https://orcid.org/0000-0002-0126-2868
Figures
Abstract
This paper examines the estimation of model parameters, reliability, and hazard rate functions of the inverted exponentiated Rayleigh distribution under progressive hybrid Type-I censoring. Three estimation methodologies, maximum likelihood, maximum product of spacing, and Bayesian approaches, are explored. The classical perspective employs maximum likelihood and maximum product of spacing approaches for estimating unknown parameters, reliability, hazard rate functions, and computing approximate confidence intervals. Bayesian estimation is formulated using the squared-error and LINEX loss functions, predicated on independent gamma priors. Owing to the complex nature of the joint posterior distribution, Bayes estimates are evaluated by generating samples from the whole conditional distributions via Markov Chain Monte Carlo methods. The highest posterior density credible intervals are also established for each unknown parameter, reliability, and hazard rate functions. The efficacy of the proposed strategies is evaluated through a simulated study. To assess the efficacy of the estimation techniques, a comprehensive simulation study is conducted, encompassing various scenarios with diverse sample sizes and progressive censoring schemes. Furthermore, the practical applicability of the proposed methods is demonstrated through the analysis of real-world datasets taken from the medical field. This data represents the relief time (in hours) of arthritis patients receiving a fixed dose of this drug. Numerical investigations reveal that Bayes estimates employing the LINEX loss function exhibit superior performance compared to other estimation methods, underscoring their preference due to heightened accuracy and robustness.
Citation: Nassr SG, Abo-Kasem O, Khashab RH, Alshawarbeh E, Alshqaq SS, Elharoun NM (2025) Reliability analysis of inverted exponentiated Rayleigh parameters via progressive hybrid censoring data with applications in medical data. PLoS One 20(12): e0336169. https://doi.org/10.1371/journal.pone.0336169
Editor: Inés P. Mariño, Universidad Rey Juan Carlos, SPAIN
Received: March 14, 2025; Accepted: October 21, 2025; Published: December 30, 2025
Copyright: © 2025 Nassr et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: The author(s) received no specific funding for this work.
Competing interests: We have no conflicts of interest to disclose.
1 Introduction
Researchers frequently do reliability and life testing tests in diverse practical investigations, including survival analysis, clinical trials, and industrial or mechanical applications. These studies collect observable data to infer unknown values of interest, like failure rates, quantiles, and reliability characteristics. The efficacy of these conclusions depends on the information included in the observed data. Nonetheless, the results observed in these tests are censored in numerous instances. censoring transpires when comprehensive information about the event of interest is inaccessible. Data censoring occurs for various reasons, including the predetermined duration of trials or the non-occurrence of specific events at the time of analysis. Numerous censoring approaches have been developed in literature to analyze suppressed data appropriately. Type I and Type II censoring systems are the most employed procedures. Type I censoring occurs when a life testing experiment is performed for a specified duration, referred to as T. In the experiment, a specific quantity of items, referred to as n, undergoes examination. No additional observations are documented for any object once the experiment reaches the predetermined time point T. Type I censoring permits researchers to examine the recorded failure times of items until time T. This allows them to deduce the pertinent unknown values and estimate failure probabilities, quantiles, reliability characteristics, and other associated metrics from the available censored data.
Type II censoring continues the experiment until m failures (less than or equal to n) are observed. However, type I censoring may conclude the experiment before gathering enough failure observations, while type II censoring may prolong the experiment. Type I hybrid censoring, suggested by Epstein [1], addresses these restrictions by combining type I and type II censoring. This hybrid technique balances managing experimental time and collecting enough failure observations. In type I hybrid censoring, the experiment ends at a random time T0, which is the minimum of the m-th failure time (Xm) and a pre-specified time T. The experiment stops when m failures are seen or T is reached, whichever comes first. However, Childs et al. [2] proposed type II hybrid censoring, where the experiment ceases when either (m) failure times are observed or T is reached. Here, is the experiment’s end time. These hybrid censoring schemes randomly generate failure observations. Existing censoring schemes, including type I and type II hybrid censoring, do not allow live units to be removed from a life test at any point other than the final termination point.
In certain experimental scenarios, there arises a necessity to remove units throughout the duration of the test. Balakrishnan and Aggarwala [3] have highlighted the benefits of such flexibility in specific contexts. Allowing for unit removal at points other than the final termination enhances experimental efficiency while ensuring observation of extreme lifetimes. This flexibility proves advantageous when some surviving units can be repurposed or when practical constraints necessitate early removal, such as accidents or loss of contact with subjects. Progressive censoring emerges as a viable solution in situations where such unpredictability occurs. It permits the removal of surviving units before the test’s conclusion, thus accommodating unforeseen events during the experiment. Kundu and Joarder [4] and Childs et al. [5] have integrated type I hybrid censoring and progressive censoring into the progressive hybrid type I censoring scheme (PHT-ICS). This scheme combines the merits of both censoring approaches, aiming to optimize experimental design and data collection processes.
The PHT-ICS censoring system is , where m is the predefined number of failures and n is the total number of units initially placed on the life test. This scheme’s main feature is that some surviving units can be removed before the trial ends. Take a life test with n test units and a progressive censoring method
before the experiment. A predetermined time point T is also set in advance. Upon the first failure X1:m:n, R1 surviving units are randomly picked and removed from the experiment. During the second failure X2:m:n, R2 units are removed from the remaining
units. The procedure repeats, removing Ri units from the pool of
surviving units at each failure. The experiment continues until reaching the termination point T*, which is the minimum of the predefined time T and the time of the m-th failure Xm:m:n. If the m-th failure happens before the predetermined time T, the observed failures in the life test are
. When the m-th failure occurs, the test is ended and the remaining units are eliminated. The number of units to be deleted is Rm, computed as
. However, if the time T is reached before the m-th failure (Xm:m:n>T), the observed life test failures are
(where j < m). At time T, the test ends and the remaining units are eliminated. At this point, the number of units to be deleted is estimated as
(see Fig 1).
Significant contributions to statistical inference regarding PHT-ICS have been made by various authors, including references [Kayal et al. [6], Park et al. [7]. For a comprehensive understanding of this topic, valuable resources can be found in the monograph by Balakrishnan and Cramer [8] and the review artile by Balakrishnan and Kundu [9]. These references provide up-to-date accounts and insights into the PHT-I censoring scheme. Additionally, for further inferential results and related information on this scheme and other schemes of importance in life testing, references such as [Hemmati and Khorram [10], Lin and Huang [11], Tomer and Panwar [12], Abu El Azm et al. [13], Nassr et al. [14], Yousef et al. [15], Nassar et al. [16], and Nassr et al. [17] are recommended. These sources offer additional useful findings and findings that can enhance one’s understanding of the PHT-ICS.
PHT-ICS offers several advantages over traditional censoring schemes by combining the fixed-time stopping rule of Type-I censoring, the failure-based stopping of Type-II, and the flexibility of progressive censoring. Its key advantages include:
- Flexibility: It allows the removal of surviving units at intermediate stages, improving efficiency and reducing experimental costs.
- Realism: It closely reflects real-world testing environments where data may be censored due to time constraints, early failures, or resource limitations.
- Improved estimation: It often yields more informative data than fixed censoring, especially in small or expensive samples.
Inverted distributions are increasingly used in statistical modeling due to their flexibility in capturing a wide range of hazard rate behaviors, including decreasing, increasing, and non-monotonic patterns. These distributions are particularly useful when the event of interest becomes less likely over a common scenario in many real-world systems. In engineering, for example, inverted distributions are applied in reliability analysis of electronic components where failure rates often decrease after initial use (known as the "infant mortality" phase). In the medical field, they are employed in modeling patient survival times under specific treatments, such as the time until recurrence of cancer after therapy, where the risk of relapse may decline over time. Economically, they serve to analyze the duration of unemployment spells, especially in situations where the probability of finding a job increases the longer an individual remains unemployed. These distributions also accommodate censored and incomplete data, making them well-suited for practical applications where full observation is not always possible. Their mathematical flexibility and practical relevance have positioned them as essential tools in modern survival and reliability analysis. Numerous authors have noted the significance and usefulness of inverted distributions in the fields of engineering, medicine, and economics [see, for example, Calabria and Pulcini [18], Sharma et al. [19], Abd AL-Fattah et al. [20], Tahir et al. [21], Hassan and Nassr [22], Hassan and Nassr [23], Nassr et al. [24], Al Mutairi et al. [25], El-Saeed et al. [26], Nassr et al. [27], Abushal et al. [28], Al Mutairi et al [29] and Alotaibi et al. [30].
The inverted exponentiated Rayleigh (IER) distribution, introduced by Ghitany et al. [31], has recently gained notable attention in the literature for its ability to model a wide variety of hazard rate behaviors, especially decreasing and non-monotonic patterns. This makes it highly suitable for modeling lifetime data in reliability engineering, medical survival studies, and economic duration analysis. He suggested IER distribution with the probability density function (PDF), cumulative distribution function (CDF), reliability function (RF), hazard rate function (HRF), and cumulative hazard function (CH) as follows:
and
where; and
are the shape and the scale parameters respectively.
As shown in Fig 2, the IER model functions demonstrate varying behavior according to parameter values, with panel (a) presenting the PDF, panel (b) the CDF, panel (c) the reliability function R(x), and panel (d) the hazard function H(x), illustrating the statistical properties of the model.
Panels (a)–(d) correspond to these functions, respectively.
The choice of the IER distribution in this study is driven by its high flexibility in modeling diverse hazard rate behaviors, including increasing, decreasing, and bathtub-shaped functions. These characteristics make it a powerful alternative in reliability and survival analysis, particularly under censoring schemes. Moreover, recent comparative studies have demonstrated its superior fitting ability over classical models when applied to complex lifetime data. Many scholars have studied the IER distribution’s theories and applications. Rastogi and Tripathi [32] used maximum likelihood and Bayesian estimation for point and interval estimations under censored data for this distribution. Kayal et al. [33] also studied hybrid censoring scenarios for point and interval predictions for one and two samples, respectively, for the IER distribution. Kohansal [34] used Gibbs sampling to build the Bayesian estimator for the IER distribution’s stress-strength reliability. Gao et al. [35] introduced important inference methods for estimating the two unknown parameters of the IER distribution using progressive censored data. Fan and Gui [36] used maximum likelihood and Bayesian methods to estimate unknown parameters of the IER distribution under joint progressively type-II censoring. Maurya et al. [37] discuss the estimation and prediction methods for an IER distribution under progressive first-failure censoring, using maximum likelihood and Bayesian approaches. Panahi and Moradi [38] examined maximum likelihood and Bayesian approaches for the IER distribution based on an adaptive progressive hybrid censoring scheme. Anwar et al. [39] examines stress-strength reliability estimation for the IER distribution using unified progressive hybrid censoring, employing maximum likelihood estimation via the stochastic EM algorithm and Bayesian approaches with Gibbs/Metropolis-Hastings sampling. Salem et al. [40] examine point and interval estimates for IER distribution under a general progressive Type-II censoring scheme, employing both maximum likelihood and Bayesian approaches. Recent studies have also explored the inverted exponentiated Rayleigh (IER) distribution under various censoring schemes.
For instance, Wang et al. [41] investigated estimation and prediction under a modified progressive hybrid censoring scheme, whereas our study adopts the standard progressive hybrid Type-I censoring scheme (PHT-ICS). Hashem et al. [42] applied empirical Bayes estimation to progressively hybrid censored medical data, but assumed the scale parameter to be known. In contrast, we jointly estimate both shape and scale parameters using full likelihood-based and Bayesian methods (under squared-error and LINEX loss functions), with a comprehensive assessment that includes simulation, model diagnostics, and real-data validation. These distinctions broaden the applicability and rigor of the proposed framework.
The remaining sections of the paper are structured in the following manner: The maximum likelihood estimators (MLEs) and asymptotic confidence intervals (ACIs) for the unknown parameters and H(x) are examined in Sect 2. The point estimators and ACIs using the maximum product of spacing estimation (MPSE) are derived in Sect 3. Sect 4 discusses the application of Bayesian estimating. Sect 5 contains a detailed presentation of the simulation technique and its findings. Sect 6 presents the study of an actual data set that pertains to the field of medicine. Sect 7 contains the concluding remarks.
2 Maximum likelihood estimation
Suppose a test employing the PHT-ICS involves n units sampled from the IER distribution. The observed data may fall into one of two possible scenarios regarding the censoring scheme:
Case I: , if Xm:m:n<T,
Case II: , if
.
Then, the likelihood function with PHT-ICS is
where; r = m, , RT = 0 in case I, while in case II, r = d, C = T,
and
, based on the observed data, the likelihood function can be expressed as:
Then, the log likelihood function, denoted by can be written as
The first partial derivative for the unknown parameters as follow:-
From Eq. (7) the maximum likelihood estimates of is expressed by
where; .
Consequently, by substituting into Eq. (8), the system equation reduced to one nonlinear equation as follows:
It is seen that there is no analytical solution for Eq. (10). As a result, it is impossible to obtain the MLEs in their explicit form. Some numerical methods, such the Newton-Raphson method, can be used to get the necessary estimations to get around this problem; for more information about the ML approach. After obtaining the MLE
, the MLE
can be calculated using Eq. (9) by replacing
with their MLE. Following the derivation of
and
, the MLEs of R(x) and H(x) may be obtained from Eqs. (3) and (4), respectively. This is accomplished by utilizing the invariance property of the MLEs
and
, as described in the following manner:
and
To derive asymptotic confidence intervals (ACIs) for the unknown parameters , we rely on the asymptotic properties of the MLEs. According to large sample theory, the MLEs
follow a normal distribution with a mean of
and a variance-covariance matrix
. We utilize the asymptotic variance-covariance matrix
to estimate
, which is obtained by inverting the observed Fisher information matrix; for further details on Fisher information and related information measures, see Husseiny et al. [43], Barakat et al. [44], and Alawady et al. [45]. In this scenario, the asymptotic variance-covariance matrix is expressed as:
where the hat implies that the derivatives are evaluated at and
Utilizing the asymptotic normality of the MLEs, the ACIs of
can be constructed, respectively, as
and
,
where is the upper
the percentile points of the standard normal distribution.
It is necessary to estimate the variances of the estimators of R(x) and H(x) to construct the ACIs of both of these variables. A notable strategy used to approximate the variance of unknown parametric functions is the delta method, one of the most used and significant strategies. The approximated variances of the estimators of R(x) and H(x) can be derived in the following manner, respectively, by applying the delta method:
First, we must obtain and
as follows:
Therefore, by employing the confidence level , the two-sided ACIs for R(x) and H(x) are constructed in the following manner:
and
.
3 Maximum product of spacing estimation
The MPSEs are achieved by selecting parameter values that optimize the product of the distances between the CDF values at adjacent ordered points. The MPSEs demonstrate superior efficacy for small sample sizes compared to the MLEs, rendering the MPS approach increasingly attractive in reliability and life testing studies. Numerous authors have regarded the MPS technique for estimating the unknown parameters of lifespan models; refer to Basu et al. [46]. This section proposes the MPS approach to derive point estimates and ACIs of the IER distribution based on PHT-ICS samples. The observed data may conform to one of two potential scenarios about the censoring scheme:
case I: , if Xm:m:n<T,
case II: , if
.
However, we can write the MPSE function of the PHT-ICS as follows:
where B is the normalizing constant, and by convention r = m,
, RT = 0 in case I while case II, r = d, C = T,
and
. Then, from Eqs. (2) and (12), The MSPE, without the constant term, can be expressed as
Let denoted by the natural logarithm of Eq. (13) as
The MPSEs, represented by , can be obtained by maximizing the objective function of Eq. (14) concerning
. Simultaneously solving the following three nonlinear equations allows us to get the requisite estimators
where, ,
,
and
.
Similar to the MLEs, numerical methods should be employed to resolve Eqs. (15) and (16) in order to ascertain the MPSEs of . Cheng and Traylor [47] assert that the MPSEs exhibit the same invariance property as the MLEs. Consequently, we can get the MPSEs of R(x) and H(x) utilizing this condition as follows
and
It may be noted that the MPSEs are represented as cannot be obtained in closed phrases. Consequently, any numerical method may be utilized to resolve Eqs. (15) and (16). We may derive the ACIs for the unknown parameters by utilizing the asymptotic qualities of MPSEs, analogous to our approach with MLEs. The asymptotic distribution of the MPSEs
follows a normal distribution with a mean of
and an asymptotic variance-covariance matrix
. We examine
to approximate
, with the components of
detailed as follows
where, , and
After obtaining the estimated variances of , denoted by
and
which are the main diagonal elements of
, the
% ACIs of
can be obtained, respectively, as follows
, and
where is the upper
th percentile point of the standard normal distribution.
By approximating the estimated variances of the RF and HRF using the delta method, we can obtain the % ACIs of R(x) and H(x), respectively, as follows
and
,
where and
are evaluated at the MPSEs of
and
as defined in Eq. (11).
4 Bayesian estimation
This section will derive the Bayes estimate using the square error loss function, assuming a gamma prior for the unknown parameters of the IER distribution. The estimation will be based on PHT-ICS. Bayesian estimation is performed assuming that the random variables and
are independently distributed with a gamma prior distribution. Gamma priors are chosen for
and
due to their support on
, conjugacy with exponential family likelihoods, and the ability to encode prior mean/variance flexibly. They are a standard choice in Bayesian reliability (e.g., Nagy et al. [48]). The gamma prior distribution is characterised by known shape and scale parameters a, c, b, and d, and has a probability density function as
and
The joint prior density of unknown parameters is thus stated as
Combining Eqs. (6) and (20) to obtain the posterior density of take the following form:
where is the normalized constant and is provided by
Bayesian analysis relies on the loss function to assess overestimation and underestimation. Symmetric and asymmetric loss functions are frequent. The symmetric loss function weights overestimation and underestimation equally, whereas the asymmetric loss function weights them differently. We examine the popular symmetric SE loss function and the asymmetric LINEX loss function in this paper. The posterior mean is the Bayes estimator for SE loss functions. Under SE loss function, the posterior mean is the best estimate. However, the LINEX loss function lets you weight overestimation and underestimation differently, representing their relative relevance.
Varian [49] created the asymmetric LINEX loss function, which mixes exponential and linear growth on either side of zero. It seeks to capture overestimation and underestimation asymmetry. Based on the premise that the minimal loss occurs at , the LINEX loss function as
where, ,
is an estimate of
. The constant parameter
determines the shape of the loss function. If the parameter a is greater than 0 and the error
is considered.The LINEX loss function has an almost exponential behaviour for positive mistakes and an almost linear behaviour for negative errors. In such scenarios, over-estimations pose a more significant issue than under-estimations. If a<0, underestimation is prioritised over overestimation. When the absolute value of a is small, the loss function is almost symmetric and behaves similarly to the squared error (SE) loss function. The Bayes estimates of
under the LINEX loss function are provided by:
provided that exists, and is finite (see Zellner [50]).
Therefore, the Bayes estimators of the unknown parameters under PHT-ICS based on SE and LINEX loss functions, denoted by
and
respectively, can be obtained as follow:
and
where is the joint posterior distribution given by Eq. (21). Analytically calculating Bayes estimators using Eqs. (22) and (23) are impossible owing to integration difficulty. We recommend using Markov Chain Monte Carlo (MCMC) to provide Bayes estimates of unknown parameters and highest posterior density (HPD) credible intervals. To use the MCMC technique, we must determine the entire conditional distributions of the parameters
. Based on Eq. (21), we can determine the necessary full conditional distributions as follows:
The conditional posterior densities of are as follows:
and
By obtaining these full conditional distributions, we can employ MCMC methods, such as Gibbs sampling or Metropolis-Hasting’s algorithm, to iteratively sample from these distributions and obtain posterior samples of the parameters. The Bayes estimates and HPD credible intervals for unknown parameters can be estimated using these samples. We know the full conditional distributions for each parameter, but their forms are unknown, making direct sampling difficult. We create samples from these distributions using the Metropolis-Hastings (M-H) technique to address this.
The M-H sample proposal distribution is the normal distribution, which we use to calculate Bayesian estimates and HPD credible intervals. Using the entire conditional distributions in Eqs. (25) and (26), we can sketch the M-H algorithm steps:
Step 1. Initialization: Start with the first chain .
Step 2. Set the initial values of to
.
Step 3. Simulate using Eq. (25) and the Metropolis-Hastings (M-H) algorithm. Use a normal proposal distribution with mean
and variance
.
Step 4. Simulate using Eq. (26) and the M-H algorithm. Use a normal proposal distribution with mean
and variance
.
Step 5. Replace and
in Eqs. (3) and (4) with their respective
and
to compute R(j)(x) and H(j)(x) for x>0 .
Step 6. Increment j: Set j = j + 1.
Step 7. Repeat Steps 3 to 6 a total of N times to obtain , R(j)(x) and H(j)(x) for
.
The process entails sequentially sampling from the proposal distribution, determining the acceptance or rejection of the suggested samples based on the acceptance probability, and subsequently updating the parameter values. By iterating this process sufficiently, we can acquire a collection of posterior samples. From these samples, we may derive Bayesian estimates and establish the HPD credible intervals for the unknown values. To guarantee convergence and mitigate the impact of starting values, it is standard procedure to exclude the initial B samples produced by the MCMC chain. In this instance, we have acquired samples of , R(j)(x), and H(j)(x) for
, where N denotes the total number of iterations. By eliminating the early B samples, we acquire a resultant sample that relies on a suitably substantial B. The resulting sample can be regarded as an approximate posterior sample, suitable for calculating Bayes estimates and the HPD credible intervals.
Using this approximate posterior sample, we can estimate the Bayes estimates for the unknown parameters by, for example, taking the mean or median of the sample. Additionally, we can construct the HPD credible intervals by identifying the range of parameter values that contain a specified proportion of the sample, such as the central 95% or 99% of the values. By discarding the initial samples and utilizing the resulting approximate posterior sample, we can obtain reliable estimates and credible intervals for the parameters of interest.
We can calculate the SE loss function based on Bayes estimate of can be calculated as
Similarly, we can use the following formula to obtain the Bayes estimate of based on the LINEX loss function.
To calculate the HPD credible intervals of the parameters and
, denoted as
, we can follow these steps:
Step 1. Arrang the samples of , denoted as
, in ascending order:
.
Step 2. Define the index j* as follows: , such that:
Step 3. The HPD credible of
is then given by:
where denotes the highest number less than or equal to
.
5 Monte Carlo simulations
Extensive Monte Carlo simulations are conducted to assess the performance of the suggested point and interval estimators for the life parameters ,
, R(x), and H(x). To achieve this purpose, the true parameter values of (
,
) were set at (0.4,0.8), and the progressive hybrid Type-I censoring was reproduced 1000 times. At mission time t = 0.5, the survival time estimations of R(x) and H(x) are assessed, with their actual values being 0.9834913 and 0.2175712, respectively. Each unknown parameter is estimated using several values of (T,n,m), specifically:
,
, and m is defined as a failure percentage for each n, so that
and 80%. Additionally, for each pair (n,m), various censoring strategies are examined, specifically
To run the experiment according to PHT-ICS from the proposed IER distribution, do the following procedure:
- Step 1. Set the parameter values of
and
.
- Step 2. Generate an ordinary progressive Type-II censored sample as:
- Generate
independent observations of size m as
.
- For specific values of n, m and
, set
.
- Set
for
. Hence,
is a progressive Type-II censored sample of size m from U(0,1) distribution.
- Set
is generated progressive Type-II censored sample from
.
- Generate
- Step 3. Upon PHT-ICS, the sample data will consists of one of the following forms:
- If Xm<T, the experiment stops at Xm with observed failures
and censoring
, that is Case-I.
- If T<Xm, the experiment stops at T with observed failures
and censoring
, that is Case-II.
- If Xm<T, the experiment stops at Xm with observed failures
Upon the collection of 1,000 PHT-ICS, the maximum likelihood estimates (MLEs) and the corresponding 95% asymptotic confidence intervals (ACI) for ,
, R(x), and H(x) are computed using the R 4.2.2 programming language, following the installation of the ’maFxLik’ package developed by Henningsen and Toomet [51].To evaluate the impact of the proposed gamma priors on the Bayesian analysis, based on the prior mean and prior variance criteria established by Kundu [52], two distinct sets of informative hyperparameters
are employed, designated as Prior-1:
and Prior-2:
. The values of a, b, c, and d are selected to ensure that the prior average equals the expected value of the unknown target parameter. In this instance, we conducted all Bayesian assessments utilizing informative priors, as the absence of prior information regarding the unknown parameter(s) renders the maximum likelihood approach more favorable than the Bayesian approach, which is more computationally intensive; for further details, refer to Dey and Elshahhat [53]. Using the R 4.2.2 programming language and the ’coda’ package recommended by Plummer et al. [54], we produce 12,000 MCMC samples, with the initial 2,000 iterations designated as burn-in. Consequently, the remaining 10,000 MCMC samples are employed to compute the Bayes point estimates for the SE and LINEX loss functions (for
), along with the 95% HPD credible interval estimates. The average point estimates (APEs) of
are numerically represented as
where is the estimate of
at ith sample.
Comparison between point estimates of is conducted based on their root mean squared-errors (RMSEs) and mean relative absolute biases (MRABs) accordingly
and
On the other hand, the comparison between interval estimates of is made based on their average confidence lengths (ACLs) and coverage percentages (CPs) as
and
respectively, where is the indicator function,
denote the (lower,upper) bounds of
asymptotic (or Bayes) interval estimate of
. Clearly, in a similar fashion, the APEs, RMSEs, MRABs, ACLs, and CPs of
, R(x), and H(x) can be easily obtained. All simulation results of
,
, R(x) and H(x) are reported in Tables 1–6.
From Tables 1–6, in terms of the lowest RMSE, MRAB, and ACL values as well as the highest CP values, the following observations are drawn:
- All suggested point/interval estimates of
,
, R(x) and H(x) have better performances; it is our general note.
- Among point estimators, Bayes-Linex (q = + 2) consistently achieves the lowest RMSE and MRAB, followed by Bayes-SE, Bayes-Linex (q = −2), with MPS either outperforming or matching MLE.
- As n or m increases-or as
decreases-estimates of
,
, R(x), and H(x) improve markedly, confirming the consistency of the proposed methods.
- As T increases, in most cases, the simulated RMSE, MRAB, and ACL values of
,
, R(x), and H(x) decrease while their CPs increase.
- While no single scheme consistently dominates across all criteria, Scheme-1 frequently provides more accurate estimates in a broad range of scenarios.
- Comparing the point estimation methods of
,
, R(x) or H(x), in terms of the lowest RMSEs and MRABs, the Bayes MCMC estimates performed better against the LINEX loss than the SE loss and both are more favorable compared to the classical estimates. It is an anticipated result due to the Bayes MCMC estimates having included additional prior information.
- Interval estimation hierarchy: HPD < ACI-NA, with MPS-based ACI-NA yielding shorter ACLs than MLE-based; overall, Bayesian intervals (HPD) outperform frequentist ones thanks to the gamma prior.
- Comparing the priors 1 and 2, the Bayesian (point/interval) estimates using Prior-2 provide better results than Prior-1 for all unknown parameters. This result is due to the fact that the variance of Prior-2 is lower than the variance of Prior-1.
- As a result, the Bayes MCMC estimation method is recommended to estimate the parameters or its reliability characteristics of the inverted exponentiated Rayleigh distribution in the presence of data obtained from PHT-ICS.
6 Real data analysis
To illustrate the adaptability and usefulness of the proposed estimation methodologies to a real-life phenomenon, in this section, we shall present the analysis of a data set taken from the medicine area. This data, reported by Wingo [55] and reanalyzed by Soliman et al. [56], represents a relief time (in hours) for fifty arthritic patients receiving a fixed dosage of this medication. In Table 7, for computational convenience, each relief time point is divided by ten.
Before proceeding further, we need to check the validity of the IER distribution for arthritic patients data. Therefore, the Kolmogorov–Smirnov (KS) statistic (along its P-value) is calculated. For this purpose, using Table 7, the MLEs and
of
and
, respectively, must be calculated first. However, the MLEs (along their standard errors (SEs)) of
and
are 55.334(8.3398) and 3.8129(0.9978), respectively, while the K-S(P-value) is 0.127(0.394). It shows that the IER distribution fits the arthritic patients data quite well.
To examine the existence and uniqueness of the computed MLEs and
, based on the arthritic patients data set, the contour plot of the joint log-likelihood function for various choices of
and
is plotted and depicted in Fig 3(a). It shows that the MLEs
and
exist and are unique. It is also evident, to carried out any upcoming evaluations, that the most useful starting values of
and
are close to 55.334 and 3.8129, respectively. Additionally, the plot of estimated/empirical reliability function of the IER lifetime model is displayed in Fig 3(b). It also supports the numerical goodness-of-fit findings.
From the complete arthritic patients data, by taking m = 25 with different choices of R and T, four different artificial PHT-ICS are generated and reported in Table 8. For brevity, is used as
.
For each censored sample presented in Table 8, the maximum likelihood estimates (together with their standard errors) and the corresponding 95% ACI-NA/ACI-NL estimates (with their interval lengths) of ,
, R(x), and H(x) are calculated. Due to the absence of prior information regarding the unknown IER parameters
and
from the dataset of arthritic patients, the Bayes estimates (together with their standard errors) and the HPD interval estimates (with their interval lengths) are assessed using inappropriate gamma priors. In numerical logic, all unspecified hyperparameters a, b, c, and d are assigned a value of 0.001. The MCMC sampler is executed 50,000 times, with the initial 10,000 iterations disregarded as burn-in, to derive the Bayesian point and interval estimates. Tables 9 and 10 exhibit the computed point and interval estimates of
,
, R(x), and H(x) (at t = 5), respectively. Table 9 clearly indicates that the MCMC estimates of
,
, R(x), and H(x) exhibit superior performance for the minimal standard errors and interval lengths. The point (or interval) estimates derived from the maximum likelihood and Bayesian estimation approaches are notably similar.
To assess the convergence of MCMC procedure, trace plots based on 40,000 MCMC values of ,
, R(x) and H(x) are plotted in Fig 4. Furthermore, based on the same 40,000 MCMC values, the marginal PDFs with their histograms using the Gaussian kernel of
,
, R(x) and H(x) are plotted in Fig 4. For distinguish, in each trace plot, the sample mean (has soled (-) line), two bounds of 95% BCI (has dashed (- - -) line) and two bounds of 95% HPD interval (has dotted (
) line). Also, in each histogram plot, the sample mean is referred by vertical dotted-dashed line. It is clear that the MCMC algorithm converges satisfactorily and shows that ignoring the first 10,000 samples is an appropriate size to erase the effect of the initial guesses. It is observed, from Fig 5, that the generated posteriors of
,
, R(x) and H(x) for
are fairly-symmetrical. In addition, for all generated samples listed in Table 8, different useful statistics for MCMC draws of
,
, R(x) and H(x) after bun-in, namely: mean, mode, quartiles (
), standard deviation (SD) and skewness are also obtained and reported in Table 11. It supports the same findings reported in Tables 9 and 10.
Finally, based on the arthritic patients data, we can draw the decision that the proposed estimation methodologies provide a good explanation of the IER lifetime model when a sample generated from the PHT-ICS.
7 Concluding remarks
This study investigates the utilization of ML, MPS and Bayesian inference in the IER distribution through a PHT-ICS. Numerical methods are employed to obtain the ML and MPS estimates of the unknown parameters, reliability and hazard rate function. Utilizing the asymptotic properties of the ML estimates, we derive the approximate confidence intervals for the unknown parameters, as well as the reliability and hazard rate functions. Alternatively, we evaluate Bayesian estimation using two loss functions: the squared error and LINEX loss functions. We obtain estimates with the MCMC methodology. Concurrently, we compute the Bayes credible intervals with the highest posterior density for the different unknown parameters. We did simulation study to evaluate the efficacy of various estimations. This study’s simulation part computes the average root mean square errors and mean relative absolute biases to evaluate the precision of the point estimations. The average interval lengths and coverage probability are examined to assess the dependability of the interval estimates. The simulation results indicate that Bayesian estimation, with informative priors, is more credible than ML and MPS estimates across all scenarios. The Bayesian estimates employing the LINEX loss function surpass all alternative estimates. Moreover, the highest posterior density Bayes credible intervals demonstrate the most concise average interval lengths and superior coverage probability relative to the approximation confidence intervals. The genuine dataset was acquired from the medical domain, specifically from a cohort of fifty arthritis patients who received a uniform dosage of this medication. The data was studied to demonstrate the viability of the several methodologies assessed. Future study may necessitate the expansion of the proposed approaches to include the competing risk model or accelerated life testing. Future research may concentrate on, among other aspects, (1) investigating the estimation of entropy measures within a competing risks model; (2) addressing estimation challenges of entropy measures in accelerated life tests; and (3) exploring alternative estimation techniques beyond maximum likelihood, including weighted least squares, and least squares.
References
- 1. Epstein B. Truncated life tests in the exponential case. Ann Math Statist. 1954;25(3):555–64.
- 2. Childs A, Chandrasekar B, Balakrishnan N, Kundu D. Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Ann Inst Stat Math. 2003;55(2):319–30.
- 3.
Balakrishnan N, Aggarwala R. Progressive censoring: Theory, methods, and applications. Springer Science & Business Media; 2000.
- 4. Kundu D, Joarder A. Analysis of Type-II progressively hybrid censored data. Computat Stat Data Anal. 2006;50(10):2509–28.
- 5.
Childs A, Chandrasekar B, Balakrishnan N. Exact likelihood inference for an exponential parameter under progressive hybrid censoring schemes. Statistical Models and Methods for Biomedical and Technical Systems. Birkhäuser Boston; 2008. p. 319–30. https://doi.org/10.1007/978-0-8176-4619-6_23
- 6. Kayal T, Tripathi YM, Rastogi MK, Asgharzadeh A. Inference for Burr XII distribution under Type I progressive hybrid censoring. Commun Stat – Simul Computat. 2017;46(9):7447–65.
- 7. Park S, Balakrishnan N, Kim SW. Fisher information in progressive hybrid censoring schemes. Statistics. 2011;45(6):623–31.
- 8.
Balakrishnan N, Cramer E. Art of progressive censoring. Springer; 2014.
- 9. Balakrishnan N, Kundu D. Hybrid censoring: Models, inferential results and applications. Computat Stat Data Anal. 2013;57(1):166–209.
- 10. Hemmati F, Khorram E. Statistical analysis of the log-normal distribution under type-II progressive hybrid censoring schemes. Commun Stat – Simul Computat. 2013;42(1):52–75.
- 11. Lin C-T, Huang Y-L. On progressive hybrid censored exponential distribution. J Stat Computat Simul. 2012;82(5):689–709.
- 12. Tomer SK, Panwar MS. Estimation procedures for Maxwell distribution under type-I progressive hybrid censoring scheme. J Stat Computat Simul. 2013;85(2):339–56.
- 13. Azm WSAE, Aldallal R, Aljohani HM, Nassr SG. Estimations of competing lifetime data from inverse Weibull distribution under adaptive progressively hybrid censored. Math Biosci Eng. 2022;19(6):6252–75. pmid:35603400
- 14. Nassr SG, Almetwally EM, El Azm WSA. Statistical inference for the extended weibull distribution based on adaptive type-II progressive hybrid censored competing risks data. Thailand Statistician. 2021;19(3):547–64.
- 15. Yousef MM, Alsultan R, Nassr SG. Parametric inference on partially accelerated life testing for the inverted Kumaraswamy distribution based on Type-II progressive censoring data. Math Biosci Eng. 2023;20(2):1674–94. pmid:36899503
- 16. Nassar M, Nassr SG, Dey S. Analysis of burr type-XII distribution under step stress partially accelerated life tests with type-I and adaptive type-II progressively hybrid censoring schemes. Ann Data Sci. 2017;4(2):227–48.
- 17. Nassr SG, Taher TS, Alballa T, Elharoun NM. Reliability analysis of the Lindley distribution via unified hybrid censoring with applications in medical survival and biological lifetime data. MATH. 2025;10(6):14943–74.
- 18. Calabria R, Pulcini G. On the maximum likelihood and least-squares estimation in the inverse Weibull distribution. Statistica applicata. 1990;2(1):53–66.
- 19. Sharma VK, Singh SK, Singh U, Agiwal V. The inverse Lindley distribution: A stress-strength reliability model with application to head and neck cancer data. J Ind Prod Eng. 2015;32(3):162–73.
- 20. Abd AL-Fattah A, El-Helbawy A, Al-Dayian GR. Inverted Kumaraswamy distribution: Properties and estimation. Pak J Stat. 2017;33(1).
- 21. Tahir MH, Cordeiro GM, Ali S, Dey S, Manzoor A. The inverted Nadarajah–Haghighi distribution: Estimation methods and applications. J Stat Computat Simul. 2018;88(14):2775–98.
- 22. Hassan AS, Nassr SG. Parameter estimation of an extended inverse power Lomax distribution with Type I right censored data. CSAM. 2021;28(2):99–118.
- 23. Hassan AS, Nassr SG. The inverse Weibull generator of distributions: Properties and applications. J Data Sci. 2021;16(4):723–42.
- 24. Nassr SG, Hassan AS, Alsultan R, El-Saeed AR. Acceptance sampling plans for the three-parameter inverted Topp-Leone model. Math Biosci Eng. 2022;19(12):13628–59. pmid:36654061
- 25. Al Mutairi A, Hassan AS, Alshqaq SS, Alsultan R, Gemeay AM, Nassr SG, et al. Inverse power Ramos–Louzada distribution with various classical estimation methods and modeling to engineering data. AIP Adv. 2023;13(9).
- 26. El-Saeed AR, Hassan AS, Elharoun NM, Al Mutairi A, Khashab RH, Nassr SG. A class of power inverted Topp-Leone distribution: Properties, different estimation methods & applications. J Radiat Res Appl Sci. 2023;16(4):100643.
- 27. Nassr SG, Hassan AS, Almetwally EM, Al Mutairi A, Khashab RH, ElHaroun NM. Statistical inference of the inverted exponentiated Lomax distribution using generalized order statistics with application to COVID-19. AIP Adv. 2023;13(10).
- 28. A. Abushal T, S. Hassan A, R. El-Saeed A, G. Nassr S. Power inverted Topp-Leone distribution in acceptance sampling plans. Comput Mater Contin. 2021;67(1):991–1011.
- 29. Al Mutairi A, Alrashidi A, Al-Sayed NT, Behairy SM, Elgarhy M, Nassr SG. Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution. Open Phys. 2023;21(1).
- 30. Alotaibi R, Almetwally EM, Rezk H. Optimal test plan of discrete alpha power inverse Weibull distribution under censored data. J Radiat Res Appl Sci. 2023;16(2):100573.
- 31. Ghitany ME, Tuan VK, Balakrishnan N. Likelihood estimation for a general class of inverse exponentiated distributions based on complete and progressively censored data. J Stat Computat Simul. 2012;84(1):96–106.
- 32. Rastogi MK, Tripathi YM. Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring. J Appl Stat. 2014;41(11):2375–405.
- 33. Kayal T, Tripathi YM, Rastogi MK. Estimation and prediction for an inverted exponentiated Rayleigh distribution under hybrid censoring. Commun Stat – Theory Meth. 2017;47(7):1615–40.
- 34. Kohansal A. Large estimation of the stress-strength reliability of progressively censored inverted exponentiated Rayleigh distributions. J Appl Math Stat Inform. 2017;13(1):49–76.
- 35. Gao S, Yu J, Gui W. Pivotal inference for the inverted exponentiated Rayleigh distribution based on progressive type-II censored data. Am J Math Manag Sci. 2020;39(4):315–28.
- 36. Fan J, Gui W. Statistical inference of inverted exponentiated Rayleigh distribution under joint progressively type-II censoring. Entropy (Basel). 2022;24(2):171. pmid:35205466
- 37. Maurya RK, Tripathi YM, Rastogi MK. Estimation and prediction for a progressively first-failure censored inverted exponentiated Rayleigh distribution. J Stat Theory Pract. 2019;13(3).
- 38. Panahi H, Moradi N. Estimation of the inverted exponentiated Rayleigh distribution based on adaptive type II progressive hybrid censored sample. J Computat Appl Math. 2020;364:112345.
- 39. Anwar S, Ahmad Lone S, Khan A, Almutlak S. Stress-strength reliability estimation for the inverted exponentiated Rayleigh distribution under unified progressive hybrid censoring with application. ERA. 2023;31(7):4011–33.
- 40. Salem MOAA, Mohamed MAE. Parametric inference for inverted exponentiated Rayleigh distribution under general progressive type II censoring scheme. J Bus Res. 2025;47(2):270–303.
- 41. Wang L, Wu S-J, Tripathi YM, Dey S. Estimation and prediction of modified progressive hybrid censored data from inverted exponentiated Rayleigh distribution. Qual Technol Quant Manag. 2023;21(4):502–24.
- 42. Hashem AF, Alyami SA, Yousef MM. Utilizing empirical Bayes estimation to assess reliability in inverted exponentiated Rayleigh distribution with progressive hybrid censored medical data. Axioms. 2023;12(9):872.
- 43. Husseiny IA, Barakat HM, Taher TS, Alawady MA. Fisher information in order statistics and their concomitants for Cambanis bivariate distribution. Math Slovaca. 2024;74(2):501–20.
- 44. Barakat HM, Alawady MA, Taher TS, Husseiny IA. Second-order concomitants of order statistics from bivariate Cambanis family: Some information measures–estimation. Commun Stat – Simul Computat. 2024;54(8):3195–221.
- 45. Alawady MA, Barakat HM, Taher TS, Husseiny IA. Measures of extropy for k-record values and their concomitants based on Cambanis family. J Stat Theory Pract. 2025;19(1).
- 46. Basu S, Singh SK, Singh U. Bayesian inference using product of spacings function for Progressive hybrid Type-I censoring scheme. Statistics. 2017;52(2):345–63.
- 47. Cheng RCH, Traylor L. Non-regular maximum likelihood problems. J R Stat Soc Ser B: Stat Methodol. 1995;57(1):3–24.
- 48. Nagy M, Barakat HM, Alawady MA, Husseiny IA, Alrasheedi AF, Taher TS, et al. Inference and other aspects for q-Weibull distribution via generalized order statistics with applications to medical datasets. MATH. 2024;9(4):8311–38.
- 49.
Varian HR. A Bayesian approach to real estate assessment. Studies in Bayesian econometrics and statistics in honor of Leonard J Savage; 1975.
- 50. Zellner A. Bayesian estimation and prediction using asymmetric loss functions. J Am Stat Assoc. 1986;81(394):446–51.
- 51. Henningsen A, Toomet O. maxLik: A package for maximum likelihood estimation in R. Comput Stat. 2010;26(3):443–58.
- 52. Kundu D. Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring. Technometrics. 2008;50(2):144–54.
- 53. Dey S, Elshahhat A. Analysis of Wilson-Hilferty distribution under progressive Type-II censoring. Qual Reliab Eng. 2022;38(7):3771–96.
- 54. Plummer M, Best N, Cowles K, Vines K. CODA: Convergence diagnosis and output analysis for MCMC. R News. 2006;6(1):7–11.
- 55. Wingo DR. Maximum likelihood methods for fitting the Burr type XII distribution to life test data. Biometrical J. 1983;25(1):77–84.
- 56. Soliman AA, Abd Ellah AH, Abou-Elheggag NA, Modhesh AA. Estimation from Burr type XII distribution using progressive first-failure censored data. J Stat Computat Simul. 2013;83(12):2270–90.