Open AccessArticle

Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application

Mahmoud M. Ramadan

¹,

Rashad M. EL-Sagheer

^2,3,*

and

Amel Abd-El-Monem

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt

Mathematics Department, Faculty of Science, Al-Azhar University, Naser City 11884, Cairo, Egypt

High Institute of Computer and Management Information System, First Statement, New Cairo 11865, Cairo, Egypt

Author to whom correspondence should be addressed.

Axioms 2024, 13(12), 822; https://doi.org/10.3390/axioms13120822

Submission received: 12 September 2024 / Revised: 12 November 2024 / Accepted: 21 November 2024 / Published: 25 November 2024

(This article belongs to the Special Issue Reliability and Risk of Complex Systems: Modelling, Analysis and Optimization)

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Abstract

This paper investigates statistical methods for estimating unknown lifetime parameters using a progressive first-failure censoring dataset. The failure mode’s lifetime distribution is modeled by the odd-generalized-exponential–inverse-Weibull distribution. Maximum-likelihood estimators for the model parameters, including the survival, hazard, and inverse hazard rate functions, are obtained, though they lack closed-form expressions. The Newton–Raphson method is used to compute these estimations. Confidence intervals for the parameters are approximated via the normal distribution of the maximum-likelihood estimation. The Fisher information matrix is derived using the missing information principle, and the delta method is applied to approximate the confidence intervals for the survival, hazard rate, and inverse hazard rate functions. Bayes estimators were calculated with the squared error, linear exponential, and general entropy loss functions, utilizing independent gamma distributions for informative priors. Markov-chain Monte Carlo sampling provides the highest-posterior-density credible intervals and Bayesian point estimates for the parameters and reliability characteristics. This study evaluates these methods through Monte Carlo simulations, comparing Bayes and maximum-likelihood estimates based on mean squared errors for point estimates, average interval widths, and coverage probabilities for interval estimators. A real dataset is also analyzed to illustrate the proposed methods.

Keywords:

odds-generalized-exponential–inverse-Weibull distribution; progressive first-failure censoring; Bayesian and non-Bayesian approaches; MCMC technique; Gibbs sampler within Metropolis–Hasting algorithm

MSC:

62F10; 62F12; 62F15

1. Introduction

The odd-generalized-exponential–inverse-Weibull distribution (OGE-IWD), with a more detailed view, is a flexible statistical model combining the features of both the exponential and the Weibull distributions, tailored for various applications. Its probability density function incorporates parameters that control the shape and scale, allowing it to model a wide range of data types, from life testing to reliability analysis. This distribution is particularly useful in fields where data exhibit both exponential decay and Weibull-like tail behavior, such as engineering, survival analysis, and environmental studies. By adapting to different data patterns, the OGE-IWD enables researchers to better capture underlying phenomena and make more accurate predictions. Its versatility makes it the ultimate tool for analyzing complex datasets where traditional models fall short.

Hassan et al. [1] introduced the OGE-IWD as a generalization of the inverse Weibull distribution (IWD) and obtained some of its statistical properties. Also, they estimated the unknown parameters of the OGE-IWD using maximum likelihood (ML), percentiles, and least square methods based on complete samples. Moreover, the Bayes and E-Bayes estimators for the unknown parameters of the OGE-IWD based on Type-II censored samples under the balanced squared error and linear exponential loss functions were derived by Mohamed et al. [2]. If a continuous random variable X follows the OGE-IWD, denoted by OGE-IWD

(α, β, λ)

, then the cumulative distribution function (CDF), probability density function (PDF), survival function (SF), hazard rate function (HRF), and inverse hazard rate function (IHRF) are given, respectively, as

F (x; α, β, λ) = 1 - exp [- \frac{λ}{exp [β x^{- α}] - 1}], x > 0, α, β, λ > 0,

(1)

\begin{matrix} f (x; α, β, λ) = α β λ x^{- α - 1} exp [- β x^{- α}] {(1 - exp [- β x^{- α}])}^{- 2} exp [- \frac{λ}{exp [β x^{- α}] - 1}], \\ x > 0, α, β, λ > 0, \end{matrix}

(2)

S (t; α, β, λ) = exp [- \frac{λ}{exp [β t^{- α}] - 1}], t > 0, α, β, λ > 0,

(3)

h (t; α, β, λ) = α β λ t^{- α - 1} exp [- β t^{- α}] {(1 - exp [- β t^{- α}])}^{- 2}, t > 0, α, β, λ > 0,

(4)

and

r (t; α, β, λ) = \frac{α β λ t^{- α - 1} exp [- β t^{- α}] {(1 - exp [- β t^{- α}])}^{- 2}}{exp [\frac{λ}{exp [β t^{- α}] - 1}] - 1}, t > 0, α, β, λ > 0,

(5)

where

α,

β

are the shape parameters and

λ

is the scale parameter. It is clear that the OGE-IWD tends toward odd-generalized-exponential–inverse-exponential-distribution (OGE-IED) when

α = 1,

and the odd-generalized-exponential–inverse-Rayleigh distribution (OGE-IRD) when

α = 2

. Also, they observed that the PDF of OGE-IWD for a range of parameter values, as displayed in Figure 1, can be unimodal or decreasing in shape and right-skewed, and they showed that the shape of the HRF of the OGE-IWD is increasing, decreasing, and right-skewed; see Figure 2.

The inclusion of failure times for both cases would indeed enrich the discussion, as they provide a foundation for understanding why the OGE-IWD is a suitable choice for modeling lifetime data. Here are some key points to clarify this.

1.

i.: The importance of OGE-IWD in capturing failure characteristics: The OGE-IWD distribution offers flexibility in modeling skewed lifetime data and can capture the heavy tails or asymmetry often observed in failure times. This capability makes it particularly useful for datasets with progressive first-failure censoring, where failures may occur early or late in a product’s lifetime, depending on the external stressors or quality variances.
ii.: Modeling both early and late failures: Discussing cases of both early failures and late failures can illustrate how the OGE-IWD accommodates a wide range of failure behaviors. Early failures may be common in systems with a “burn-in” period, while late failures might indicate wear-out phenomena. The OGE-IWD’s structure allows it to adapt to these different modes, improving model accuracy across varying failure patterns.
iii.: OGE-IWD’s flexibility compared to traditional distributions: Traditional lifetime models, like the Weibull or exponential, may fall short in adequately modeling the dual nature of early and late failures due to their more restrictive shapes. The OGE-IWD adds flexibility through additional shape parameters, which enable it to fit a broader spectrum of real-world lifetime data characteristics, particularly for complex failure patterns.

In life testing experiments, obtaining complete lifespan data for every component can be challenging, leading researchers to use two main types of censoring: Type-I and Type-II. Type-I censoring is applied when the duration of the experiment is predetermined, making the number of units that fail be a random variable. In contrast, Type-II censoring is used when the experiment’s duration is a random variable but the number of units that fail is fixed in advance. Several authors have studied these two types on many different statistical distributions, including Noor and Aslam [3], who explored Bayesian inference for the inverse Weibull mixture distribution under Type-I censoring, highlighting challenges and methodologies for parameter estimation; Basu et al. [4], who focused on parameter estimation for the inverse Lindley distribution using Type-I censored data, offering insights into model fitting and estimation techniques; Joarder et al. [5], who provided a comprehensive analysis of Weibull parameters with conventional Type-I censoring, emphasizing conventional inference methods; Kundu and Howlader [6], who shifted to Type-II censoring, discussing Bayesian inference and predictions for the inverse Weibull distribution; Kundu and Raqab [7], who extended this to Bayesian inference on order statistics under Type-II censoring for Weibull distributions; Singh et al. [8], who addressed Bayesian estimation for flexible Weibull models with Type-II censoring, focusing on model flexibility and predictions; Panahi and Sayyareh [9], who investigated parameter estimation for the Burr Type-XII distribution with Type-II censoring, expanding the scope to more complex distributions; Asgharzadeh et al. [10], who studied the Lindley model under Type-II censoring, detailing statistical inference and prediction strategies; Xin et al. [11], who conducted reliability inference for the three-parameter Burr Type-XII distribution with Type-II censoring, highlighting its applications in reliability analysis; Goyal et al. [12], compared classical and Bayesian approaches for a new lifetime model with Type-II censoring, providing a comprehensive view of different inference techniques; and Arabi Belaghi et al. [13], who examined the Poisson-Exponential distribution under Type-II censoring, discussing estimation and prediction methodologies in this context.

Both censoring types ensure that any remaining components cannot be removed until the end of the test. To address this, a progressive Type-II censoring (PT-2C) method, as outlined in Balakrishnan and Aggarwala [14], seeks to optimize resource use and cut costs by incorporating multiple stages of censoring. This approach allows for the possibility of removing some surviving units during the period of the test according to a predefined scheme, which is specifically useful when the experimental units being tested are very expensive. Also, it is a useful scheme in which a a specific fraction of individuals at risk may be removed from the experiment at several ordered failure times; see Cohen [15]. Under this scheme, a set of n independent units is subjected to a life testing experiment, and failure times are observed as a progressive sample

X_{1 : m : n} < X_{2 : m : n} < \dots < X_{m : m : n}

, where m (with

m < n

) denotes the number of observed failure units. A censoring plan

R = (R_{1}, R_{2}, \dots, R_{m})

is predetermined. This process unfolds as follows: upon the occurrence of the first failure

X_{1}

R_{1}

of the

n - 1

surviving units are randomly selected and removed from the test. At the second failure

X_{2}

R_{2}

n - R_{1} - 2

surviving units are randomly selected and removed, and this pattern continues. At the

m^{t h}

failure, the remaining surviving units

R_{m} = n - m - (R_{1} + R_{2} + \dots + R_{m - 1})

are removed, ending the test.

In recent studies, PT-2C has emerged as a crucial censor for obtaining reliable statistical inference across various distributions, including Brito et al. [16]; who explored inference techniques for the very flexible Weibull distribution, focusing on PT-2C to enhance model accuracy and estimation methods; Abo-Kasem et al. [17], who extended this concept to the Kumaraswamy distribution, emphasizing optimal sampling strategies under similar censoring conditions; Dey and Al-Mosawi [18], who delved into both classical and Bayesian inference for the unit Gompertz distribution using PT-2C data, highlighting the versatility and application of different inferential approaches; Kumar et al. [19], who further contributed by investigating inference methods for the Li–Li Rayleigh distribution, showcasing the practical implications of PT-2C in real-world applications; and Choudhary et al. [20], who examined estimation techniques in the generalized uniform distribution under PT-2C samples, demonstrating the robustness of this censoring scheme across diverse distributions. Although the PT-2C improves experimental efficiency, the testing time remains interesting, due to the long life of the units. For this reason, Johnson [21] introduced a life test that divided the test units into many groups with the same number of units in each group and then ran all the test units simultaneously until the occurrence of the first failure in each group. This type of censoring is called first-failure censoring (FFC). For more details on FFC, see Wu et al. [22] and Wu and Yu [23]. The FFC does not enable the experimenter to remove some groups of test units before observing the first failure in these groups. For this reason, Wu and Kus [24] proposed a life-testing scheme, which combines FFC with a PT-2C called a progressive first-failure censoring (PFFC) scheme.

The PFFC scheme extends and improves progressive censoring; its adaptability makes it a common choice in experimental design. In this scheme, consider a life test involving n independent groups, each containing k units. When the first failure occurs (denoted as

X_{1 : m : n : k}^{R}

R_{1}

groups, along with the group where the failure was observed

R_{X_{1}}

, are randomly removed from the test. Similarly, when the second failure happens (denoted as

X_{2 : m : n : k}^{R}

R_{2}

groups and the group where this failure occurred,

R_{X_{2}}

, are randomly removed from the test, and so on until the

m^{t h}

failure (denoted as

X_{m : m : n : k}^{R}

), at which point

R_{m}

groups and the group with this failure

R_{X_{m}}

are removed. Consequently, the failure times

X_{1 : m : n : k}^{R} < X_{2 : m : n : k}^{R} < \dots < X_{m : m : n : k}^{R}

represent the PFFC order statistics under the progressive censoring scheme

R

. It follows that

n = m + R_{1} + R_{2} + \dots + R_{m}

. Figure 3 shows a simple diagram of the description of the PFFC scheme. One benefit of this censoring scheme is that it shortens the test time and reduces the test cost of the experiment; more items can be utilized, yet only m out of

n \times k

units fail. If the failure times of the

n \times k

units come from a continuous distribution with CDF

F (x)

and PDF

f (x)

, then the joint PDF for the order statistics

X_{1 : m : n : k}^{R} < X_{2 : m : n : k}^{R} < \dots < X_{m : m : n : k}^{R}

can be specified as

f (x_{1 : m : n : k}^{R}, x_{2 : m : n : k}^{R}, \dots, x_{m : m : n : k}^{R}) = C_{R} k^{m} \prod_{i = 1}^{m} f (x_{i : m : n : k}^{R}) {[1 - F (x_{i : m : n : k}^{R})]}^{k (R_{i} + 1) - 1},

(6)

where

C_{R}

is a progressive normalizing constant and is defined as

C_{R} = n (n - 1 - R_{1}) (n - 2 - R_{1} - R_{2}) \dots (n - m + 1 - R_{1} - R_{2} - \dots - R_{m - 1})

. It is important to note that various censoring schemes can be represented within this framework. For instance, the complete sample scenario can be captured when

k = 1

and

R_{i} = 0

for all

i = 1, 2, . . ., m

. The FFC case is applicable when

k \neq 1

and

R_{i} = 0

for all

i = 1, 2, . . ., m

. PT-2C is represented when

k = 1

. Additionally, the standard Type-II censored sample can be modeled by setting

k = 1

and

R_{i} = 0

for

i = 1, 2, 3, . . ., m - 1

. Additionally, it is evident that

X_{1 : m : n : k}^{R} < X_{2 : m : n : k}^{R} < \dots < X_{m : m : n : k}^{R}

can be considered as a sample from a PT-2C scheme, drawn from a population with a CDF,

1 - {(1 - F (x))}^{k .}

. Consequently, findings related to PT-2C-order statistics can be directly applied to PFFC order statistics with relative cases.

Recent advancements in the statistical analysis of PFFC data reveal a broad array of innovative methodologies and theoretical frameworks aimed at enhancing reliability estimation and model accuracy. Kumar et al. [25] delved into reliability estimation using the inverse Pareto distribution, providing a robust approach for dealing with censored life data. Abd-El-Monem et al. [26] introduced a partially accelerated life test model utilizing a power hazard distribution, offering a refined theoretical framework for handling censored data under accelerated conditions. Elshahhat et al. [27] contributed to the field by applying beta-binomial removals to analyze censored data, which enhances the modeling of failure rates. Kumar et al. [28] extended the discussion to Shannon’s entropy within the Maxwell distribution, incorporating entropy measures into the analysis of censored data. Saini [29] addressed the challenge of multi-stress scenarios by estimating strength reliability using a generalized inverted exponential distribution. Shi and Shi [30] focused on stress-strength reliability for the beta log Weibull distribution, providing insights into system performance under stress conditions. Fathi et al. [31] offered a detailed examination of the Weibull inverted exponential distribution, integrating constant-stress partially accelerated life test models. Eliwa et al. [32] proposed a theoretical framework for fitting extreme data using the modified Weibull distribution in a PFFC context, enhancing the analysis of extreme values. Recently, Gong et al. [33] focused on the generalized Rayleigh distribution and applied a PFFC model to enhance the accuracy of reliability estimations.

Additionally, researchers have investigated various Bayesian and maximum likelihood estimation methods across different censoring scenarios and model assumptions. However, a significant gap exists in the literature concerning the use of the OGE-IW lifespan distribution with any type of censoring. Hence, the main aim of the paper is to investigate and compare statistical methods for estimating lifetime parameters from datasets with PFFC, under OGE-IWD. The study seeks to derive and compute maximum likelihood estimators for the model parameters, including survival, hazard rate, and inverse hazard rate functions, despite the lack of closed-form expressions. It employs the Newton–Raphson (NR) method for estimation and approximates confidence intervals for the parameters of OGE-IWD using normal distribution of MLs and the delta method for survival, hazard rate, and inverse hazard rate functions of OGE-IWD. Additionally, the paper aims to develop Bayes estimators under various loss functions with informative gamma priors and to use Markov-chain Monte Carlo sampling to provide credible intervals (CRIs) and Bayesian point estimates. The effectiveness of these methods is evaluated through Monte Carlo simulation and applied to a real dataset to demonstrate their practical utility.

Reliable parameter estimation is crucial in practical scenarios, especially in fields like reliability engineering, medical research, and risk assessment, where understanding the lifespan and failure behavior of components, systems, or biological processes is essential. Accurate estimates of lifetime parameters allow practitioners to make informed decisions on maintenance schedules, warranty policies, safety assessments, and resource allocation, ultimately minimizing risks and optimizing performance.

In the context of this study, the reliable estimation of parameters in the odd-generalized-exponential–inverse-Weibull distribution provides a robust tool for modeling complex lifetime data with flexible hazard shapes, accommodating various failure modes and censoring types. Progressive first-failure censoring schemes are particularly valuable for situations where testing all items to failure is impractical due to time, cost, or ethical constraints, such as in medical trials or high-cost product testing.

There are many potential applications, such as reliability assessment in engineering systems, where failure modes need nuanced modeling, or in biomedical research where patient survival data often include censored observations. Moreover, follow-up work could explore alternative estimation techniques for scenarios with smaller sample sizes or extreme censoring, which often challenge classical methods. Future research could also examine how these estimation methods perform with other complex lifetime distributions or investigate computational improvements for Bayesian estimation, particularly in high-dimensional parameter spaces. For more details on the different data models, see for example, He et al. [34], Ran and Bai [35], Zhuang et al. [36], and Xu et al [37].

The structure of this paper is as follows: Section 2 examines maximum-likelihood estimators (MLEs) and their approximate confidence intervals (ACIs) for the parameters of the OGE-IWD. In Section 3, we utilize the Markov-chain Monte Carlo (MCMC) method to obtain Bayesian estimates for these parameters, as well as for survival, hazard rate, and inverse hazard rate functions, using independent gamma priors. We also provide the highest posterior density credible intervals based on MCMC results. Section 4 conducts a Monte Carlo simulation to evaluate the proposed estimates in terms of mean squared error, average interval width, and coverage probability. Section 5 analyzes real-world datasets and a simulation example for illustration. Finally, Section 6 wraps up with concluding remarks.

2. Maximum-Likelihood Estimation

Maximum-likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function. This approach seeks to find parameter values that make the observed data most probable under the given model. The MLE possesses several desirable properties: it is asymptotically unbiased, meaning that as the sample size grows, the estimator converges to the true parameter value. It is also consistent, indicating that with larger samples, the estimates tend to become more accurate. Additionally, the MLE is efficient in the sense that it achieves the Cramér–Rao lower bound, which provides the minimum variance of unbiased estimators. Under regularity conditions, MLEs are asymptotically normal, facilitating inference using standard statistical techniques. However, MLE can be sensitive to model misspecification and may produce biased estimates if the model does not fit the data well. Despite these limitations, the MLE remains a widely used and powerful tool in statistical analysis. This section focuses on applying MLE to obtain both point and interval estimates for the unknown parameters

α

β

, and

λ

, as well as for the survival

S (t)

, hazard rate

h (t)

, and inverse hazard rate

r (t)

functions, within the context of PFFC data.

Let

X_{1 : m : n : k}^{R} < X_{2 : m : n : k}^{R} < \dots < X_{m : m : n : k}^{R}

denote the order statistics of the PFFC sample drawn from OGE-IW

(α, β, λ)

with a pre-fixed censoring scheme

(R_{1}, R_{2}, . . ., R_{m})

. Henceforth, instead of using

(X_{1 : m : n : k}^{R} < X_{2 : m : n : k}^{R} < \dots < X_{m : m : n : k}^{R})

, we will use

\underset{̲}{x} = (x_{1} < x_{2} < \dots < x_{m :})

. Equations (1), (2), and (6) are used to create the likelihood function (LF), which is shown as follows:

L \propto α^{m} β^{m} λ^{m} \prod_{i = 1}^{m} x_{i}^{- α - 1} exp [- β x_{i}^{- α}] {(1 - exp [- β x_{i}^{- α}])}^{- 2} {(exp [- \frac{λ}{exp [β x_{i}^{- α}] - 1}])}^{k (R_{i} + 1)} .

(7)

If the additive constant is excluded, the log-likelihood function, denoted by ℓ, can be written as

\begin{matrix} ℓ \propto m log (α β λ) - (α + 1) \sum_{i = 1}^{m} log (x_{i}) - β \sum_{i = 1}^{m} x_{i}^{- α} - 2 \sum_{i = 1}^{m} log (1 - exp [- β x_{i}^{- α}]) \\ - λ k \sum_{i = 1}^{m} [\frac{(R_{i} + 1)}{exp [β x_{i}^{- α}] - 1}] . \end{matrix}

(8)

Upon computing the first partial derivatives of Equation (8) concerning

α

β

, and

λ

, the partial derivatives of the log-likelihood can be obtained in the following manner:

\begin{matrix} \frac{\partial ℓ}{\partial α} = \frac{m}{α} - \sum_{i = 1}^{m} log (x_{i}) + β \sum_{i = 1}^{m} x_{i}^{- α} log (x_{i}) + 2 β \sum_{i = 1}^{m} \frac{x_{i}^{- α} exp [- β x_{i}^{- α}] log (x_{i})}{1 - exp [- β x_{i}^{- α}]} \\ - k β λ \sum_{i = 1}^{m} \frac{x_{i}^{- α} exp [β x_{i}^{- α}] (R_{i} + 1) log (x_{i})}{{(exp [β x_{i}^{- α}] - 1)}^{2}} = 0, \end{matrix}

(9)

\frac{\partial ℓ}{\partial β} = \frac{m}{β} - \sum_{i = 1}^{m} x_{i}^{- α} - 2 \sum_{i = 1}^{m} \frac{x_{i}^{- α} exp [- β x_{i}^{- α}]}{1 - exp [- β x_{i}^{- α}]} + k λ \sum_{i = 1}^{m} \frac{x_{i}^{- α} exp [β x_{i}^{- α}] (R_{i} + 1)}{{(exp [β x_{i}^{- α}] - 1)}^{2}} = 0,

(10)

and

\frac{\partial ℓ}{\partial λ} = \frac{m}{λ} - k \sum_{i = 1}^{m} \frac{(R_{i} + 1)}{exp [β x_{i}^{- α}] - 1} = 0 .

(11)

To improve clarity for readers, we give a brief practical interpretation of Equations (9)–(11) and why these derivatives are necessary for estimating the model parameters

α

β

, and

λ

as follows: Equation (9) represents how the log-likelihood function changes concerning

α

. Intuitively, this derivative captures the sensitivity of the likelihood of changes in

α

, considering the dependence on the observed values

x_{i}

and their logarithms. The terms involving

log (x_{i})

indicate that this parameter controls how the scaling of the data affects the model fit, allowing us to estimate

α

as the shape parameter in the distribution. Equation (10) shows the rate of change of the log-likelihood with respect to

β

, providing insights into how this parameter influences the distribution’s shape. It includes terms involving

x_{i}^{- α}

, which link it closely with

α

. As a result,

β

primarily adjusts the data’s spread, and its partial derivative helps determine the optimal shape to maximize the likelihood. Equation (11) shows the influence of this parameter on the likelihood. In this model,

λ

might represent an external constraint or regularization effect. This equation reflects the impact of

λ

on the likelihood, where the term involving

R_{i}

suggests it relates to external adjustments in the model.

Simultaneously solving the complex nonlinear equations

\frac{\partial ℓ}{\partial α} = 0

\frac{\partial ℓ}{\partial β} = 0

and

\frac{\partial ℓ}{\partial λ} = 0

enables the calculation of the MLEs for

α

β

, and

λ

. Deriving closed-form solutions for Equations (9)–(11) is difficult. Therefore, numerical methods like the NR algorithm are used to determine the MLEs for unknown parameters. The subsequent steps outline the detailed procedures involved in this iterative method with a set threshold

10^{- 6}

S1.: The starting values for $Ω = (α, β, λ)$ need to be specified with $l = 0$ ; that is, $Ω^{(0)} = (α^{(0)}, β^{(0)}, λ^{(0)})$ .
S2.: In the $l^{t h}$ iteration, compute ${(\frac{\partial ℓ}{\partial α}, \frac{\partial ℓ}{\partial β}, \frac{\partial ℓ}{\partial λ})}^{T} ∣_{(α = α^{(l)}, β = β^{(l)}, λ = λ^{(l)})}$ and $I = I (α^{(l)}, β^{(l)}, λ^{(l)})$ , where

$I = I (α^{(l)}, β^{(l)}, λ^{(l)}) = (\begin{matrix} - ℓ_{11} & - ℓ_{12} & - ℓ_{13} \\ - ℓ_{21} & - ℓ_{22} & - ℓ_{23} \\ - ℓ_{31} & - ℓ_{32} & - ℓ_{33} \end{matrix}) .$

(12)

The matrix representing the observed information for the parameters $α$ , $β$ , and $λ$ is referred to as I. The components of this matrix, I, are detailed as follows:

$\begin{matrix} ℓ_{11} = \frac{\partial^{2} ℓ}{\partial α^{2}} = - \frac{m}{α^{2}} - β \sum_{i = 1}^{m} x_{i}^{- α} log {(x_{i})}^{2} \\ + 2 β \sum_{i = 1}^{m} \frac{x_{i}^{- α} (β x_{i}^{- α} + exp [- β x_{i}^{- α}] - 1) exp [- β x_{i}^{- α}] log {(x_{i})}^{2}}{{(1 - exp [- β x_{i}^{- α}])}^{2}} \\ - k β λ \sum_{i = 1}^{m} \frac{x_{i}^{- α} φ (x, α, β, λ) exp [β x_{i}^{- α}] (R_{i} + 1) log {(x_{i})}^{2}}{{(exp [β x_{i}^{- α}] - 1)}^{3}}, \end{matrix}$

(13)

$\begin{matrix} ℓ_{12} = ℓ_{21} = \frac{\partial^{2} ℓ}{\partial α \partial β} = \frac{\partial^{2} ℓ}{\partial β \partial α} = \sum_{i = 1}^{m} x_{i}^{- α} log (x_{i}) \\ - 2 \sum_{i = 1}^{m} \frac{x_{i}^{- α} (β x_{i}^{- α} + exp [- β x_{i}^{- α}] - 1) exp [- β x_{i}^{- α}] log (x_{i})}{{(1 - exp [- β x_{i}^{- α}])}^{2}} \\ + k λ \sum_{i = 1}^{m} \frac{x_{i}^{- α} φ (x, α, β, λ) exp [β x_{i}^{- α}] (R_{i} + 1) log (x_{i})}{{(exp [β x_{i}^{- α}] - 1)}^{3}}, \end{matrix}$

(14)

where

$φ (x, α, β, λ) = β x_{i}^{- α} exp [β x_{i}^{- α}] + β x_{i}^{- α} - exp [β x_{i}^{- α}] + 1$

$ℓ_{13} = ℓ_{31} = \frac{\partial^{2} ℓ}{\partial α \partial λ} = \frac{\partial^{2} ℓ}{\partial λ \partial α} = - k β \sum_{i = 1}^{m} \frac{x_{i}^{- α} exp [β x_{i}^{- α}] (R_{i} + 1) log (x_{i})}{{(exp [β x_{i}^{- α}] - 1)}^{2}},$

(15)

$\begin{matrix} ℓ_{22} = \frac{\partial^{2} ℓ}{\partial β^{2}} = - \frac{m}{β^{2}} + 2 \sum_{i = 1}^{m} \frac{x_{i}^{- 2 α} exp [- β x_{i}^{- α}]}{{(1 - exp [- β x_{i}^{- α}])}^{2}} \\ - k λ \sum_{i = 1}^{m} \frac{x_{i}^{- 2 α} exp [β x_{i}^{- α}] (exp [β x_{i}^{- α}] + 1) (R_{i} + 1)}{{(exp [β x_{i}^{- α}] - 1)}^{3}}, \end{matrix}$

(16)

$ℓ_{23} = ℓ_{32} = \frac{\partial^{2} ℓ}{\partial β \partial λ} = \frac{\partial^{2} ℓ}{\partial λ \partial β} = k \sum_{i = 1}^{m} \frac{x_{i}^{- α} exp [β x_{i}^{- α}] (R_{i} + 1)}{{(exp [β x_{i}^{- α}] - 1)}^{2}},$

(17)

and

$ℓ_{33} = \frac{\partial^{2} ℓ}{\partial λ^{2}} = - \frac{m}{λ^{2}} .$

(18)
S3.: Assign

${(α^{(l + 1)}, β^{(l + 1)}, λ^{(l + 1)})}^{T} = G \times {(\frac{\partial ℓ}{\partial α}, \frac{\partial ℓ}{\partial β}, \frac{\partial ℓ}{\partial λ})}^{T} ∣_{(α = α^{(l)}, β = β^{(l)}, λ = λ^{(l)})},$

where $G = {(α^{(l)}, β^{(l)}, λ^{(l)})}^{T} + I^{- 1} (α^{(l)}, β^{(l)}, λ^{(l)})$ , ${(α, β, λ)}^{T}$ is the transpose of vector $(α, β, λ)$ , and $I^{- 1} (α^{(l)}, β^{(l)}, λ^{(l)})$ symbolizes the matrix’s inverse $I (α^{(l)}, β^{(l)}, λ^{(l)})$ .
S4.: By setting $l = l + 1$ , the MLEs for the parameters (represented by $\hat{α}$ , $\hat{β}$ , and $\hat{λ}$ ) can be obtained by repeatedly executing steps S2 and S3 until

$|{(α^{(l + 1)}, β^{(l + 1)}, λ^{(l + 1)})}^{T} - {(α^{(l)}, β^{(l)}, λ^{(l)})}^{T}| < δ,$

where $δ$ is a predetermined threshold value.

In optimization methods like the NR or Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, the convergence criteria are critical for deciding when to stop iterations. Two commonly used convergence criteria are as follows.

1.

(a)

Tolerance level for stopping: This is a threshold value that determines how close the estimated parameters need to be to the optimal solution before the algorithm halts. This tolerance can apply to either of the following

Gradient norm tolerance: The algorithm stops when the norm (magnitude) of the gradient of the objective function falls below a specified small value, indicating that the function’s slope is almost flat and a minimum has been reached. Typical values for the gradient norm tolerance might be in the range of $10^{- 5}$ to $10^{- 8}$ .
Parameter change tolerance: The algorithm also stops when consecutive iterations yield parameter estimates that differ by less than a certain tolerance. This indicates that the estimates are no longer changing significantly and have likely converged. Common values for parameter change tolerance are similar, often set around $10^{- 5}$ to $10^{- 8}$ .

(b)

BFGS method convergence: The BFGS method, which is a quasi-Newton method, utilizes an approximation of the Hessian matrix of second derivatives. This approximation is updated iteratively, so each iteration requires the Hessian to be closer to the true value. The stopping tolerance for BFGS is typically set based on the following.

Gradient norm: As with Newton–Raphson, the gradient norm is checked to see if it has fallen below the tolerance.
Objective function convergence: The algorithm also stops if changes in the objective function value (e.g., likelihood or error) between iterations fall below a set threshold, such as $10^{- 6}$ or smaller. This threshold is often used in conjunction with gradient tolerance to confirm that the method has reached an optimal point.

Choosing these tolerance levels involves a trade-off between accuracy and computational cost. Stricter (lower) tolerance levels generally yield more accurate solutions but require more iterations and computation time. Conversely, higher tolerance levels yield faster convergence but may stop before reaching a true minimum. For many statistical estimation tasks, tolerance values around

10^{- 5}

10^{- 8}

for both gradient norm and objective function convergence are common in practice.

By using the obtained point estimates and the invariance property of MLEs, we can calculate the estimates for the survival, hazard rate, and inverse hazard rate functions based on Equations (3)–(5), which are detailed as follows:

\hat{S} (t; \hat{α}, \hat{β}, \hat{λ}) = exp [- \frac{\hat{λ}}{exp [\hat{β} t^{- \hat{α}}] - 1}], t > 0,

(19)

\hat{h} (t; \hat{α}, \hat{β}, \hat{λ}) = \hat{α} \hat{β} \hat{λ} t^{- \hat{α} - 1} exp [- \hat{β} t^{- \hat{α}}] {(1 - exp [- \hat{β} t^{- \hat{α}}])}^{- 2}, t > 0,

(20)

and

\hat{r} (t; \hat{α}, \hat{β}, \hat{λ}) = \frac{\hat{α} \hat{β} \hat{λ} t^{- \hat{α} - 1} exp [- \hat{β} t^{- \hat{α}}] {(1 - exp [- \hat{β} t^{- \hat{α}}])}^{- 2}}{exp [\frac{\hat{λ}}{exp [\hat{β} t^{- \hat{α}}] - 1}] - 1}, t > 0 .

(21)

2.1. Existence and Uniqueness of MLEs

The existence and uniqueness of MLEs are fundamental aspects to consider in statistical inference. In this subsection, we discuss the necessary and sufficient conditions for the existence of the ML estimators for arbitrary PFFC data under OGE-IWD. To this end, we examine the behavior of Equations (9)–(11) on the positive real line

(0, \infty) .

▸: For (9), when $α \to 0^{+}$ , we have $lim_{α \to 0^{+}} \frac{\partial ℓ}{\partial α} = + \infty$ , but when $α \to + \infty$ , we have $lim_{α \to + \infty} \frac{\partial ℓ}{\partial α} = -$ $\sum_{i = 1}^{m} log (x_{i}) < 0$ , for $x_{i} > 0;$ $x_{i} \neq 1, i = 1, 2, \dots, m .$
▸: Similarly, for (10), when $β \to 0^{+}$ , we have $lim_{β \to 0^{+}} \frac{\partial ℓ}{\partial β} = + \infty$ , but when $β \to + \infty$ , we have $lim_{β \to + \infty} \frac{\partial ℓ}{\partial β} = -$ $\sum_{i = 1}^{m} x_{i}^{- α} < 0$ , for $x_{i} > 0;$ $α > 0, i = 1, 2, \dots, m .$
▸: Similarly, for (11), when $λ \to 0^{+}$ , we have $lim_{λ \to 0^{+}} \frac{\partial ℓ}{\partial λ} = + \infty$ , but when $λ \to + \infty$ , we have $lim_{λ \to + \infty} \frac{\partial ℓ}{\partial λ} = -$ $k \sum_{i = 1}^{m} \frac{(R_{i} + 1)}{exp [β x_{i}^{- α}] - 1} < 0$ , for $x_{i} > 0;$ $α > 0, i = 1, 2, \dots, m .$

Therefore, on

(0, \infty)

for

Ω = (α, β, λ)

, there exists at least one positive root for

\frac{\partial ℓ}{\partial Ω} = 0

. Additionally, we find that the second partial derivatives of ℓ with respect to

α, β

and

λ

are always negative; Equations (9)–(11) have a unique solution; and this solution represents the MLEs of

α

β

and

λ

. Consequently, we deduce that

\frac{\partial ℓ}{\partial Ω}

is a continuous function on

(0, \infty)

, and it monotonically decreases from ∞ to negative values. This demonstrate the existence and uniquencess of the MLEs of

α

β

and

λ

2.2. Approximate Confidence Intervals

Approximate confidence intervals (ACIs) for a parameter can be constructed using the observed Fisher information matrix, which quantifies the amount of information that an observable random variable carries about an unknown parameter. To derive these intervals, one typically calculates the observed Fisher information matrix (FIM), which is the negative of the second derivative of the log-likelihood function with respect to the parameter. This matrix is then inverted to estimate the variance–covariance matrix of the parameter estimates. Assuming the estimates are approximately normally distributed, the square root of the diagonal elements of this variance–covariance matrix provides the standard errors of the parameter estimates. These standard errors can be used to construct ACIs, typically by assuming a normal distribution and using a critical value from the standard normal distribution to determine the range within which the true parameter value is likely to lie with a specified level of confidence. This method relies on the asymptotic properties of MLEs, which suggest that for large sample sizes, the parameter estimates are approximately normally distributed and the FIM provides a good approximation of their variability. In this subsection, we derive the ACIs for the vector of the parameters

Ω = (α, β, λ)

, as well as for the

S (t)

h (t)

and

r (t)

by using the asymptotic normality of the MLEs. This method relies on the fact that, under regularity conditions, the MLEs

\hat{Ω} = (\hat{α}, \hat{β}, \hat{λ})

are approximately normally distributed with the true parameters

Ω

as their means and the inverse of the FIM at the MLEs as their variance–covariance matrix. Or, equivalently,

(\hat{α} - α, \hat{β} - β, \hat{λ} - λ) \sim N (0, I^{- 1} (\hat{α}, \hat{β}, \hat{λ})),

(22)

where, from (12),

I^{- 1} (\hat{α}, \hat{β}, \hat{λ}) = {(\begin{matrix} - ℓ_{11} & - ℓ_{12} & - ℓ_{13} \\ - ℓ_{21} & - ℓ_{22} & - ℓ_{23} \\ - ℓ_{31} & - ℓ_{32} & - ℓ_{33} \end{matrix})}_{\hat{Ω} = Ω}^{- 1} = (\begin{matrix} V a r (\hat{α}) & C o v (\hat{α}, \hat{β}) & C o v (\hat{α}, \hat{λ}) \\ C o v (\hat{β}, \hat{α}) & V a r (\hat{β}) & C o v (\hat{β}, \hat{λ}) \\ C o v (\hat{λ}, α) & C o v (\hat{λ}, \hat{β}) & V a r (\hat{λ}) \end{matrix}),

(23)

is the inverse of the observed FIM. Moreover, the quantity

ℓ_{i j}

, where i,

j = 1, 2, 3

, is specified by Equation (13) through (18). Hence, the

100 (1 - γ) %

ACIs for the parameters

α

β

, and

λ

are established by

(\hat{α} \pm Z_{γ / 2} \sqrt{V a r (\hat{α})}), (\hat{β} \pm Z_{γ / 2} \sqrt{V a r (\hat{β})}), (\hat{λ} \pm Z_{γ / 2} \sqrt{V a r (\hat{λ})}),

(24)

where

Z_{γ / 2}

is the value from the standard normal distribution

N (0, 1)

that corresponds to the upper

γ / 2

quantile.

2.3. Delta Method

The Delta method (DM) is a statistical technique used to approximate the confidence intervals for complex functions of parameters, such as survival, hazard rate, and inverse hazard rate functions. When dealing with such functions, the direct estimation of their confidence intervals can be challenging due to their non-linear nature. The DM simplifies this by utilizing a first-order Taylor expansion around the estimated parameters. It approximates the variance–covariance matrix of the function of interest by linearizing the function using the gradient (or Jacobian) of the function with respect to the parameters. This approach relies on the assumption that the parameter estimates are approximately normally distributed; see Greene [38]. In applying this method, the approximate variances of

S (t)

h (t),

and

r (t)

are computed by the following steps:

S1.: Define $Φ_{1}$ , $Φ_{2}$ , and $Φ_{3}$ as three quantities with the following specific forms:

$\begin{matrix} Φ_{1} & = & (\frac{\partial S (t)}{\partial α}, \frac{\partial S (t)}{\partial β}, \frac{\partial S (t)}{\partial λ}), Φ_{2} = (\frac{\partial h (t)}{\partial α}, \frac{\partial h (t)}{\partial β}, \frac{\partial h (t)}{\partial λ}), \\ Φ_{3} & = & (\frac{\partial r (t)}{\partial α}, \frac{\partial r (t)}{\partial β}, \frac{\partial r (t)}{\partial λ}), \end{matrix}$

where according to Equations (3), (4), and (5),

$\frac{\partial S (t)}{\partial α} = - \frac{β λ t^{- α} exp [β t^{- α} - \frac{λ}{exp [β t^{- α}] - 1}] log (t)}{{(exp [β t^{- α}] - 1)}^{2}},$

$\frac{\partial S (t)}{\partial β} = \frac{λ t^{- α} exp [β t^{- α} - \frac{λ}{exp [β t^{- α}] - 1}]}{{(exp [β t^{- α}] - 1)}^{2}},$

$\frac{\partial S (t)}{\partial λ} = - \frac{exp [- \frac{λ}{exp [β t^{- α}] - 1}]}{exp [β t^{- α}] - 1},$

$\frac{\partial h (t)}{\partial α} = \frac{1}{4} β λ t^{- 2 α - 1} \csc h {[\frac{β t^{- α}}{2}]}^{2} (α β coth [\frac{β t^{- α}}{2}] log (t) + t^{α} (1 - α log (t))),$

$\frac{\partial h (t)}{\partial β} = \frac{1}{4} α λ t^{- 2 α - 1} (t^{α} - β coth [\frac{β t^{- α}}{2}]) \csc h {[\frac{β t^{- α}}{2}]}^{2},$

$\frac{\partial h (t)}{\partial λ} = \frac{1}{4} α β t^{- α - 1} \csc h {[\frac{β t^{- α}}{2}]}^{2},$

$\frac{\partial r (t)}{\partial α} = - \frac{β λ t^{- 2 α - 1} exp [β t^{- α}] (α β Λ (t; α, β, λ) log (t) + t^{α} {(exp [β t^{- α}] - 1)}^{2} D (α log (t) - 1))}{{(exp [β t^{- α}] - 1)}^{4} D^{2}},$

$\frac{\partial r (t)}{\partial β} = \frac{α λ t^{- 2 α - 1} exp [β t^{- α}] (t^{α} {(exp [β t^{- α}] - 1)}^{2} D + β Λ (t; α, β, λ))}{{(exp [β t^{- α}] - 1)}^{4} D^{2}},$

and

$\frac{\partial r (t)}{\partial λ} = - \frac{α β t^{- α - 1} exp [β t^{- α}] (exp [β t^{- α}] - exp [\frac{λ}{exp [β t^{- α}] - 1}] (exp [β t^{- α}] - λ - 1) - 1)}{{(exp [β t^{- α}] - 1)}^{3} D^{2}},$

where $Λ (t; α, β, λ) = exp [2 β t^{- α}] - exp [\frac{λ}{exp [β t^{- α}] - 1}] (exp [2 β t^{- α}] - λ exp [β t^{- α}] - 1) - 1 .$ and

$D = exp [\frac{λ}{exp [β t^{- α}] - 1}] - 1 .$
S2.: Utilize the formulas provided to calculate the approximate variances for $S (t)$ , $h (t),$ and $r (t)$ :

$\begin{matrix} V a r (\hat{S} (t)) ⋍ {[Φ_{1}^{T} V a r (Ω) Φ_{1}]}_{Ω = \hat{Ω}}, V a r (\hat{h} (t)) ⋍ {[Φ_{2}^{T} V a r (Ω) Φ_{2}]}_{Ω = \hat{Ω}}, \\ V a r (\hat{r} (t)) ⋍ {[Φ_{3}^{T} V a r (Ω) Φ_{3}]}_{Ω = \hat{Ω}}, \end{matrix}$

(25)

where $V a r (\hat{Ω})$ is derived from Equation (23) for $\hat{Ω} = (\hat{α}, \hat{β}, \hat{λ})$ .
S3.: Compute the $100 (1 - γ) %$ ACIs for $S (t)$ , $h (t),$ and $r (t)$ by applying the following formula:

$(\hat{S} (t) \pm Z_{γ / 2} \sqrt{V a r (\hat{S} (t))}), (\hat{h} (t) \pm Z_{γ / 2} \sqrt{V a r (\hat{h} (t))}), (\hat{r} (t) \pm Z_{γ / 2} \sqrt{V a r (\hat{r} (t))}) .$

(26)

2.4. Log-Normal ACIs

In cases where the lower bound of the asymptotic confidence intervals might fall below

0,

which conflicts with the requirement that

Ω > 0

, this issue can be addressed using a log transformation combined with the DM. Specifically, the log-transformed MLEs satisfy

ln \hat{Ω} \sim N (ln Ω, V a r (ln \hat{Ω}))

, where

V a r (ln \hat{Ω}) = \frac{V a r (\hat{Ω})}{{\hat{Ω}}^{2}}

; see Meeker and Escobar [39]. Consequently, the asymptotic confidence intervals at the

100 (1 - γ) %

level for the log-transformed vector of the parameters

Ω = (α, β, λ)

are computed as follows:

[\hat{Ω} exp \{- Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{Ω})}{{\hat{Ω}}^{2}}}\}, \hat{Ω} exp \{Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{Ω})}{{\hat{Ω}}^{2}}}\}] .

(27)

In the same manner, the

100 (1 - γ) %

log-transformed approximate confidence intervals (LACIs) for

S (t)

h (t),

r (t)

can be expressed as follows:

[\hat{S} (t) exp \{- Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{S} (t))}{{[\hat{S} (t)]}^{2}}}\}, \hat{S} (t) exp \{Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{S} (t))}{{[\hat{S} (t)]}^{2}}}\}],

(28)

[\hat{h} (t) exp \{- Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{h} (t))}{{[\hat{h} (t)]}^{2}}}\}, \hat{h} (t) exp \{Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{h} (t))}{{[\hat{h} (t)]}^{2}}}\}],

(29)

and

[\hat{r} (t) exp \{- Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{r} (t))}{{[\hat{r} (t)]}^{2}}}\}, \hat{h} (t) exp \{Z_{(γ / 2)} \sqrt{\frac{V a r (\hat{r} (t))}{{[\hat{r} (t)]}^{2}}}\}] .

(30)

3. Bayesian Estimation

Bayes estimation, grounded in Bayes’ theorem, is a powerful statistical approach that updates the probability estimate for a hypothesis as additional evidence is acquired. At its core, Bayes’ theorem combines prior distributions with observed data to produce a posterior distribution, which represents a revised probability of the hypothesis given the new evidence. The prior distribution reflects the initial beliefs or knowledge about the parameters before observing the data, while the likelihood function represents the probability of the observed data given the parameters. By integrating these components, Bayes estimation allows for a dynamic adjustment of parameter estimates based on both prior knowledge and current data, thus providing a coherent framework for incorporating uncertainty and updating beliefs as more information becomes available. This methodology is particularly useful in scenarios where data are limited or uncertain, as it enables the incorporation of prior information to make more informed and probabilistic estimates.

The literature on Bayesian parameters estimation is rich and diverse, reflecting advancements across various fields and methodologies. Santos et al. [40] provided a comprehensive exploration of Bayesian estimation techniques applied to decay parameters within Hawkes processes, showcasing innovative approaches in modeling temporal events. In a complementary vein, Han [41] offered a multifaceted examination of hierarchical Bayesian estimation and presented a range of perspectives that deepen the understanding of parameter estimation complexities in different contexts. Vaglio et al. [42] extended Bayesian techniques to astrophysical phenomena, specifically in the parameter estimation of boson–star binary signals, incorporating advanced models like coherent inspiral templates and spin-dependent quadrupolar corrections. Meanwhile, Bangsgaard et al. [43] addressed Bayesian parameter estimation in the realm of medical science, focusing on phosphate dynamics during hemodialysis. Their work underscores the versatility of Bayesian methods in handling complex biological systems and highlights the method’s applicability beyond traditional statistical fields. Collectively, these studies illustrate the broad applicability and continuous evolution of Bayesian estimation methods across diverse scientific disciplines. In this section, we introduce the posterior density function for the OGE-IWD using samples from the PFFC model. We then employ Bayesian inference techniques to derive estimates for parameters,

Ω = (α, β, λ)

as well as

S (t)

h (t),

and

r (t)

, considering both symmetric and asymmetric loss functions.

3.1. Prior and Posterior Distributions

In Bayes’s estimation, the initial step involves assessing how the prior distribution affects the estimation. To do this, we use a gamma prior distribution for our analysis. Gamma prior distribution is often used as a prior for parameters that are positive and continuous, particularly in the context of modeling rate parameters in Poisson processes or the precision of normal distributions. The gamma distribution is characterized by two parameters, shape and scale, which together control the distribution’s form. It is particularly useful because it is conjugate to the exponential likelihoods, meaning that when used as a prior in these models, the posterior distribution will also be gamma-distributed. This conjugacy simplifies the process of updating the prior with new data, leading to straightforward analytical solutions. Additionally, the gamma distribution’s flexibility in adjusting its shape and scale makes it a versatile tool in Bayesian inference, allowing practitioners to encode various degrees of prior belief about the parameter’s likely values. This combination of practical computational benefits and flexibility in modeling prior knowledge makes the gamma distribution a preferred choice in Bayesian analysis. In this scenario, it is assumed that the parameters

α

β

, and

λ

are mutually independent, and each adheres to a gamma distribution:

π_{1} (α) \propto α^{a_{1} - 1} exp [- b_{1} α], π_{2} (β) \propto β^{a_{2} - 1} exp [- b_{2} β], π_{3} (λ) \propto λ^{a_{3} - 1} exp [- b_{3} λ]; α, β, λ > 0 .

(31)

In this context, it is assumed that each hyper-parameter, represented as

a_{i}, b_{i} > 0

for

i = 1, 2, 3

, is known and non-negative. Based on this assumption, a specific prior distribution is proposed. As a result, the joint prior distribution of

(α, β, λ)

is defined as follows:

π (α, β, λ) \propto α^{a_{1} - 1} β^{a_{2} - 1} λ^{a_{3} - 1} exp [- b_{1} α - b_{2} β - b_{3} λ] .

(32)

By integrating the prior information detailed in Equation (32) with the likelihood function outlined in Equation (7), Bayes’ theorem enables us to formulate the joint posterior distribution as follows:

\begin{matrix} π^{*} (α, β, λ | \underset{̲}{x}) = \frac{L \times π (α, β, λ)}{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} L \times π (α, β, λ) d α d β d λ}, \\ \propto α^{m + a_{1} - 1} β^{m + a_{2} - 1} λ^{m + a_{3} - 1} exp [- b_{1} α - b_{2} β - b_{3} λ] \\ \times \prod_{i = 1}^{m} (x_{i}^{- α - 1} exp [- β x_{i}^{- α} - \frac{k λ (R_{i} + 1)}{exp [β x_{i}^{- α}] - 1}] {(1 - exp [- β x_{i}^{- α}])}^{- 2}) . \end{matrix}

(33)

The Bayes estimate for any function of

α

β

, and

λ

, denoted as

η (α, β, λ)

, under the squared error (SE), linear exponential (LINEX), and general entropy (GE) loss functions, is given by the following expressions (see Varian [44] Calabria and Pulcini [45]):

{\hat{η}}_{S E} = \frac{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} η (α, β, λ) π^{*} (α, β, λ | \underset{̲}{x}) d α d β d λ}{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} π^{*} (α, β, λ | \underset{̲}{x}) d α d β d λ},

(34)

{\hat{η}}_{L I N E X} = \frac{- 1}{ω} log [\frac{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} exp [- ω η (α, β, λ)] π^{*} (α, β, λ | \underset{̲}{x}) d α d β d λ}{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} π^{*} (α, β, λ | \underset{̲}{x}) d α d β d λ}],

(35)

and

{\hat{η}}_{G E} = {[\frac{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} {(η (α, β, λ))}^{- ν} π^{*} (α, β, λ | \underset{̲}{x}) d α d β d λ}{\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} π^{*} (α, β, λ | \underset{̲}{x}) d α d β d λ}]}^{\frac{- 1}{ν}} .

(36)

Since the Bayes estimators cannot be explicitly derived for the SE, LINEX, or GE loss functions, it is necessary to solve them numerically. Therefore, we suggest using the Markov-chain Monte Carlo (MCMC) technique to compute the Bayes estimators for

α

β

, and

λ

3.2. MCMC Techniques

MCMC techniques are essential in Bayesian estimation for approximating complex posterior distributions. At the heart of these techniques is the idea of constructing a Markov chain that converges to the desired distribution, allowing us to estimate it by sampling. Key types include Metropolis–Hastings (MH), which proposes new states based on a proposal distribution and accepts them with a probability that ensures convergence, and Gibbs sampling (GS), which updates one variable at a time while keeping others fixed, suitable for models with conditional distributions. Both techniques showcase MCMC’s flexibility in handling various problem structures and data types, as detailed in the studies by Geman and Geman [46], Metropolis et al. [47], and Hastings [48]. Additional advancements encompass the Hamiltonian Monte Carlo (HMC), which uses gradient information to propose samples and is effective for high-dimensional spaces. Lastly, importance sampling (IMS) is a technique used to estimate the properties of a particular distribution while sampling from a different one. The core concept involves selecting samples from a proposal distribution. This is achieved by calculating a weight for each sample based on the ratio of the target distribution’s density to the proposal distribution’s density. These methods collectively provide powerful tools to delve into Bayesian models and obtain ultimate insights into parameter estimates. At the outset, the fully conditional posterior distributions for the parameters

α

β

, and

λ

can be expressed in the following way:

\begin{matrix} π_{1}^{*} (α | β, λ, \underset{̲}{x}) \propto α^{m + a_{1} - 1} exp [- b_{1} α] \\ \times \prod_{i = 1}^{m} (x_{i}^{- α} exp [- β x_{i}^{- α} - \frac{k λ (R_{i} + 1)}{exp [β x_{i}^{- α}] - 1}] {(1 - exp [- β x_{i}^{- α}])}^{- 2}), \end{matrix}

(37)

π_{2}^{*} (β | α, λ, \underset{̲}{x}) \propto β^{m + a_{2} - 1} exp [- b_{2} β] \underset{i = 1}{\prod^{m}} (exp [- β x_{i}^{- α} - \frac{k λ (R_{i} + 1)}{exp [β x_{i}^{- α}] - 1}] {(1 - exp [- β x_{i}^{- α}])}^{- 2}),

(38)

and

π_{3}^{*} (λ | α, β, \underset{̲}{x}) \propto λ^{m + a_{3} - 1} exp [- λ (b_{3} + \frac{k (R_{i} + 1)}{exp [β x_{i}^{- α}] - 1})] .

(39)

Equation (39) conforms to a gamma distribution, allowing for the easy sampling of

λ

with any gamma-generating routine. In contrast, Equations (37) and (38) do not fit standard distributions, requiring MCMC methods for sampling. The algorithm will utilize GS and the MH algorithm in sequence to produce samples from the posterior distribution by using a proposal density. The algorithm advances through the following phases:

1.

Establish the starting values for the parameters

(α^{(0)}, β^{(0)}, λ^{(0)}) = (\hat{α}, \hat{β}, \hat{λ})

and initialize

i = 1

2.

Generate

λ^{(i)}

from gamma distribution

π_{3}^{*} (λ | α, β, \underset{̲}{x})

3.

Employ the following MH algorithm to generate

α^{(i)}

from

π_{1}^{*}

(α^{(i - 1)} | β^{(i - 1)},

λ^{(i)}, \underset{̲}{x})

and

β^{(i)}

from

π_{2}^{*}

(β^{(i - 1)} | α^{(i - 1)}, λ^{(i)}, \underset{̲}{x})

, using normal proposal distributions

N (α^{(i - 1)}, V a r (\hat{α}))

and

N (β^{(i - 1)}, V a r (\hat{β}))

, respectively. Execute the following tasks:

(a): Generate a proposal $α^{*}$ from $N (α^{(i - 1)}, V a r (\hat{α}))$ and $β^{*}$ from $N (β^{(i - 1)}, V a r (\hat{β}))$ .
(b): Assess the acceptance probabilities:

$ζ_{α} = min (1, \frac{π_{1}^{*} (α^{*} | β^{(i - 1)}, λ^{(i)}, \underset{̲}{x})}{π_{1}^{*} (α^{(i - 1)} | β^{(i - 1)}, λ^{(i)}, \underset{̲}{x})}), ζ_{β} = min (1, \frac{π_{2}^{*} (β^{*} | α^{(i)}, λ^{(i)}, \underset{̲}{x})}{π_{3}^{*} (β^{(i - 1)} | α^{(i)}, λ^{(i)}, \underset{̲}{x})}) .$
(c): Obtain values $ρ_{1}$ and $ρ_{2}$ from a iniform $(0, 1)$ distribution.
(d): If $ρ_{1} ⩽ ζ_{α}$ , accept the proposal and set $α^{(i)} = α^{*}$ . Otherwise, retain the previous value by setting $α^{(i)} = α^{(i - 1)}$ .
(e): If $ρ_{2} ⩽ ζ_{β}$ , accept the proposal and set $β^{(i)} = β^{*}$ . Otherwise, retain the previous value by setting $β^{(i)} = β^{(i - 1)}$ .

4.

For a given t, compute the survival, hazard rate, and the inverse hazard rate functions:

S^{(i)} (t) = exp [- \frac{λ^{(i)}}{exp [β^{(i)} t^{- α^{(i)}}] - 1}], t > 0,

h^{(i)} (t) = α^{(i)} β^{(i)} λ^{(i)} t^{- (α^{(i)} + 1)} exp [- β^{(i)} t^{- α^{(i)}}] {(1 - exp [- β^{(i)} t^{- α^{(i)}}])}^{- 2}, t > 0,

and

r^{(i)} (t) = \frac{α^{(i)} β^{(i)} λ^{(i)} t^{- (α^{(i)} + 1)} exp [- β^{(i)} t^{- α^{(i)}}] {(1 - exp [- β^{(i)} t^{- α^{(i)}}])}^{- 2}}{exp [\frac{λ^{(i)}}{exp [β^{(i)} t^{- α^{(i)}}] - 1}] - 1}, t > 0 .

5.

Set

i = i + 1

6.

Repeat steps

(2 - 4)

M times to collect the required number of samples. After removing the initial

M_{0}

burn-in samples, use the remaining

M - M_{0}

samples to compute the Bayesian estimates.

7.

It is now feasible to compute the Bayes estimate of

η = η (α, β, λ)

for the SE, LINEX, and GE loss functions as follows:

{\hat{η}}_{S E} = \frac{1}{M - M_{0}} \sum_{i = M_{0} + 1}^{M} η (α^{(i)}, β^{(i)}, λ^{(i)}),

{\hat{η}}_{L I N E X} = \frac{- 1}{ω} log [\frac{1}{M - M_{0}} \sum_{i = M_{0} + 1}^{M} exp [- ω η (α^{(i)}, β^{(i)}, λ^{(i)})]],

and

{\hat{η}}_{G E} = {[\frac{1}{M - M_{0}} \sum_{i = M_{0} + 1}^{M} {(η (α^{(i)}, β^{(i)}, λ^{(i)}))}^{- ν}]}^{\frac{- 1}{ν}},

where

η = η (α, β, λ)

represents the parameters

α

β

λ

S (t)

h (t)

and

r (t)

8.

Arrange

η^{(M_{0} + 1)}, η^{(M_{0} + 2)}, \dots, η^{(M)}

in ascending order as

η_{[1]} < η_{[2]} < \dots < η_{[M]}

. Accordingly,

(η_{[(M - M_{0}) \frac{γ}{2}]}, η_{[(M - M_{0}) (1 - \frac{γ}{2})]}),

provide the

100 (1 - γ) %

Bayesian CRIs for

η

4. Simulation Study

Simulation studies is essential for comparing and evaluating different estimation methods because they provide a versatile and controlled framework for analyzing their performance across a variety of scenarios. By simulating a wide range of conditions and generating synthetic data, researchers can test how well different estimation techniques perform when faced with known benchmarks and varying complexities. This approach allows for a detailed assessment of each method’s accuracy, reliability, and computational efficiency, offering insights into their strengths and weaknesses. Consequently, simulations help identify the most effective estimation strategies, refine existing methods, and ensure that the chosen techniques are robust and applicable to real-world problems. Through this iterative process, simulations enhance the overall quality and credibility of estimation practices. In this section, we assess the performance of the proposed Bayes estimators in comparison with the MLEs through a Monte Carlo simulation study. This study involves testing various combinations of sample sizes (n), numbers of groups (m), and censoring schemes (R) with different values of

R_{i}

. We use the algorithm developed by Balakrishnan and Sandhu [49], applying the CDF

1 - {(1 - F (x))}^{k}

where k is the same number of units within each group, to generate a set of PFFC samples from the OGE-IWD with parameters

(α, β, λ)

set to

(1.75, 0.5, 1)

. We then compute the true values of

S (t)

h (t)

and

r (t)

at time

t = 0.3

, obtaining results

0.9835

0.4061

and

24.1811

. The performance of the estimators is evaluated using mean square error (MSE), calculated as

M S E = \frac{1}{N} \sum_{i = 1}^{N} {({\hat{μ}}_{j}^{(i)} - μ_{j})}^{2}

, where

N = 1000

, with

μ_{j} = α

β

λ

S (t)

h (t)

r (t)

j = 1, 2, \dots, 6

for the point estimates. Additionally, we examine the average confidence interval widths (ACIWs)/credible intervals (CRIWs), along with their coverage probabilities (CPs), for both asymptotic and the highest posterior density interval estimates. The Bayes estimates and CRIs are derived from M = 12,000 MCMC samples, discarding the initial

M_{0} = 2000

as “burn-in”. Informative gamma priors are used for

α

β

, and

λ

, with hyper-parameters

a_{1} = 1

b_{1} = 2, a_{2} = 2

b_{2} = 3

a_{3} = 2

and

b_{3} = 2

. Furthermore,

95 %

CRIs are calculated for each simulation, considering two different group sizes,

k = 3

and

k = 5

, and various censoring schemes (CS): CSI:

R_{1} = n - m,

R_{i} = 0,

for

i \neq 1

. CSII:

R_{\frac{m}{2}} = R_{\frac{m}{2} + 1} = \frac{n - m}{2}, R_{i} = 0

for

i \neq \frac{m}{2}

and

i \neq \frac{m}{2} + 1

if m even;

R_{\frac{m + 1}{2}} = n - m, R_{i} = 0

for

i \neq \frac{m + 1}{2}

if m odd. Finally, CSIII:

R_{m} = n - m,

R_{i} = 0

for

i \neq m

. This means that, if (n,m) = (30,15), then CSI

= (15, 0^{14}),

CSII

= (0^{7}, 15, 0^{7})

and CSIII

= (0^{14}, 15)

. If (n,m) = (30,20), then CSI

= (10, 0^{19}),

CSII

= (0^{9}, 5, 5, 0^{9})

and CSIII

= (0^{19}, 10),

such that,

0^{5}

= (0,0,0,0,0). The outcomes are detailed in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. Based on these results, several notable observations can be made:

1.: As anticipated from Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, increasing the sample sizes $(n, m)$ results in a reduction in MSEs and ACIWs/CRIWs.
2.: Among the various methods, Bayes estimates exhibit the smallest MSEs and AWs for the parameters $α$ , $β$ , $λ$ , $S (t)$ , $h (t)$ and $r (t)$ , indicating superior performance compared to MLEs.
3.: The Bayes estimate utilizing the GE loss function with $ω = 2$ delivers more accurate estimates for $α$ , $β$ , $λ$ , $S (t)$ , $h (t)$ and $r (t)$ due to its smaller MSEs.
4.: When using the LINEX and GE loss function with $ω = 2$ , Bayes estimates outperform those obtained with $ω = - 2$ , as they yield smaller MSEs.
5.: For the SE loss function, Bayes estimates for $α$ , $β$ and $λ$ are superior to those derived under the LINEX loss function with $ω = - 2$ , demonstrated by their smaller MSEs.
6.: Bayes estimates for $S (t)$ , $h (t)$ and $r (t)$ under the LINEX loss function with $ω = - 2$ show better performance compared to those under the SE loss function, as evidenced by their smaller MSEs.
7.: Analysis of all tables reveals that increasing the group size k leads to higher MSEs and AWs for $α$ , $β$ , $λ$ , $S (t)$ , $h (t)$ and $r (t)$ .
8.: Given fixed sample sizes and observed failures, the CSI scheme proves to be the most effective, offering the smallest MSEs and AWs.
9.: In general, all the point estimates are completely effective because the average biases are very small. Average bias tends to zero when n and m increase.
10.: The MLE and Bayesian methods provide very similar estimates, with both having ACIs with high CPs of around $0.95$ . However, Bayesian CRIs achieve the highest CPs.

5. Application to Real-Life Data

The practical application of theoretical studies using real data is essential for ensuring that abstract concepts are grounded in reality and truly effective. Theories provide the foundational principles and frameworks, but real data reveal how these theories function in actual situations. This application allows for the testing and refinement of theoretical models, making them more accurate and relevant. In this section, to illustrate the inference methods covered earlier, we use real-life data on bladder cancer patients. Specifically, we analyze the dataset provided by Lee and Wang [50], which was also utilized in Hassan et al. [1]. This dataset includes remission times, measured in months, for a random sample of 128 bladder cancer patients.

To evaluate the goodness of fit, we calculated the K-S distance between the empirical distribution and the fitted distribution functions, which turned out to be

0.05914955

, with a corresponding p-value of

0.7617021

in the presence of the estimated parameters

(\hat{α}, \hat{β}, \hat{λ}) = (0.8699, 0.3355, 0.05653)

. This p-value is the highest among the values obtained for these data, indicating that the OGE-IWD model provides the best fit. The empirical, Q-Q, TTT, SF, PDF and P-P plots displayed in Figure 4 further confirm that the OGE-IWD model aligns closely with the data. The data are randomly partitioned into 64 groups, each containing 2 items. The groups are organized as follows: {(0.08, 0.31), (0.19, 0.39), (0.2, 0.4), (0.51, 0.52), (0.73, 0.81), (0.22, 0.5), (1.76, 2.02), (2.26, 2.64), (0.26, 0.4), (0.62, 0.66), (0.82, 1.35), (0.96, 2.02), (0.9, 1.46), (1.05, 2.07), (2.69, 2.75), (2.69, 2.83), (2.87, 3.02), (1.26, 2.09), (3.88, 4.18), (2.33, 3.25), (3.7, 4.23), (3.48, 3.57), (2.46, 3.46), (5.41, 7.28), (5.41, 8.66), (5.49, 7.93), (5.62, 12.63), (2.54, 4.33), (3.28, 4.26), (26.31, 32.15), (3.36, 4.34), (3.36, 4.4), (4.5, 5.32), (4.87, 5.34), (4.98, 5.32), (4.51, 5.17), (5.09, 23.63), (5.71, 11.98), (7.32, 9.74), (5.06, 7.59), (12.07, 13.11), (5.85, 7.62), (6.25, 7.63), (6.39, 7.87), (9.02, 9.47), (6.54, 10.06), (14.77, 16.62), (6.76, 10.34), (17.36, 18.1), (6.94, 10.66), (10.75, 11.25), ( 6.97, 11.64), (7.09, 11.79), (14.76, 17.12), (7.26, 12.03), (8.37, 13.29), (8.53, 13.8), (19.13, 22.69), (8.65, 14.38), (12.02, 17.14), (34.26, 43.01), (14.24, 46.12), (20.28, 25.74), (36.66, 79.05)}. A PFFC sample, denoted as

x^{(1)},

is generated as follows: the original data are divided into

n = 64

groups, with each group containing

k = 2

items. The resulting groups are organized according to

R =

(2, 1, 2, 1, 1, 1, 0, 2, 3, 0, 2, 0, 2, 0, 2, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3), where 34 first-failure times are recorded

(m = 34)

and 30 groups are excluded

(n - m = 30)

. Consequently, the resulting PFFC sample is

x^{(1)} =

{0.08, 0.19, 0.2, 0.22, 0.26, 0.96, 1.05, 1.26, 2.33, 2.46, 2.54, 3.28, 3.36, 3.36, 4.87, 4.98, 5.06, 5.09, 5.85, 6.25, 6.39, 6.54, 6.76, 6.94, 6.97, 7.09, 7.26, 8.37, 8.53, 8.65, 12.02, 14.24, 20.28, 36.66}. Based on the data

x^{(1)}

, the MLEs and

95 %

ACIs for

α

β

λ

S (t)

h (t)

and

r (t)

are listed in Table 10 and Table 11. To obtain Bayesian estimates, we need to define prior distributions for these parameters. In the absence of prior information, we use non-informative gamma priors for

α

β

and

λ

, with hyper-parameters

a_{i} = 0.0001

and

b_{i} = 0.0001

i = 1, 2, 3

. The posterior analysis was carried out using the MCMC technique, specifically combining the MH algorithm with the GS. As outlined in Section 3, the MLEs were used as initial values for

α

β

and

λ

, and 12,000 MCMC samples were generated, with the first 2000 samples discarded as “burn-in” to mitigate initial value effects. Table 10 and Table 12 present the Bayesian estimates and

95 %

CRIs for

α

β

λ

S (t)

h (t)

and

r (t)

. From the results presented in Table 10 and Table 11, we can make the following observations: The model performs exceptionally well in fitting the data, as evident from the figures. The estimates from the two methods are quite similar, with only minor differences, which provides a favorable impression. The approximate confidence intervals are appropriate, with all point estimates falling within them. Additionally, there are minor variations in the interval widths, as anticipated.

6. Conclusions

In this study, we developed two distinct methods using a PFFC scheme to estimate the unknown parameters of the OGE-IWD. We utilized the Fisher information matrix to create approximate confidence intervals (ACIs) for

α

β

, and

λ

, while employing the delta method to compute ACIs for SF, HRF, and IHRF. The complexity of the posterior distribution equations for these parameters, particularly in the context of Bayesian estimates, made analytical reductions challenging. To address this, we used MCMC techniques to calculate Bayesian estimators for the SE, LINEX, and GE loss functions. The study began by comparing various methodologies in a simulated environment, concluding that the Bayes method is effective for estimating and constructing ACIs for unknown parameters with PFFC data from the OGE-IWD. The MCMC algorithm showed better performance than the MLE method. Applying the OGE-IWD to real-world medical data demonstrated its ability to model current data accurately, suggesting its potential for similar applications in the medical field. The study also identifies future research directions, including optimizing censoring schemes and expanding statistical inference methods for accelerated life testing models with multiple failure factors.

One critical aspect that requires further examination is the sample size necessary to achieve desirable statistical, such as property unbiasedness, consistency, and asymptotic normality of the estimators. In MLE, these properties generally hold asymptotically as the sample size approaches infinity. However, for practical applications, particularly with progressive first-failure censoring, finite-sample performance may vary significantly depending on the sample size and censoring scheme. Smaller samples can lead to biased or inefficient estimators, and confidence intervals may exhibit lower-than-expected coverage probabilities.

For the proposed model, the non-closed-form nature of MLEs and the use of iterative methods like Newton–Raphson introduce further dependency on sample size for convergence and stability. To ensure unbiasedness and consistency, it is crucial to determine an appropriate minimum sample size, particularly as the censoring level increases. In progressive censoring schemes, fewer observed failures can reduce the effective sample size, impacting the asymptotic properties and potentially leading to biased estimates.

Additionally, for Bayesian estimation, small sample sizes can lead to posterior distributions that are overly influenced by the prior distributions. When informative priors are employed, as with the gamma priors used in this study, the posterior estimates might exhibit significant shrinkage towards the prior mean. Monte Carlo simulations could be utilized to investigate how sample size affects Bayesian point estimates and credible interval coverage rates under different censoring levels, providing practical guidance on sample size requirements for reliable inference.

Author Contributions

Conceptualization, R.M.E.-S. and M.M.R.; methodology, A.A.-E.-M. and R.M.E.-S.; software, M.M.R., A.A.-E.-M. and R.M.E.-S.; validation, R.M.E.-S., M.M.R. and A.A.-E.-M.; formal analysis, A.A.-E.-M.; investigation, M.M.R. and R.M.E.-S.; resources, M.M.R.; data curation, R.M.E.-S.; writing-original draft preparation, A.A.-E.-M.; writing-review and editing, R.M.E.-S.; visualization, M.M.R.; supervision, R.M.E.-S. and M.M.R.; project administration, R.M.E.-S.; funding acquisition, A.A.-E.-M., M.M.R. and R.M.E.-S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data that support the findings of this study are available within the article.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. PDF for OGE-IWD.

Figure 2. HRF for OGE-IWD.

Figure 3. Description of the PFFC scheme.

Figure 4. The KD, box, TTT, Q-Q, P-P, SF, PDF, and violin plots for the data set.

Table 1. RMSE (first row) and bias (second row) of estimates for the parameter

α

Table 1. RMSE (first row) and bias (second row) of estimates for the parameter

α

$(k, n, m)$	CS	MLE	Bayes
			SE	LINEX		GE
				$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$(3, 30, 15)$	CSI	0.5956	0.5809	0.5906	0.5616	0.5748	0.5277
		0.0250	0.0134	0.0146	0.0129	0.0132	0.0114
	CSII	0.6129	0.5925	0.6020	0.5752	0.5993	0.5384
		−0.0199	0.0174	0.0186	0.0156	0.0163	0.0142
	CSIII	0.6353	0.6136	0.6209	0.6001	0.6160	0.5544
		−0.0176	0.0187	0.0203	0.0165	0.0177	0.0158
$(3, 30, 20)$	CSI	0.5446	0.5239	0.5332	0.5047	0.5152	0.4860
		−0.0021	0.0022	0.0023	0.0020	0.0023	0.0019
	CSII	0.5705	0.5501	0.5716	0.5351	0.5616	0.5047
		0.0029	0.0027	0.0029	0.0024	0.0028	0.0021
	CSIII	0.6039	0.588	0.6045	0.5476	0.5967	0.5288
		0.0041	0.0039	0.0040	0.0036	0.0038	0.0026
$(3, 50, 35)$	CSI	0.4742	0.447	0.4595	0.4174	0.4532	0.3919
		0.0016	0.0015	0.0016	0.0013	0.0014	0.0009
	CSII	0.5057	0.4715	0.4999	0.4422	0.4851	0.4096
		−0.0253	0.0022	0.0022	0.0023	0.0022	0.0018
	CSIII	0.5354	0.5063	0.5147	0.4691	0.5261	0.4423
		−0.0278	0.0025	0.0024	0.0021	0.0023	0.0014
$(5, 30, 15)$	CSI	0.6652	0.6508	0.6605	0.6304	0.6554	0.5997
		−0.0283	0.0236	0.0265	0.0225	0.0246	0.0174
	CSII	0.6966	0.6811	0.6905	0.6627	0.6855	0.6322
		−0.0235	0.0250	0.0271	0.0242	0.0259	0.0189
	CSIII	0.7174	0.7105	0.7244	0.6864	0.7164	0.6495
		−0.0286	0.0263	0.0295	0.0255	0.0271	0.0194
$(5, 30, 20)$	CSI	0.6353	0.6170	0.6296	0.5956	0.6208	0.5471
		−0.0334	0.0321	0.0354	0.0314	0.0325	0.0264
	CSII	0.6724	0.6608	0.6707	0.6263	0.6660	0.6065
		−0.0314	0.0387	0.0390	0.0368	0.0354	0.0297
	CSIII	0.6978	0.6810	0.6893	0.6643	0.6855	0.6244
		0.0456	0.0442	0.0453	0.0412	0.0428	0.0355
$(5, 50, 35)$	CSI	0.5610	0.5454	0.5744	0.5222	0.5610	0.4616
		−0.0019	0.0018	0.0019	0.0014	0.0013	0.0007
	CSII	0.5956	0.5803	0.5915	0.5445	0.5897	0.4911
		−0.0017	0.0021	0.0023	0.0020	0.0021	0.0008
	CSIII	0.6279	0.6093	0.6217	0.5784	0.6163	0.5471
		0.0013	0.0012	0.012	0.0010	0.0011	0.0009

Table 2. MSE of estimates for the parameter

α

Table 2. MSE of estimates for the parameter

α

$(k, n, m)$	CS	MLE	Bayes
			SE	LINEX		GE
				$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$(3, 30, 15)$	CSI	0.3547	0.3375	0.3488	0.3154	0.3304	0.2785
	CSII	0.3756	0.3511	0.3624	0.3308	0.3592	0.2899
	CSIII	0.4036	0.3765	0.3855	0.3601	0.3795	0.3074
$(3, 30, 20)$	CSI	0.2966	0.2745	0.2843	0.2547	0.2654	0.2362
	CSII	0.3255	0.3026	0.3267	0.2863	0.3154	0.2547
	CSIII	0.3647	0.3457	0.3654	0.2999	0.3561	0.2796
$(3, 50, 35)$	CSI	0.2249	0.1998	0.2111	0.1742	0.2054	0.1536
	CSII	0.2557	0.2223	0.2499	0.1955	0.2353	0.1678
	CSIII	0.2866	0.2563	0.2649	0.2201	0.2768	0.1956
$(5, 30, 15)$	CSI	0.4425	0.4236	0.4362	0.3974	0.4295	0.3596
	CSII	0.4852	0.4639	0.4768	0.4392	0.4699	0.3997
	CSIII	0.5147	0.5048	0.5247	0.4712	0.5133	0.4219
$(5, 30, 20)$	CSI	0.4036	0.3807	0.3964	0.3547	0.3854	0.2993
	CSII	0.4521	0.4366	0.4499	0.3922	0.4436	0.3678
	CSIII	0.4869	0.4637	0.4752	0.4413	0.4699	0.3899
$(5, 50, 35)$	CSI	0.3147	0.2975	0.3299	0.2727	0.3147	0.2131
	CSII	0.3547	0.3368	0.3499	0.2965	0.3478	0.2412
	CSIII	0.3942	0.3713	0.3865	0.3345	0.3798	0.2993

Table 3. MSE of estimates for the parameter

β

Table 3. MSE of estimates for the parameter

β

$(k, n, m)$	CS	MLE	Bayes
			SE	LINEX		GE
				$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$(3, 30, 15)$	CSI	0.0293	0.0274	0.0289	0.0235	0.0282	0.0199
	CSII	0.0346	0.0325	0.0338	0.0264	0.0331	0.0225
	CSIII	0.0415	0.0388	0.0401	0.0296	0.0395	0.0257
$(3, 30, 20)$	CSI	0.0223	0.0204	0.0217	0.0187	0.0214	0.0154
	CSII	0.0264	0.0245	0.0260	0.0222	0.0253	0.0186
	CSIII	0.0313	0.0296	0.0311	0.0254	0.0302	0.0213
$(3, 50, 35)$	CSI	0.0164	0.0145	0.0160	0.0129	0.0159	0.0099
	CSII	0.0202	0.0183	0.0195	0.0155	0.0191	0.0127
	CSIII	0.0251	0.0233	0.0249	0.0185	0.0242	0.0149
$(5, 30, 15)$	CSI	0.0353	0.0332	0.0341	0.0277	0.0339	0.0241
	CSII	0.0391	0.0375	0.0387	0.0335	0.0381	0.0296
	CSIII	0.0442	0.0425	0.0432	0.0376	0.0429	0.0332
$(5, 30, 20)$	CSI	0.0292	0.0273	0.0283	0.0225	0.0278	0.0189
	CSII	0.0331	0.0315	0.0326	0.0257	0.0321	0.0231
	CSIII	0.0367	0.0351	0.0362	0.0285	0.0357	0.0259
$(5, 50, 35)$	CSI	0.0234	0.0215	0.0234	0.0176	0.0225	0.0139
	CSII	0.0285	0.0264	0.0279	0.0223	0.0271	0.0187
	CSIII	0.0334	0.0321	0.0331	0.0267	0.0328	0.0235

Table 4. MSE of estimates for the parameter

λ

Table 4. MSE of estimates for the parameter

λ

$(k, n, m)$	CS	MLE	Bayes
			SE	LINEX		GE
				$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$(3, 30, 15)$	CSI	0.1825	0.1597	0.1693	0.1355	0.1622	0.1176
	CSII	0.2354	0.2143	0.2261	0.1524	0.2199	0.1369
	CSIII	0.2677	0.2581	0.2652	0.1744	0.2607	0.1532
$(3, 30, 20)$	CSI	0.1163	0.0956	0.1123	0.0846	0.1069	0.0723
	CSII	0.1456	0.1178	0.1357	0.0936	0.1243	0.0855
	CSIII	0.1932	0.1699	0.1756	0.1235	0.1722	0.1062
$(3, 50, 35)$	CSI	0.0986	0.0775	0.0869	0.0734	0.0824	0.0593
	CSII	0.1235	0.0992	0.0111	0.0846	0.0105	0.0692
	CSIII	0.1577	0.1347	0.1453	0.1067	0.1422	0.0899
$(5, 30, 15)$	CSI	0.2546	0.2348	0.2453	0.1911	0.2398	0.1637
	CSII	0.2865	0.2645	0.2733	0.2299	0.2675	0.1894
	CSIII	0.3155	0.2943	0.3057	0.2654	0.2956	0.2278
$(5, 30, 20)$	CSI	0.1736	0.1564	0.1635	0.1157	0.1598	0.0969
	CSII	0.2239	0.2077	0.1911	0.1356	0.1863	0.1175
	CSIII	0.2536	0.2394	0.2456	0.1752	0.2431	0.1406
$(5, 50, 35)$	CSI	0.1235	0.1155	0.1237	0.0975	0.1196	0.0786
	CSII	0.1536	0.1463	0.1501	0.1237	0.1498	0.0991
	CSIII	0.1837	0.1694	0.1732	0.1423	0.1702	0.1195

Table 5. MSE of estimates for the parameter

α

Table 5. MSE of estimates for the parameter

α

$(k, n, m)$	CS	MLE	Bayes
			SE	LINEX		GE
				$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$(3, 30, 15)$	CSI	0.0225	0.0205	0.0209	0.0097	0.0198	0.0092
	CSII	0.0268	0.0243	0.0257	0.0112	0.0235	0.0099
	CSIII	0.0312	0.0276	0.0286	0.0145	0.2647	0.0115
$(3, 30, 20)$	CSI	0.0175	0.0156	0.0167	0.0079	0.0145	0.0068
	CSII	0.0213	0.0204	0.0211	0.0085	0.0194	0.0074
	CSIII	0.0254	0.0239	0.0246	0.0102	0.0231	0.0087
$(3, 50, 35)$	CSI	0.0116	0.0107	0.0111	0.0068	0.0099	0.0058
	CSII	0.0147	0.0129	0.0132	0.0076	0.0125	0.0066
	CSIII	0.0205	0.0195	0.0199	0.0083	0.0187	0.0075
$(5, 30, 15)$	CSI	0.0273	0.0255	0.0258	0.0117	0.0245	0.0109
	CSII	0.0324	0.0305	0.0309	0.0159	0.0295	0.0134
	CSIII	0.0367	0.0351	0.0367	0.0178	0.0346	0.0156
$(5, 30, 20)$	CSI	0.0213	0.0202	0.0207	0.0093	0.0195	0.0082
	CSII	0.0276	0.0259	0.0261	0.0099	0.0256	0.0089
	CSIII	0.0347	0.0327	0.0332	0.0117	0.0325	0.0096
$(5, 50, 35)$	CSI	0.0169	0.0148	0.0156	0.0088	0.0145	0.0077
	CSII	0.0198	0.0182	0.0189	0.0094	0.0175	0.0083
	CSIII	0.0235	0.0214	0.0227	0.0105	0.0205	0.0095

Table 6. MSE of estimates for the parameter

h (t)

Table 6. MSE of estimates for the parameter

h (t)

$(k, n, m)$	CS	MLE	Bayes
			SE	LINEX		GE
				$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$(3, 30, 15)$	CSI	0.0321	0.0097	0.0107	0.0088	0.0091	0.0073
	CSII	0.0435	0.0109	0.0135	0.0092	0.0098	0.0079
	CSIII	0.0512	0.0123	0.0154	0.0106	0.0110	0.0084
$(3, 30, 20)$	CSI	0.0275	0.0091	0.0094	0.0081	0.0084	0.0067
	CSII	0.0346	0.0096	0.0103	0.0086	0.0093	0.0072
	CSIII	0.0429	0.0111	0.0124	0.0093	0.0098	0.0079
$(3, 50, 35)$	CSI	0.0198	0.0083	0.0086	0.0065	0.0075	0.0058
	CSII	0.0236	0.0087	0.0092	0.0073	0.0081	0.0064
	CSIII	0.0319	0.0094	0.0105	0.0084	0.0089	0.0071
$(5, 30, 15)$	CSI	0.0521	0.0124	0.0137	0.0096	0.0112	0.0085
	CSII	0.0583	0.0147	0.0153	0.0111	0.0135	0.0094
	CSIII	0.0622	0.0179	0.0185	0.0138	0.0165	0.0109
$(5, 30, 20)$	CSI	0.0446	0.0097	0.0101	0.0087	0.0092	0.0078
	CSII	0.0492	0.0113	0.0125	0.0093	0.0099	0.0084
	CSIII	0.0536	0.0128	0.0139	0.0098	0.0117	0.0089
$(5, 50, 35)$	CSI	0.0255	0.0085	0.0089	0.0075	0.0081	0.0068
	CSII	0.0376	0.0093	0.0097	0.0081	0.0087	0.0076
	CSIII	0.0427	0.0105	0.0119	0.0094	0.0093	0.0082

Table 7. MSE of estimates for the parameter

r (t)

Table 7. MSE of estimates for the parameter

r (t)

$(k, n, m)$	CS	MLE	Bayes
			SE	LINEX		GE
				$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$(3, 30, 15)$	CSI	0.9145	0.8841	0.9023	0.7992	0.8755	0.7254
	CSII	0.9542	0.9253	0.9344	0.8257	0.9128	0.7623
	CSIII	0.9993	0.9547	0.9637	0.8644	0.9456	0.7952
$(3, 30, 20)$	CSI	0.8347	0.7962	0.8027	0.7442	0.7834	0.6791
	CSII	0.8725	0.8355	0.8462	0.7725	0.8253	0.7248
	CSIII	0.9166	0.8723	0.8923	0.7999	0.8641	0.7634
$(3, 50, 35)$	CSI	0.7825	0.7481	0.7533	0.6992	0.7354	0.6223
	CSII	0.8264	0.7835	0.7964	0.7296	0.7753	0.6545
	CSIII	0.8736	0.8364	0.8432	0.7542	0.8255	0.7039
$(5, 30, 15)$	CSI	1.2147	1.0849	1.0009	0.8999	1.1556	0.8556
	CSII	1.2965	1.2667	1.2795	0.9536	1.2534	0.9145
	CSIII	1.3462	1.3148	1.3254	0.9978	1.3015	0.9652
$(5, 30, 20)$	CSI	1.0984	0.9782	0.9965	0.8546	0.9547	0.7966
	CSII	1.1568	1.0999	1.1174	0.9147	1.0987	0.8469
	CSIII	1.2547	1.1799	1.1867	0.9648	1.1365	0.8893
$(5, 50, 35)$	CSI	0.9278	0.8865	0.8964	0.7967	0.8795	0.7361
	CSII	0.9645	0.9257	0.9345	0.8342	0.9144	0.7564
	CSIII	1.1326	0.9724	0.9831	0.8824	0.9637	0.7992

Table 8. The AWs and CPs of

95 %

ACIs and CRIs for the parameters

α

and

β

Table 8. The AWs and CPs of

95 %

ACIs and CRIs for the parameters

α

and

β

$(k, n, m)$	CS	$α$				$β$
		MLE		Bayes		MLE		Bayes
		ACIWs	CPs	CRIWs	CPs	ACIWs	CPs	CRIWs	CPs
$(3, 30, 15)$	CSI	1.2654	0.912	0.7936	0.941	0.6635	0.918	0.3523	0.939
	CSII	1.3145	0.925	0.8234	0.942	0.7456	0.915	0.4536	0.941
	CSIII	1.3869	0.919	0.8736	0.939	0.8215	0.922	0.4962	0.938
$(3, 30, 20)$	CSI	1.1652	0.925	0.6645	0.949	0.5863	0.941	0.2987	0.951
	CSII	1.2235	0.918	0.7144	0.951	0.6241	0.939	0.3649	0.952
	CSIII	1.2967	0.922	0.7869	0.947	0.6863	0.938	0.4325	0.949
$(3, 50, 35)$	CSI	0.9984	0.936	0.5763	0.955	0.4496	0.954	0.2147	0.954
	CSII	1.1362	0.941	0.6347	0.949	0.5147	0.947	0.2869	0.957
	CSIII	1.2147	0.939	0.7264	0.951	0.5636	0.952	0.3468	0.959
$(5, 30, 15)$	CSI	1.4666	0.923	0.8563	0.941	0.8345	0.927	0.5524	0.945
	CSII	1.5123	0.921	0.9364	0.950	0.8994	0.925	0.6347	0.937
	CSIII	1.6325	0.931	1.0567	0.939	0.9253	0.919	0.7221	0.941
$(5, 30, 20)$	CSI	1.2954	0.936	0.7236	0.952	0.7236	0.954	0.4362	0.961
	CSII	1.3358	0.942	0.8147	0.947	0.7893	0.942	0.5124	0.942
	CSIII	1.4125	0.941	0.9362	0.955	0.8475	0.938	0.5766	0.948
$(5, 50, 35)$	CSI	1.1647	0.955	0.6472	0.961	0.6625	0.950	0.3654	0.963
	CSII	1.2113	0.949	0.7554	0.952	0.7334	0.949	0.4355	0.954
	CSIII	1.2954	0.951	0.8233	0.955	0.7993	0.947	0.5067	0.955

Table 9. The AWs and CPs of

95 %

ACIs and CRIs for the parameter

λ

and

S (t)

Table 9. The AWs and CPs of

95 %

ACIs and CRIs for the parameter

λ

and

S (t)

$(k, n, m)$	CS	$α$				$β$
		MLE		Bayes		MLE		Bayes
		ACIWs	CPs	CRIWs	CPs	ACIWs	CPs	CRIWs	CPs
$(3, 30, 15)$	CSI	1.8932	0.912	0.8495	0.948	0.4231	0.939	0.2341	0.956
	CSII	1.9968	0.923	0.9635	0.946	0.4763	0.928	0.2536	0.951
	CSIII	2.1345	0.919	1.2145	0.939	0.4963	0.937	0.2847	0.955
$(3, 30, 20)$	CSI	1.6456	0.925	0.7145	0.951	0.3765	0.947	0.1963	0.959
	CSII	1.7473	0.934	0.8566	0.947	0.3954	0.951	0.2245	0.949
	CSIII	1.9745	0.939	0.9932	0.949	0.4165	0.946	0.2568	0.957
$(3, 50, 35)$	CSI	1.5536	0.951	0.6389	0.952	0.2965	0.955	0.1565	0.962
	CSII	1.6477	0.938	0.7456	0.955	0.3215	0.951	0.1848	0.957
	CSIII	1.8632	0.946	0.8763	0.949	0.3647	0.947	0.2269	0.958
$(5, 30, 15)$	CSI	2.3937	0.925	1.3496	0.951	0.6155	0.923	0.3997	0.946
	CSII	2.4568	0.931	1.4658	0.955	0.6745	0.933	0.4421	0.951
	CSIII	2.5661	0.929	1.6377	0.949	0.7231	0.941	0.4936	0.948
$(5, 30, 20)$	CSI	1.9998	0.945	1.0965	0.953	0.4236	0.939	0.2863	0.951
	CSII	2.2357	0.944	1.1354	0.954	0.4968	0.941	0.3367	0.944
	CSIII	2.3699	0.939	1.3659	0.948	0.5543	0.938	0.3875	0.941
$(5, 50, 35)$	CSI	1.7586	0.951	0.7963	0.961	0.3767	0.952	0.2365	0.961
	CSII	1.8743	0.948	0.8672	0.957	0.4362	0.945	0.2966	0.957
	CSIII	1.9521	0.952	0.9935	0.959	0.4839	0.949	0.3452	0.952

Table 10. The AWs and CPs of

95 %

ACIs and CRIs for

h (t)

and

r (t)

Table 10. The AWs and CPs of

95 %

ACIs and CRIs for

h (t)

and

r (t)

$(k, n, m)$	CS	$α$				$β$
		MLE		Bayes		MLE		Bayes
		ACIWs	CPs	CRIWs	CPs	ACIWs	CPs	CRIWs	CPs
$(3, 30, 15)$	CSI	0.5172	0.925	0.3481	0.939	5.1654	0.942	3.6473	0.952
	CSII	0.5563	0.928	0.3952	0.932	5.9268	0.931	4.1287	0.955
	CSIII	0.6247	0.921	0.4328	0.934	6.7475	0.935	5.2473	0.951
$(3, 30, 20)$	CSI	0.4436	0.933	0.2868	0.945	4.2587	0.942	2.7849	0.949
	CSII	0.4863	0.929	0.3314	0.942	5.0625	0.951	3.6147	0.951
	CSIII	0.5361	0.918	0.3799	0.939	5.8697	0.947	4.4693	0.948
$(3, 50, 35)$	CSI	0.3961	0.941	0.2358	0.952	3.6544	0.949	2.0247	0.955
	CSII	0.4315	0.939	0.2863	0.955	4.5152	0.950	2.7965	0.961
	CSIII	0.4799	0.942	0.3157	0.961	5.1337	0.946	3.6473	0.957
$(5, 30, 15)$	CSI	0.6344	0.939	0.4583	0.955	6.2653	0.939	4.5472	0.941
	CSII	0.6894	0.928	0.4892	0.949	6.9112	0.933	5.1237	0.939
	CSIII	0.7155	0.941	0.5213	0.957	7.5334	0.921	5.8624	0.951
$(5, 30, 20)$	CSI	0.5467	0.955	0.3661	0.961	5.5632	0.938	3.4251	0.949
	CSII	0.5993	0.947	0.4055	0.958	6.2754	0.941	4.6557	0.951
	CSIII	0.6342	0.948	0.4512	0.953	6.9651	0.937	5.1125	0.944
$(5, 50, 35)$	CSI	0.4463	0.951	0.2994	0.962	4.3322	0.948	2.5757	0.961
	CSII	0.5083	0.946	0.3466	0.953	4.8693	0.945	3.3145	0.954
	CSIII	0.5637	0.949	0.3937	0.955	5.6624	0.949	4.1956	0.957

Table 11. ML and Bayes estimates for

α

β

λ

S (t)

h (t)

and

r (t)

Table 11. ML and Bayes estimates for

α

β

λ

S (t)

h (t)

and

r (t)

Parameter	MLE	Bayes
		SE	LINEX		GE
			$ω = - 2$	$ω = 2$	$ν = - 2$	$ν = 2$
$α$	0.2835	0.4866	0.4723	0.4548	0.4437	0.4266
$β$	1.0837	1.0337	1.2134	1.0145	1.1965	0.9976
$λ$	0.9474	0.9599	0.9644	0.9343	0.9475	0.9291
$S (0.3)$	0.7338	0.9783	0.9455	0.9341	0.9552	0.9369
$h (0.3)$	0.4095	0.2944	0.3541	0.3225	0.3478	0.2899
$r (0.3)$	1.1288	2.5387	2.5541	1.7644	1.9658	1.6634

Table 12.

95 %

ACIs and CRIs of

α

β

λ

S (t)

h (t)

and

r (t)

Table 12.

95 %

ACIs and CRIs of

α

β

λ

S (t)

h (t)

and

r (t)

Parameter	ACIs		CRIs
	[Lower, Upper]	Width	[Lower, Upper]	Width
$α$	[0.1652, 0.3758]	0.2106	[0.4131, 0.8026]	0.3895
$β$	[1.0651, 1.1026]	0.0375	[0.9120, 1.5972]	0.6852
$λ$	[0.9270, 0.9683]	0.0414	[0.9181, 0.9735]	0.0554
$S (0.3)$	[0.6915, 0.7760]	0.0845	[0.9299, 0.9917]	0.0618
$h (0.3)$	[0.3382, 0.4809]	0.1428	[0.2517, 0.4052]	0.1535
$r (0.3)$	[0.7646, 1.4930]	0.7284	[1.5219, 2.6354]	1.1064

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Ramadan, M.M.; EL-Sagheer, R.M.; Abd-El-Monem, A. Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application. Axioms 2024, 13, 822. https://doi.org/10.3390/axioms13120822

AMA Style

Ramadan MM, EL-Sagheer RM, Abd-El-Monem A. Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application. Axioms. 2024; 13(12):822. https://doi.org/10.3390/axioms13120822

Chicago/Turabian Style

Ramadan, Mahmoud M., Rashad M. EL-Sagheer, and Amel Abd-El-Monem. 2024. "Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application" Axioms 13, no. 12: 822. https://doi.org/10.3390/axioms13120822

APA Style

Ramadan, M. M., EL-Sagheer, R. M., & Abd-El-Monem, A. (2024). Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application. Axioms, 13(12), 822. https://doi.org/10.3390/axioms13120822

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Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application

Abstract

1. Introduction

2. Maximum-Likelihood Estimation

2.1. Existence and Uniqueness of MLEs

2.2. Approximate Confidence Intervals

2.3. Delta Method

2.4. Log-Normal ACIs

3. Bayesian Estimation

3.1. Prior and Posterior Distributions

3.2. MCMC Techniques

4. Simulation Study

5. Application to Real-Life Data

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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