.3693
	0.1007	0.0045	0.0038	0.0034	0.0029	0.0025
(20, 50, 40)	0.3452	0.4067	0.4133	0.4000	0.4201	0.3855
	0.0910	0.0033	0.0039	0.0029	0.0015	0.0014
(15, 50, 15)	0.2775	0.3828	0.3931	0.3721	0.4039	0.3410
	0.0278	0.0014	0.0018	0.0012	0.0015	0.0010
(20, 50, 20)	0.3743	0.3637	0.3756	0.3515	0.3893	0.3137
	0.0104	0.0004	0.0009	0.0006	0.0005	0.0003

and

Table 16. The estimated values of $h\left(t\right)$ under various censoring schemes are presented, with their means and MSEs highlighted in bold.

$(\mathit{r},\mathit{k})$	Classical Estimates	Bayes Estimates
		SE	LINEX		GE
			$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$	$\mathsf{\epsilon }=-\mathbf{2}$	$\mathsf{\epsilon }=\mathbf{2}$
(15, 25, 30)	0.5496	0.5148	0.5795	0.4689	0.5865	0.4195
	0.5042	0.0581	0.0448	0.0483	0.0456	0.0540
(20, 25, 30)	0.5814	0.4818	0.5350	0.4427	0.5515	0.3916
	0.4248	0.0496	0.0494	0.0383	0.0339	0.0297
(15, 50, 50)	0.5458	0.7354	0.8027	0.6829	0.7987	0.6509
	0.4125	0.0343	0.0309	0.0368	0.0286	0.0213
(20, 50, 50)	0.5095	0.5144	0.5466	0.4874	0.5626	0.4505
	0.3020	0.0239	0.0230	0.0263	0.0228	0.0204
(15, 25, 24)	0.5672	0.4775	0.5370	0.4338	0.5594	0.3695
	0.2943	0.0223	0.0262	0.0256	0.0221	0.0129
(20, 25, 24)	0.5026	0.5138	0.6133	0.4522	0.6028	0.3905
	0.2026	0.0125	0.0177	0.0195	0.0108	0.0112
(15, 25, 25)	0.4791	0.6072	0.6977	0.5434	0.7088	0.4632
	0.1959	0.0119	0.0112	0.0114	0.0112	0.0111
(20, 25, 25)	0.5873	0.4276	0.4968	0.3815	0.4992	0.3275
	0.1013	0.0118	0.0105	0.0110	0.0109	0.0106
(15, 50, 40)	0.5556	0.5401	0.5831	0.5048	0.5998	0.4592
	0.1001	0.0095	0.0096	0.0089	0.0064	0.0054
(20, 50, 40)	0.6578	0.5153	0.5547	0.4828	0.5725	0.4368
	0.0197	0.0057	0.0057	0.0083	0.0056	0.0047
(15, 50, 15)	0.7015	0.4334	0.5093	0.3840	0.5453	0.2912
	0.0121	0.0045	0.0037	0.0035	0.0031	0.0029
(20, 50, 20)	0.5328	0.5712	0.6732	0.5047	0.6874	0.4165
	0.0108	0.0035	0.0022	0.0023	0.0015	0.0012

. The corresponding 95% ACIs are provided in

Table 17. 95% CI and 95% CRI for $\mu$.

Classical Estimates			Bayes Estimates
Lower	Upper	Length	Lower	Upper	Length
1.05455	1.68031	0.62576	1.26680	1.75768	0.49088
1.10535	1.85124	0.74590	1.05817	1.73516	0.67699
1.10498	1.68590	0.58093	1.82021	2.27832	0.45811
1.05727	1.63845	0.58118	1.30727	1.74375	0.43648
1.02840	1.82119	0.79279	0.99725	1.74940	0.75215
0.94039	1.64029	0.69990	1.22607	1.81203	0.58596
0.91815	1.56806	0.64991	1.58292	2.12813	0.54521
1.07262	1.85746	0.78485	0.9088	1.53141	0.62261
1.09453	1.75775	0.66322	1.20150	1.84221	0.64071
1.26685	2.00227	0.73542	0.96548	1.77265	0.80717
1.20589	2.11093	0.90504	1.03232	1.69677	0.66446
0.96276	1.75368	0.79092	1.29477	2.08252	0.79775

,

Table 18. 95% CI and 95% CRI for $\upsilon$.

Classical Estimates			Bayes Estimates
Lower	Upper	Length	Lower	Upper	Length
0.81404	1.3748	0.56076	1.09402	1.09443	0.00041
0.98474	1.66114	0.6764	1.32243	1.32287	0.00044
0.92909	1.55874	0.62965	1.24402	1.24447	0.00045
1.01833	1.7084	0.69007	1.36371	1.36442	0.00071
0.88031	1.49216	0.61184	1.18624	1.18692	0.00068
0.87163	1.47839	0.60676	1.17497	1.1753	0.00033
0.89322	1.51494	0.62172	1.2039	1.2043	0.00041
0.87625	1.48406	0.60781	1.17961	1.18003	0.00042
0.96515	1.62099	0.65584	1.2929	1.29345	0.00055
0.98020	1.64670	0.66649	1.31343	1.31391	0.00048
0.75273	1.30371	0.55098	1.02806	1.02861	0.00055
0.88715	1.51465	0.62750	1.20094	1.20186	0.00092

,

Table 19. 95% CI and 95% CRI for $S\left(t\right)$.

Classical Estimates			Bayes Estimates
Lower	Upper	Length	Lower	Upper	Length
0.27291	0.49392	0.22101	0.34351	0.51855	0.17505
0.27695	0.47477	0.19782	0.24173	0.39856	0.15684
0.31092	0.51801	0.20709	0.33554	0.49283	0.15729
0.23964	0.47564	0.23600	0.30291	0.49539	0.19247
0.26394	0.49604	0.23210	0.30598	0.48429	0.17831
0.28705	0.51502	0.22797	0.24077	0.45579	0.21502
0.23134	0.4601	0.22876	0.35377	0.51184	0.15807
0.27614	0.4886	0.21246	0.30385	0.47616	0.17231
0.24139	0.44906	0.20767	0.32088	0.48993	0.16905
0.16464	0.39052	0.22588	0.27195	0.48399	0.21204
0.16464	0.39052	0.22588	0.27195	0.48399	0.21204
0.25009	0.49863	0.24853	0.24804	0.47599	0.22796

and

Table 20. 95% CI and 95% CRI for $h\left(t\right)$.

Classical Estimates			Bayes Estimates
Lower	Upper	Length	Lower	Upper	Length
0.38736	0.7119	0.32454	0.32275	0.75689	0.43414
0.39102	0.77197	0.38095	0.29916	0.71398	0.41482
0.39702	0.69462	0.29761	0.51484	1.00161	0.48677
0.36318	0.6559	0.29271	0.35245	0.70678	0.35432
0.3622	0.77227	0.41007	0.27996	0.73356	0.4536
0.32418	0.68117	0.35699	0.3007	0.78593	0.48522
0.31449	0.64383	0.32935	0.35703	0.92476	0.56773
0.38337	0.79129	0.40793	0.25188	0.65135	0.39946
0.38602	0.72517	0.33915	0.35692	0.76554	0.40862
0.4649	0.85085	0.38595	0.33929	0.72751	0.38821
0.45884	0.94424	0.4854	0.22287	0.72245	0.49957
0.32906	0.73668	0.40762	0.31868	0.90105	0.58237

. Bayesian estimates were obtained using three loss functions (SE, LINEX, and GE) under an informative prior, with hyperparameters $a_i=2.5$ and $b_i=3.5$ for $i=1,2$. Table 13, Table 14, Table 15 and Table 16 display the Bayesian estimates and the associated MSEs. The 95% CRIs are listed in Table 17, Table 18, Table 19 and Table 20.

Figure 12, Figure 13, Figure 14 and Figure 15 present the MCMC trace plots of the posterior samples for the unknown parameters, as well as for the survival function $S\left(t\right)$ and hazard function $h\left(t\right)$. These plots are used to evaluate the convergence and mixing behavior of the MCMC algorithm.

7. Conclusions

The methods developed in this paper are directly relevant to the Special Issue theme, particularly in the context of data-driven modeling under censoring and uncertainty using real-life data. This study investigated the estimation of the parameters, reliability function, and hazard function of the G-IID under UHCS. Both classical and Bayesian estimation methods were considered. Bayesian estimates were obtained under three different loss functions: SE, LINEX, and GE. A Monte Carlo simulation study was conducted to evaluate and compare the performance of the proposed estimators.

Several important conclusions can be drawn from the findings:

• Bayesian estimation methods produce significantly narrower CRIs compared to the classical asymptotic confidence intervals, indicating that Bayesian techniques provide higher precision in parameter estimation.
• Bayesian estimates based on the GE loss function consistently achieve the lowest MSE values when compared to those based on SE and LINEX loss functions. This demonstrates the superior performance of the GE-based Bayesian approach, particularly in terms of efficiency and robustness.
• Increasing the values of the censoring parameters r and k leads to a noticeable decrease in the MSEs for the estimates of $\mu$, $\upsilon$, $S\left(t\right)$, and $h\left(t\right)$. This highlights the importance of larger observed sample sizes in improving estimation accuracy.
• Bayesian estimators outperform maximum likelihood estimators across all scenarios analyzed. They produce smaller MSEs for both the parameters and the reliability-related functions. This confirms the advantages of incorporating prior information and a loss function framework in the estimation process.
• The CRIs resulting from Bayesian methods are not only narrower but also more stable across different censoring schemes. This reliability enhances the confidence in Bayesian interval estimation, especially in highly censored or small sample environments.
• The simulation study revealed that while both classical and Bayesian methods benefit from larger sample sizes, Bayesian methods show a stronger and more consistent improvement in estimation accuracy.
• Under extreme censoring conditions, where traditional methods tend to suffer from large variances and unstable behavior, Bayesian estimation maintains reliable and consistent performance, reinforcing its applicability in practical situations involving incomplete data.
• Among the three loss functions considered, the GE loss function proved to be the most efficient, yielding the smallest MSEs in almost all scenarios.
• Overall, the simulation results strongly support the use of Bayesian estimation techniques, particularly under the GE loss function, when analyzing lifetime data modeled by the G-IID under unified hybrid censoring schemes.

The estimated reliability function $S\left(t\right)$ and hazard function $h\left(t\right)$ provide critical insights into the dynamics of COVID-19 mortality. Understanding these functions can help public health officials and researchers make informed decisions regarding interventions and resource allocation. The reliability function $S\left(t\right)$ represents the probability that an individual survives beyond time t. In the context of COVID-19, it indicates the likelihood of survival for patients diagnosed with the virus over time.

Epidemiological Meaning: A higher $S\left(t\right)$ value at a given time indicates better survival rates, suggesting effective treatment protocols or lower virulence of the virus.

Monitoring changes in $S\left(t\right)$ over time can help assess the impact of public health measures, such as vaccination campaigns or social distancing, on mortality rates.

If $S\left(t\right)$ decreases significantly, it may signal the emergence of more severe variants or the need for updated treatment strategies.

The hazard function $h\left(t\right)$ quantifies the instantaneous risk of death at time t given that the individual has survived up to that time. It reflects the rate at which deaths occur among the population at risk.

Epidemiological Meaning:

A higher $h\left(t\right)$ indicates an increased risk of mortality, which can inform healthcare providers about the urgency of care needed for patients at different stages of the disease.

Analyzing $h\left(t\right)$ can help identify critical time points when patients are most vulnerable, guiding the timing of interventions.

Changes in $h\left(t\right)$ can also reflect the effectiveness of public health responses, such as the introduction of new treatments or changes in healthcare capacity.