Bayesian Inference for Inverse Power Exponentiated Pareto Distribution Using Progressive Type-II Censoring with Application to Flood-Level Data Analysis

: Progressive type-II (Prog-II) censoring schemes are gaining traction in estimating the parameters, and reliability characteristics of lifetime distributions. The focus of this paper is to enhance the accuracy and reliability of such estimations for the inverse power exponentiated Pareto (IPEP) distribution, a flexible extension of the exponentiated Pareto distribution suitable for modeling engineering and medical data. We aim to develop novel statistical inference methods applicable under Prog-II censoring, leading to a deeper understanding of failure time behavior, improved decision-making, and enhanced overall model reliability. Our investigation employs both classical and Bayesian approaches. The classical technique involves constructing maximum likelihood estimators of the model parameters and their bootstrap covariance intervals. Using the Gibbs process constructed by the Metropolis–Hasting sampler technique, the Markov chain Monte Carlo method provides Bayesian estimates of the unknown parameters. In addition, an actual data analysis is carried out to examine the estimation process’s performance under this ideal scheme.


Introduction
In survival analysis and reliability studies, censoring occurs when it is not feasible to collect comprehensive data on the sample.When analyzing lifetime data in such cases, censored samples are used.Analyzing data from various life-testing experiments is receiving more attention.In order to analyze lifetime data, several censoring schemes have been introduced in the literature.Type-I and type-II censoring schemes are the two most widely used applications in reliability analysis.However, during the experiment, none of these censoring schemes permit the removal of experimental units.Progressive Type-I and Type-II censoring schemes permit the removal of experimental units while the experiment is being performed.In the last ten to twelve years, statistical literature studies have focused on progressive censoring schemes due to their flexibility.Type-I censoring establishes an end time for experiments regardless of the number of failures, while Type-II censoring terminates an experiment following a particular number of failures.Type-II censoring is a method that 'stops' an experiment once a pre-established number of failures has occurred, regardless of the time that has passed since the failure occurred.There is no way to remove live units from the experiment at any other time than the point where it ends, which is one of the scheme's defects.As a solution to this drawback, Cohen [1] introduced progressive censoring schemes.For reliability experiments, progressive censoring has a significant impact on setting up duration experiments.It is common in industrial experiments to immediately terminate experiments and limit the overall number of failures due to different causes (such as when expensive items have to be destroyed when it takes a long time to complete experiments).Progressive censoring techniques cut down on testing times and costs by permitting the removal of survival elements at various stages of the experiment.
In this study, three parameters for IPEP distribution were proposed by Khalifa et al. [2].The random variable, y, has an IPEP distribution, given that its pdf and cdf are as follows: where θ and η are the shape parameters and γ is the scale parameter.
The cdf is defined as follows: whereas the S(y) and h(y) functions for the IPEP distribution, respectively, are obtained by the following: Khalifa et al. [2] explained the IPEP model, a more accurate and flexible extension of the exponentiated Pareto distribution for fitting engineering and medical data, and introduced some statistical characteristics.Furthermore, this dataset, which explores a novel area of application, has been evaluated for weather phenomena (rainfall, floods, droughts, etc.); flood data provided by Dumonceaux and Antle [3] have been used for application.The data are components of a sequence of inundation layers designed to establish a connection between the observations and forecasts obtained from the river gauge and a graphical depiction of the areas affected by high water levels.The purpose of this paper is to investigate how to select models that represent river floods.Our focus is on methods that use the maximum annual flood series, as well as some features of the most commonly used parametric models in hydrology.Classical goodness-of-fit procedures have limitations, and we propose a simple method based on the "flood rate" in order to screen alternative models.
As a result, the remainder of this work is arranged as follows: In Section 2, Prog-II censoring is presented.In Section 3, the literature review is discussed.In Section 4, the MLEs of θ, η and γ, and the Fisher information matrix are formulated.In Section 5, Bootstrap CIs are introduced.In Section 6, for various loss functions, including SEL and LINEX functions, Bayes estimates for the previously mentioned parameters and reliability functions are additionally obtained.In Section 7, the M-H algorithm is introduced.In Section 8, the dataset is studied.In Section 9, a simulation study is performed.Lastly, the conclusion is provided in Section 10.

Progressive Type-II Censoring
A brief description of a Prog-II censored experiment is as follows.Select R 1 , . . ., R m , m non-negative integers so that Consider a trial where n identical items are tested.It is supposed that the n units' lifetime distributions have the same distribution function (F) and are i.d.d random variables.When the first failure occurs, let us say that x 1:m:n , R 1 surviving items are randomly selected to be eliminated.In the same way, the surviving items are eliminated at the time of the second failure, say x 2:m:n , R 2 , and so on.Ultimately, all of the remaining items are removed at the time of the m − th failure, let us say x m:m:n ; see Balakrishnan and Cramer [4].
As a result, for a Prog-II censoring scheme, (n, m, R 1 , . . ., R m ), we observe the following sample: x 1:m:n < x 2:m:n < . . .< x m:m:n .The joint pdf for x 1:m:n < x 2:m:n < . . .< x m:m:n is as follows: where C is a constant independent of the parameters and it is determined by ; R i is the censoring scheme, and F(x i:m:n ) and f (x i:m:n ) are the cdf and probability density functions of X 1:m::n , X 2:m:n , . . . ,X m:m:n , respectively, see Figure 1.

Literature Review
Many studies, such as Mait and Kaya [5], Meeker and Escobar [6], and Cai et al. [7], have explored Prog-II censoring with differed failure time breakdowns.Additionally, Panah and Asadi [8] and Mousa and Jaheen [9] have applied Prog-II censoring to estimate parameters from different lifetime distributions.Balakrishnan [10] investigated the characteristics under progressively censored samples and reported that a collection of several advances in the inferential procedures depend on types-I and -II censored samples, Almetwally et al. [11] studied the MLE and BE method to determine the parameters of the EGE distribution under progressive censoring.Buzaridah et al. [12] and Khalifa et al. [13] studied estimations of the distribution parameters under Prog-II censored data and Chen and Gui [14] presented statistical inference of the joint Prog-II censored generalized inverted exponential distribution.
This study seeks to fill this need by pursuing three objectives.Estimating parameters: We estimate the parameters, S(y) and h(y), of the IPEP using the standard MLE technique.Further, we examine ACI and two CIs based on bootstrap methods, concerning the model parameters.As far as we know, there is currently no study available on the estimation of model parameters or reliability features of the IPEP using Prog-II censoring schemes.Bayesian estimation (BE): We calculate the BEs for the unknown IPEP parameters, together with the corresponding S(y) and h(y) functions.The Bayesian methodology is utilized for both LFs, considering two commonly used LFs in Bayesian estimation, which are the SEL and the LINEX.To determine the unknown parameters' BEs, we utilize MCMC methods.Specifically, we apply the Gibbs process inside the M-H approach.We evaluate the effectiveness of the offered approaches by conducting a comprehensive simulation analysis, specifically focusing on their simulated MSE.Another criterion is employed to compare the 95% confidence intervals derived using the asymptotic MLE and CRI.
This comparison is based on a distinct criterion.To make a comparison, CP and ACLs are utilized.

Inference Procedures
In this part, we will develop the inference processes for the parameters and reliability function of the IPEP distribution using the scheme R = (R 1 , R 2 , . . ., R m ).

Maximum Likelihood Estimator
Let us suppose that n components are tested with corresponding lifetimes that are equally distributed using the cdf from (2) and the pdf from (1).If y 1 , y 2 , y 3 , . . ., y m are denoted as Prog-II censored samples of size m, then the likelihood function is provided by using the Prog-II censored data.
) is independent of the parameters, and f (y i:m:n |θ, η, γ) is the pdf of y in (1), while F(y i:m:n |θ, η, γ) is the cdf of y in (2).Consequently, the joint pd f is as follows: Following that, the natural logarithm of the likelihood function, without the normalization constant, is as follows: ℓ(y; θ, η, γ) = n ln(θ) + n ln(η) + n ln(γ) − (γ + 1) Using the log-likelihood (8) to construct the nonlinear equations, the MLEs of the parameters θ, η and γ are obtained as follows: and The estimates should be obtained using a numerical method like Newton-Raphson since ( 9)-( 11) cannot be solved analytically.Ahmed [15] provides a clear illustration of the algorithm.

Fisher's Information Matrix
The normal approximation of the MLEs can help construct the approximate CI and conduct hypothesis testing for the parameters θ, η , and γ.I n (θ, η, γ), and the corresponding 3 × 3 observed information matrix is given as follows: Considering that the MLE of φ = ( θ, η, γ) illustrates asymptotic normality, we may express it as φ ∼ N(ϕ, I −1 n ( φ)).As a result, the 100(1 − ϑ)% approximate CIs are defined by the following: where 0 < ϑ < 1 and z ϑ 2 represent the upper ϑ 2 percentiles of the standard normal, respectively.
So, a major diagonal of I −1 n ( φ) is formed by the elements var θ , var( η) and var( γ).It is also necessary to know the variance of the parameters θ, η , and γ, to clarify the ACI of S(t) and h(t) functions.See, Greene [16] for more information on using the delta method to approximately estimate the variance of Ŝ(t) and ĥ(t).Following this technique, it can determine the variance of Ŝ(t) and ĥ(t), respectively and var( ĥ(t)) = ▽ ĥ(t) T V ▽ ĥ(t) , where Ŝ(t) and ĥ(t) for θ, η , and γ are denoted by ▽ Ŝ(t) and ▽ ĥ(t).Following that, S(t) and h(t) can have a 100(1 − ϑ)% two-sided confidence estimate.

Bootstrap Confidence Intervals
The bootstrap techniques are more likely to give CIs that are more approximative.As a consequence, we present CIs based on bootstrap techniques.

Bayesian Estimation
Now, we derive BE under the assumption that the parameters are random.We describe parameter uncertainty by methods of a joint prior distribution that was constructed before the failure data were gathered.In this situation, the Bayesian method will be helpful since it enables us to use prior information when analyzing failure data.In Bayesian analysis, the parameters are updated using data to fit the unknown numbers as random variables.Compared to SEL and LI NEX S(t) and h(t), as well as some unknown parameters, are all estimated using the BE.We assume θ, η, and γ to be three independent parameters, following the gamma prior distributions, as follows: where the parameters are assumed to be unknown and the hyperparameters a i and b i , i = 1, 2, 3 are selected to represent the prior assumption.Additionally, the posterior distribution of the parameters, indicated by π * θ, η, γ | y ¯ , approximately the same, can be generated by combining the prior with the likelihood function in (7), and applying Bayes' theorem, which can be expressed as follows: The joint posterior density function from ( 15) is used to generate samples using the MCMC approach.Estimating the value of integrals is the major objective of the MCMC technique.Within M-H samplers, we perform the MCMC method using the Gibbs procedure.The joint posterior distribution is given as follows: For θ, η, and γ, the full conditionals are obtained as follows: and It should be observed that the integral calculations in ( 17)-( 19) cannot be resolved analytically.Therefore, samples use the MCMC approach, the joint posterior density function in (16).The plots shown in Figures 2-4 indicate that, despite the difficulty in directly sampling them using conventional methods, they are comparable to normal distributions.

Loss Function
In Bayesian analysis, loss function (LF) plays an essential role in reducing the risk connected to the estimator.We consider both symmetric and asymmetric LFs in this analysis.Compared to an asymmetric LF, which provides distinct Bayes estimators as the posterior distribution, a symmetric LF is straightforward and generally used.

Symmetric Bayes Estimation
To obtain the SEL for estimates, the parameters θ, η , and γ are as follows: and where

A Symmetric Bayes Estimation
The LI NEX can be used to derive the BE of θ, η and γ, as the following: where where where In order to obtain posterior distribution samples for computing the BE of θ, η and γ of IPEP(θ, η, γ), with distribution under Prog-II censored, we suggest using the MCMC approach.Additional Metropolis-in-Gibbs samplers and Gibbs sampling are essentially classes of MCMC techniques.

Metropolis-Hastings Algorithm
A general-purpose MCMC approach is the M-H algorithm, originally presented by Metropolis et al. [18] and extended by Hastings [19].The M-H algorithm can select random samples from the known target distribution, regardless of how complicated it is.We can produce samples from the posterior distribution for unknown parameters to estimate interval approximations and Bayesian point estimators.
(ii) We estimate the probabilities of approval (iii) From a Uniform (0, 1) distribution, we produce u 1 , u 2 and u 3 .

Application to Real Data and Simulation Study
To illustrate the practical applications of the provided techniques on estimation and prediction problems, we provide numerical outcomes for the estimation of IPEP model parameters using Prog-II censoring on actual datasets.The analysis involves comparing the performance of BE with MLE by analyzing both simulated and actual data.

Flood-Level Data
Using real datasets provided by Dumonceaux and Antle [12], the 20 flood levels represent a real dataset of Susquehanna River's maximum flood levels at Harrisburg, Pennsylvania, between 1890 and 1969.The data are as follows: Prog-II samples were generated from the complete dataset, with m = 10 and a censoring scheme of R = (5 (2) , 0 (18) ), as presented in Table 1.Prior to conducting the analysis, it is crucial to evaluate the adequacy of the presented model to represent the data.The K-S distance measures the difference between both the fitted and empirical distribution functions, yielding a K-S value of 0.084 and an associated p-value of 0.95.These findings indicate that the distribution of the IPEP provides a satisfactory fit to the given dataset.Figure 5 represents the empirical quantile function of IPEP distribution for the flood-level data.Point estimates for the parameters θ, η, and γ, along with S(t) and h(t) at time t = 0.4, were obtained using real data on Prog-II censoring schemes presented in Table 1.All these point estimates, including MLE, Boot-p , Boot-t, and BE under SEL and LI NEX, are presented in Table 2.The 95% confidence intervals (CIs) for the BE estimators and MLEs are reported in Table 3.It is well established that the LI NEX approximates the behavior of the SEL function due to its symmetrical nature for values of c near to zero.Figures 6-10 display the MCMC method's posterior density function plots as well as the trace plots of the unknown parameters θ, η, γ, and S(t) and h(t) functions.Furthermore, the MLE for the parameters θ, η, γ as well as the S(t) and h(t) are unique and represent the actual maximums, as shown in Figure 11.Furthermore, a comparison of the CIs for MLEs and BEs in Table 3 demonstrates that Bayes estimators possess narrower CIs compared to their MLEs for all three parameters (θ, η , and γ).For the parameters and reliability functions, Table 3 shows the 95% ACIs for MLE and CRIs for MCMC (BE). Figure 11 visualizes the profile's likelihood, illustrating the uniqueness and similarity of the maximum values with the MLE results.In Tables 2 and 3, it is seen that BEs outperform MLEs in the Prog-II censoring samples.Compared to the approximation CIs, the Bayes CRIs have the shortest confidence lengths.This study may effectively assist in the failure analysis of the airplane's air conditioning dataset since the distribution of IPED is appropriate to the applicable data.

Simulation Study
To produce the censored Prog-II illustration with initial values θ 0 = 1.2, η 0 = 1 and γ 0 = 0.5 from the IPEP distribution, a comparison of the various techniques used by the estimators of results θ, η, γ, S(t) and h(t), at t = 0.4, has been discussed in calculating their MSE for k = 1, 2, 3, 4, 5, t = 0.4 and , where M = 10,000 represents the number of generated samples.Another criterion is used to compare the 95% CI obtained using the asymptotic MLE and CRI, compared using a different criterion.For comparison, CP and ACL are used.The CP of an interval of confidence is the percentage of times that the initial significance value is found in the interval.Under the following data created from the IPEP distribution using the quantile function of y, Monte Carlo experiments were conducted, where y follows IPEP distribution and is obtained by using F IPEP (Y, θ, η, γ) = U as follows: The random variable Y = Q(U) is given by Equation (26), if U is a uniform variate on the unit interval (0, 1).The subsequent progressive approaches are generated as follows: Tables 4-8 show the results of the estimate parameters and their MSEs, while Table 9 displays the values of the ACL and CP of CIs.

Simulation Results
From the following observations, we note the following: 1.
According to Tables 4-8, as sample sizes increase, MSEs become smaller, and BEs have the smallest MSEs for the parameters θ, η, γ, S(t) and h(t).Therefore, when all variables are considered, BEs outperform MLEs.

2.
The estimates from Bayes for θ, η, γ, S(t) , and h(t) are better in that the MSEs are smaller.

3.
For smaller MSEs with c = −2.0 and 0.0001, the LI NEX estimates with c = 2.0 are greater.4.
The performance of Scheme I is greater than Schemes I I and I I I due to the smaller MSEs for the failure time intervals of fixed-value samples n and m.

5.
The results from the CRIs are more perfect than those from the ACIs for identified failures, approaches, and sample sizes, as shown in Table 9.

Conclusions
In the field of distribution theory, there is a continuous effort to generalize current distributions.The aim of this work is to develop more reliable and adaptable models that may be used in a variety of censoring contexts.Many different kinds of techniques are being investigated to attain this goal, as indicated by the enormous amount of literature.The reliability and efficiency of the distribution used in fitting the given data have a considerable impact on the subsequent analysis and empirical results.This work focuses on the problem of estimating unknown parameters in the formulation of an IPEP distribution using a Prog-II censoring strategy.Our strategy combines both classical and Bayesian approaches.We calculated approximation CIs and bootstrap CIs for the IPEP distribution's unknown parameters.Furthermore, we used MCMC with the MH technique to calculate BEs for both LFs, across their related HPD interval estimations.According to the simulation results, BEs with the LINEX loss function exceed all of the traditional estimates.We also determined the best censoring approach for life-testing experiments using three criteria schemes, which is an important consideration for reliability researchers.As an actual data application, we used the flood dataset for all estimations in our research study.Future studies might investigate statistics applied to the IPEP distribution.In future works, we will use Joint Prog-II censored data to estimate the IPEP distribution parameters and compare them to all other censored algorithms that we will use.gradient of Ŝ(Y) ▽ ĥ(y) gradient of ĥ(Y)

Figure 1 .
Figure 1.The Prog-II censoring scheme is represented schematically.

Figure 2 .
Figure 2.For θ, we display the conditional posterior density of MCMC.

Figure 3 .
Figure 3.For η, we display the conditional posterior density of MCMC.

Figure 4 .
Figure 4.For γ, we display the conditional posterior density of MCMC.

Figure 5 .
Figure 5.The IPEPD fitted to the real dataset.

Figure 6 .
Figure 6.For θ, we display the posterior density function and trace plots.

Figure 7 .
Figure 7.For η, we display the posterior density function and trace plots.

Figure 8 .
Figure 8.For γ, we display the posterior density function and trace plots.

Figure 9 .
Figure 9.For S(t), we display the posterior density function and trace plots.

Figure 10 .
Figure 10.For h(t), we display the posterior density function and trace plots.

Figure 11 .
Figure 11.The profile log-likelihood plots for MLE for the parameters and reliability functions.

Table 1 .
Progressively censored sample based on data of 20 flood levels.

Table 2 .
MLE and Bayes MCMC estimations for real data under SEL and LI NEX.

Table 3 .
The 95% CIs of the parameters and reliability functions for real data.

Table 4 .
The biased and MSE (in brackets) of the ML and BE for θ.

Table 5 .
The biased and MSE (in brackets) of the ML and BE for η.

Table 6 .
The biased and MSE (in brackets) of the ML and BE for γ.

Table 7 .
The biased and MSE (in brackets) of the ML and BE for S(t).

Table 8 .
The biased and MSE (in brackets) of the ML and BE for h(t).

Table 9 .
The parameters and reliability functions that display the ACLs and CPs of 95% CIs.