Estimations in a Constant-Stress Partially Accelerated Life Test for Generalized Rayleigh Distribution under Type-II Hybrid Censoring Scheme

Rabie, Abdalla; Hussam, Eslam; Muse, Abdisalam Hassan; Aldallal, Ramy Abdelhamid; Alharthi, Amirah Saeed; Aljohani, Hassan M.

doi:https://doi.org/10.1155/2022/6307435

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Volume 2022 | Article ID 6307435 | https://doi.org/10.1155/2022/6307435

Estimations in a Constant-Stress Partially Accelerated Life Test for Generalized Rayleigh Distribution under Type-II Hybrid Censoring Scheme

Abdalla Rabie,¹Eslam Hussam,²Abdisalam Hassan Muse,³Ramy Abdelhamid Aldallal,⁴Amirah Saeed Alharthi,⁵and Hassan M. Aljohani⁵

Academic Editor: Melike Kaplan

Received06 Jan 2022

Accepted23 Feb 2022

Published26 Apr 2022

Abstract

The constant-stress partially accelerated life test (CSPALT) model with Type-II hybrid censoring scheme (Type-II HCS) is the subject of our research. Units have a lifetime that follows the generalized Rayleigh distribution. Bayesian and E-Bayesian estimates are derived by applying two of the loss functions, mainly the squared error loss (SEL) and LINEX loss functions. Bayesian and E-Bayesian estimates are obtained using Markov chain Monte Carlo (MCMC) methods. To prove the applicability and the importance of the subject, a test for real data will be provided. To evaluate the distribution’s effectiveness, we utilized a variety of datasets and proposed several kinds of censoring. Finally, all results are compared in order to determine the effectiveness of the proposed methods. All major findings are concluded in the conclusion section.

1. Introduction

The majority of modern items is built to work efficiently over an extended length of time. So, the reliability tests in the normal use condition need more time and cost. In such instances, accelerated life tests (ALTs) are performed in order to collect more information for the reliability of the products in the shortest period. An increase in usual performance, temperature, pressure, stress, or any combination of these components constitutes an accelerated test condition. Regarding ALT, it is presumed that the accelerating factor is known, and the relationship between life span and stress is identified using a mathematical model. These models may not exist or be extremely difficult to construct in some circumstances. As a result, in this case, it is advisable to employ partially accelerated life tests (PALTs), which are a form of ALT. PALT is also utilized when only a specific acceleration condition is required, with the data being extrapolated to the test case. Nelson [1] demonstrated that frequent forms of stress, such as step stress and constant stress, exist. CSPALT, which divides test subjects into two groups before analyzing the data, will be discussed. Group 1 is for the regular case, while Group 2 is for the accelerated case. We will continue performing the experiment until all of the units fail or the time limit for the test has expired. To learn more, see [2–5].

HCS is a censoring technique that combines Type-I and Type-II censoring. Type-I HCS, based on the time at which the test is terminated at the instant of , such that refers to the item failure time, and , can be defined as the full maximum time for the experiment under consideration. We ended the test when we reached failed items out of items where and are prefixed from the beginning, so we end the test at the time whether or has been reached, whichever is closer, while in Type-II HCS, after a random period of time, the life-testing experiment is completed. Let us say at , i.e., when the latter of two stopping rules is met, the experiment is halted. As a result, we may be certain that at least failures have occurred. We assume in this study that the data are Type-II hybrid censored using the generalized Rayleigh CSPALT model, which indicates that each of the following cases may be observed.

Under normal operating conditions: Case I: if . Case II: if .

In the event of an accelerated use condition: Case 1: if . Case 2: if .

Here, and , are the number of failures recorded before time in regular and accelerated use settings, respectively.

This article is divided into several sections. Section 2 includes an explanation of the model and the test procedures. Section 3 discusses the maximum likelihood technique. Section 4 discusses Bayesian estimation using LINEX and SEL loss functions. Section 5 discusses the E-Bayesian technique for loss functions SEL and LINEX. Section 6 discusses the MCMC approach. Section 7 illustrates real datasets and concludes the findings of the suggested approaches.

1.1. Notations

: all experiment items .

: the experiment’s maximum duration.

: termination time of the experiment’s real-life testing.

: product’s life expectancy under accelerated usage conditions.

: the duration of an item’s life span under typical usage conditions.

: the time at which the item fails which is put under accelerated use condition.

: the time at which the item fails which is put under normal use condition.

: Group 1 is assigned to typical usage, whereas Group 2 is assigned to stressed usage working conditions.

: accelerating factor .

: these are considered as the failed items from the units before censoring time .

: these are considered as the failed items from the units after censoring time .

: lifetimes of failure under typical use.

: failure’s life expectancy under accelerated usage conditions.

2. Test Procedure and Model Description

2.1. The Model of Test Units Lifetime

The lifetimes of the test units are following the generalized Rayleigh model. The generalized Rayleigh (Burr-X) distribution is one of the Burr distributions, proposed by Burr [6]. It is critical in a variety of disciplines, including economics and computational modeling, as well as in health, agriculture, and biology. Here is the probability density function (PDF) of the generalized Rayleigh distribution, which is defined as follows:and the cumulative distribution function (CDF) is given bywhere is the shape parameter.

The generalized Rayleigh reliability function R(t) has the form and the hazard rate function is given by

Recently, many studies have been conducted on generalized Rayleigh distribution, for example, [7–11].

2.2. Test Procedure

All units of size in CSPALT will be separated into two categories, and , as follows:(1) items run normally.(2) are randomly picked from items and located under accelerated use conditions.(3)At a random time, , the test will be ended,whereand is the failure time of unit in the event of a typical operating state and is the failure period of units when the items were exposed to stress usage conditions.

3. Maximum Likelihood Method

We assume that are failure lifespans observed in the state of typical use under Type-II HCS, and are observed lifetimes of failures supposing that the units are exposed to the accelerated use working situation under Type-II HCS. The lifespans of test items follow a generalized Rayleigh distribution. The PDF of a product in its normal-use state is specified in (1). The PDF for an item in the accelerated usage state is as follows:where . As a result, for , at ordinary working condition by removing the constant, we may express the likelihood function in the following manner, under Type-II HCS:where

Similarly, for , for accelerated usage, the likelihood function has the following format:where

The function representing the total likelihood function for is obtained by combining equations (7) and (9) as

Applying the natural logarithm to equation (11) yields the following formula referred to as “the log-likelihood function”:

The MLEs for the unknown parameter and the acceleration factor are obtained by equating the initial partial derivatives of equation (12) to zero with regard to both parameters and solving the following equations numerically.and then the MLEs and of and are obtained.

4. Bayesian Estimation Approach

Based on the SEL and LINEX loss functions, Bayesian estimates of and are derived in this part of study. CSPALT is used with a generalized Rayleigh Type-II hybrid censored sample. Gamma and Gamma are suggested as the prior PDFs of and , respectively. As a result, the combined prior PDF of and is written aswhere

We can write the posterior PDF of the two parameters and with the aid of equations (11) and (14) as follows:where a normalizing constant is given by

4.1. Bayesian Estimates under SEL Function

Using the SEL function as an example, the posterior mean is used to estimate , as follows:

Likewise, we may construct estimates for the other parameter using the same techniques as follows:

4.2. Bayesian Estimates under LINEX Loss Function

The Bayesian estimates of and using the LINEX loss function are as follows:

5. E-Bayesian Estimation Method

Refer to [12], the hyperparameters (, ) and are selected such that and are decreasing functions.. We must obtain the derivatives of and in terms of and which are given by

When and , , and when and , . Thus, and are diminishing functions for and , respectively. For the independent hyperparameters, we hypothesize and , , bivariate PDFs provided by

According to E-Bayesian estimations, the estimates of the distribution parameters can be computed with the aid of the following equations:where and stand for Bayesian estimated values of and under the effect of both SEL function and the LINEX loss function. For further information, we refer to [13–24].

5.1. E-Bayesian Estimate of

To derive E-Bayesian estimates of and , we consider the following prior PDFs of (, ) and to clarify the impact of prior PDFs on the estimates of E-Bayesian method for and . The prior PDFs are given by

The E-Bayesian estimated value of is obtained using the SEL function from (19), (25), and (27) as follows:

Additionally, the E-Bayesian estimate of may be produced through using LINEX loss function from (21), (25), and (27) as follows:

5.2. E-Bayesian Estimate of

Based on the SEL and LINEX loss functions, the E-Bayesian estimate is obtained by using (27).

We can use (20), (26), and (27) to obtain the estimates of as follows:and for the LINEX loss function, we can use (22), (26), and (27) to obtain the estimate of the same parameter:

We employ the MCMC methodology to generate Bayesian and E-Bayesian estimates of and because all estimators of Bayesian and E-Bayesian methods cannot be expressed explicitly as shown in the preceding equations.

6. MCMC Technique

We apply the MCMC technique when we fail to compute the estimates for the parameters and mathematically. Therefore, we turn to one of the most efficient methods of generating samples from a certain distribution, which is the Metropolis–Hastings method, and subsequently obtain Bayesian estimates. We make effective use of the Metropolis–Hastings approach, which generates randomly selected samples from any proposal distribution. The following are the posterior conditional PDFs of and , respectively.

Now we encounter a difficulty: we are unable to produce samples from the conditional posterior PDFs of and , since it is self-evident that they cannot be simplified to a well-known model as shown in equations (32) and (33), and thus we adopt a normal distribution as a proposal distribution.

	Step 1: select and as beginning values, say .
	Step 2: start the counter with count .
	Step 3: create a random sample of from (32) using normal distribution.
	Step 4: compute the rejection-acceptance condition:

	Step 5: we want to generate a random number that varies and ranges from 0 to 1, so we will turn to the uniform distribution .
	Step 6: if , accept the proposal and set ; otherwise, reject the proposal and set .
	Step 7: using the Metropolis–Hastings method, generate from (33) using normal distribution.
	Step 8: in order to find the estimate of , we should make a repetition for steps 4–6.
	Step 9: make an increment in the loop counter like .
	Step 10: repeat steps 3–9 for a total of N times.
	Step 11: under the SEL function, the Bayesian estimates regarding and have been provided by the following:


	where M is a burn-in period that can be chosen.
	Step 12: the estimates for the two parameters and under the LINEX loss function can be easily computed, respectively, using the following equations:

7. Simulation Study

As per the following algorithm, a simulation study is conducted. All calculations are carried out using Mathematica 12 programming language software.(i)Determine the values of .(ii)We start with chosen initial values for both parameters and .(iii)We want to generate a random number that varies and also ranges from 0 to 1, so we will turn to the uniform distribution .(iv)Create a generalized Rayleigh Type-II hybrid censoring sample using the CSPALT framework and the inverse function technique as described in the following: a normal use condition case is given by , and an accelerated use condition case takes the following form:(v)Construction of a Markov chain comprising 11000 values of and is accomplished via the use of the Metropolis–Hastings method, with the first 1000 discarded as “burn-in.”(vi)We construct estimated values of Bayesian estimates and under SEL function from (35) and (36), respectively, using MCMC samples.(vii)Using (37) and (38), we develop Bayesian estimates and , respectively, under the LINEX loss function.(viii)We use the SEL function to construct E-Bayesian estimates and , respectively, from (28) and (30).(ix)From (29) and (31), E-Bayesian estimates and are, respectively, obtained using LINEX loss function.(x)We compute where is the estimate of .(xi)We compute where is the estimate of .(xii)We compute mean squared error (MSE) of estimates and , respectively, by(xiii)Tables 1 and 2 are used to present the numerical results.

8. Data Analysis

This section contains a numerical example that observes the efficiency of the methods used in the paper and assesses how they operate using real data and not simulated data. Many authors used these data [25]. For seven different values of mole fraction, they used it as minority electron mobility for p-type . Tarvirdizade and Filliben [26] supplied just two sets of data for the mole fractions 0.25 and 0.30, claiming that the generalized Rayleigh distribution is well fitted for datasets 1 and 2 individually. Each dataset comprises 21 observations, as seen below (Table3).

Here, CSPALT with Type-II HCS was applied on these uncensored data, with and units chosen at random from a total of . We see the following for , , , , and .

Dataset 1:(i)When , the test is terminated at random if the usage conditions are normal. , and hence 9 items will fail from 10 items. In the scenario of accelerated use, we find that the random time ; as a result, out of 11 items, 11 items fail.

Dataset 2:(i)When , the test is ended at random time ; then, 9 items fail out of 10 items in the typical use condition. We can see that the random time regarding accelerated case, resulting in 11 items failing out of 11 units.

The same prior approaches are used to construct Bayesian and E-Bayesian estimates of and under the SEL and LINEX loss functions for real datasets. Table 4 gives the numerical results for both datasets.

9. Concluding Remarks

We estimated the distribution parameter by many methods such as classical Bayesian and E-Bayesian estimation methods and the acceleration factor of the CSPALT model from generalized Rayleigh distribution under Type-II HCS. It is noted that Bayesian estimators cannot be stated in a closed form; therefore, we utilize the MCMC technique to obtain estimates of the parameter and the acceleration factor using SEL and LINEX loss functions. In terms of MSE, the E-Bayesian estimation approach under SEL and LINEX loss functions performs better than the Bayesian estimation technique under the same loss functions for and , as shown in Tables 1 and 2. When sample size and censoring time increase, the MSE of both Bayesian and E-Bayesian estimation methods decreases. Furthermore, as shown in Table 4, all previous findings are confirmed when actual datasets (1, 2) are used, demonstrating that the suggested procedures are effective in practice. Moreover, in practice, the E-Bayesian methodology outperforms the Bayesian method. At the end, because of the importance of the competing risks model in medical studies and reliability analysis, we can suggest “Bayesian inference in a constant-stress partially accelerated competing risks model based on hybrid censoring schemes” as future work.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was funded by the Taif University Researchers Supporting Project (TURSP-2020/279), Taif University, Taif, Saudi Arabia.

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Copyright © 2022 Abdalla Rabie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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