# Single and Multiple Ramp Progressive Stress with Binomial Removal: Practical Application for Industry

Authors: Eslam Hussam; Randa Alharbi; Ehab M. Almetwally; Bader Alruwaili; Ahmed M. Gemeay; Fathy H. Riad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The objective of this article is to conduct a full study on the failure times of items when subjected to a progressive-stress accelerated life test. These failure times follow the Modified Kies exponential distribution. We performed the experiments under a type-II censoring scheme with binomial removal. The stress design has two ways to be applied either the simple ramp-stress test or the multiple ramp-stress level design of acceleration. We made a comparison between these two designs to decide which design is better. When the lifetime of test units follows the Modified Kies distributions, the cumulative exposure model has been used to apply the acceleration on the failure times to produce early failures. The simulation study was done to compare two types of progressive-stress designs. Different estimation techniques such as maximum likelihood estimation and Bayes estimation are employed to estimate the model parameters. The Metropolis Hasting method is used to derive the Bayesian estimates under symmetric and asymmetric loss functions. We also developed interval estimation as well as point estimation, and we estimated the asymptotic confidence intervals, bootstrap, and credible for the distribution parameters. We made a comparison between different estimation methods. We used real data from a practical experiment as a real data set example to assess the performance of the estimations methods. These data are analyzed to demonstrate the adequacy and superiority of the distribution to fit accelerated failure times. Also, we studied the existence and uniqueness of the roots and we used graphs to assure that the estimates used are unique and global maximum. Finally, some noteworthy conclusions are reached.


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## Body

## 1. Introduction

Collecting a sufficient number of failure times is difficult in the majority of traditional life testing and reliability experiments, even more so when the items and products are extremely dependable and are highly reliable as well as they have a long lifetime. Under ordinary situations, lifetime experiments with a predefined testing time result in extremely low failure rates. One of the most well-known processes for accelerating the incidence of failing is the accelerated life test (ALT), which subjects products to extreme stress as an example of this stress, maybe the units under experiments are exposed to a high level of voltage, or may they are accused to a higher temperature than the normal use one. We estimate the life characteristics for the proposed distribution under normal conditions and under stress conditions to determine their reliability function under both conditions.The ALT can be used in a variety of ways; the most three famous ways to apply acceleration on products are constant stress in which all the items are accused to the same level of stress; in contrast to the step stress sometimes, we called it partially accelerated as the stress is increased gradually after some pre-fixed time; the last ways are the progressive-stress ALT; see Nelson [1] for more information about various classifications, as he described all these kinds extensively.The testing is applied entirely under conditions of constant stress when the experiment goes constant-stress ALT. Numerous studies have identified the designs of constant stress as follows. Watkins and John [2] studied the constant-stress accelerated life tests under type-II censoring at one of the stress levels. Rabie et al. [3] made statistical inferences for Exponentiated Gamma distribution using constant stress partially accelerated based on generalized type-I hybrid censored data. Bantan et al. [4] discussed Bayesian analysis in partially accelerated life tests for weighted Lomax distribution. Abd El-Raheem [5] made a study on the optimal plans for the generalized half-normal distribution and the exponential distribution under the type-I censoring scheme. Abd El-Raheem [6] made a full study on the optimal multiple constant-stress accelerated life tests for the generalized exponential distribution. Guan et al. [7] estimated the optimal plans on constant-stress accelerated life tests for the extension of the exponential distribution under type-I censoring. Jaheen et al. [8] focused their attention on Bayes inference in constant partially accelerated life tests for the distribution of generalized exponential with the progressive censoring. Kim and Bai [9] made a practical experiment on the accelerated models under different kinds of stress. Abd El-Raheem et al. [10] studied accelerated life tests for Modified Kies exponential (MKEx) lifetime distribution with binomial removal.Another important way that accelerated stress can be applied is called the step stress accelerated life test (SSALT). In this type of stress, we increase the level of stress at predetermined times in order to induce early failures. Numerous studies have identified SSALT, as an example. Bai et al. [11] made a simulation study to find the optimum simple step stress accelerated life tests under censored sample [12, 13]. They studied the optimal plans under step stress accelerated life tests using type-I and type-II progressive censored samples under SSALT for many distributions. Balakrishnan et al. [14] estimated the point and interval estimation for the distribution parameters under type-II censoring. The authors in [15–21] used Bayesian and classical methods to estimate the parameters for the Weibull, the power generalized Weibull distribution, and the extension of the exponentiated exponential distribution, respectively, under SSALT using different kinds of censoring plans.Besides this, however, in contrast for progressive-stress accelerated life test (PSALT), products are subjected to stress at a growing rate over time. In other words, the stress is a function of times and grows larger as long as the time of the experiment increases. For extra information and details on this kind of applying ALT, we can refer to Yin and Sheng [22] who considered the problem of working on accelerated life testing by progressive-stress loading. Mohie El-Din et al. [23] made a statistical inference using classical and Bayesian methods using progressive-stress accelerated life testing for the extension of the exponential distribution with the progressive type-II censoring. Bai et al. [24] studied optimum simple ramp-tests for the Weibull distribution and type-I censoring. Ronghua and Heliang [25] worked on statistical inference of the Weibull distribution for the tampered model of failure rate in progressive-stress accelerated life testing; also, we can refer the work of the following authors on different kinds of ALT such as Abdel-Hamid and AL-Hussaini [26] worked on the progressive-stress accelerated life tests under finite mixture models, Abdel-Hamid and AL-Hussaini [27] made statistical inference for a progressive-stress model from Weibull distribution with the progressive type-II censoring, and also AL-Hussaini et al. [28] studied on one-sample Bayesian prediction intervals based on progressively type-II censored data set from the distribution of half-logistic under progressive-stress model. Abdel-Hamid and Abushul [29] estimated the parameters of the exponentiated exponential distributions using the inverse power law. Abd El-Raheem [30] regarded the optimization technique of progressive-stress (ALT) design for the generalized half-normal distributions. Aslam et al. [31] studied the multiple levels accelerated life test. For more details about multilevel accelerated test, see the study by Kamal [32], and for more details about competing for risk under the accelerated life test, see the study by Bai et al. [33].Considering the cost and the time required for life testing experimentation, the experimenter must terminate the experiment prior to monitoring all failures; therefore, data censoring techniques are widely used to reduce the testing time and cost. The most frequently used censoring schemes for lab tests are type-I and type-II traditional censoring where the progressive type-II censoring scheme was the most common censoring scheme (CS).Taking this into consideration as an illustration, it is presumed that n is the total number of units under examination. A lifetime test is conducted on identical units or equipment with a preset total number of failures wherem≤n. When the first failure happens at a certain point in time t1:m:n, R1 components that pass the test are discarded. At the time for failing the second working item let is say at time t2:m:n occurs we will remove a number of working units let us say it will be R2 components. The remaining items that survive after the test are removed, and so forth, until the m-th item has evidence of failure, at this failure the test reached the endpoint and, the test is revoked and the residual surviving units, Rm=n−m−R1+R2+…+Rm−1 units are eliminated. For additional research on progressive CS, please refer to the study by Balakrishnan and Aggarwala [34] where they introduced many kinds of censoring and see also the study by Balakrishnan and Cramer [35].The main aim and motivation of this article are to study the effect of PSALT under progressive type-II (PT-II) censored sample using binomial removal for the items under experiment. When the failure times are considered as Modified Kies distributed, we made the statistical inference under different stress levels, either simple ramp experiment or multiple ramp test in the simulation study. Also, we studied a practical experiment as an application on applying practically the PSALT practically and seeing the effect of this stress on the failure times, and also we studied the reliability behavior and the effect on the increase in stress on it. Such kinds of studies are used mainly in practical application in industry, that is, because manufactured items have a long lifetime and drains a lot of time and money to get failures so it is better to apply stress loading on the manufactured items to study its reliability function which has a great role in specifying the insurance period for every item, which saves a lot of money and costs to every company; we applied a true experiment in the application section, which was a true application on the progressive acceleration, that was applied practically in industry.The following sections comprise the paper: In Section2, we present the ALT model and associated test assumptions. The classical method, the maximum likelihood estimation (MLE), and how we use it to estimate the parameters under the progressive-stress model are determined in Section 3. Using the Metropolis-Hastings (MH) algorithm, the Bayesian Estimation (BE) for distribution parameters is presented in Section 4. Section 5 is devoted to obtaining the asymptotic and bootstrap CIs of the parameters of the distribution. The simulation study is included in Section 6. A practical experiment and the recorder failure times and its data analysis are presented in Section 7 to demonstrate the existing methodology. The graphs of the reliability functions and the effect of increasing the stress on the items are presented in Section 7.2. Finally, Section 8 concludes the major findings of the paper.


## 2. ALT Model and Test Assumptions

### 2.1. MKEx Distribution

Al-Babtain et al. [36] introduced a new lifetime distribution, and they are called MKEx distribution. This distribution has a lot of merits. It has a bathtub shape, increasing and decreasing failure rate. Furthermore, it has the ability to model negatively and positively skewed data. Additionally, it has a closed formula for the cumulative distribution function (CDF) and is extremely simple to handle, and the distribution is well-suited for use in a variety of fields, including life testing, quality, medical research, and statistical techniques. Aljohani et al. [37] discussed ranked set sampling with application of MKEx distribution. In this context, Al-Babtain et al. [36] used two different types of real data to show that this distribution may be a good alternative to many popular distributions such as Weibull, Marshall–Olkin exponential, Kumaraswamy exponential, beta exponential, gamma, and general exponential distribution. The CDF of the MKEx distribution is(1)Fx;α,σ=1−e−eσx−1α,x>0,α,σ>0.The corresponding probability density function (PDF) of (1) is given by(2)fx;α,σ=ασeασx1−e−σxα−1e−eσx−1α,x>0,α,σ>0,where σ is the scale parameter and α is a shape parameter.Figure1 shows a graphical representation of the PDF of the proposed model for various values of α and σ. This representation shows that the PDF of the proposed model can be right-skewed, left-skewed, unimodal, symmetric, increasing, or decreasing density function.Figure 1
Plots of the proposed model PDF for different parametric values.


### 2.2. PSALT Assumptions and Acceleration Model

In this part of the paper, we identify the assumptions used for PSALT design:(1)
A unit’s lifetime is determined by Modified Kies distributionσ,α.(2)
The relation between the progressive-stress at leveli and the time is controlled by the law, i.e., τit=uit,0<u1<u2<⋯<uk. τit increases by increasing the time ui, where ui is the ramp value, may be Volt or temperature.(3)
As shown in the following, the life characteristicθi (the scale parameter) and the stress loading τit are controlled by the following law, namely, the inverse power law:(3)θit=1aτitb,in whicha and b are unknown parameters; see the experimental section on acceleration (for more details about these acceleration ways, we can refer to Cha. 2 in [1]).(4)
The cumulative exposure models are used to determine how stress affects failure time as it progresses through one level to the next (see, e.g., Nelson [1]).(5)
Regarding the relation in (3) and by making some mathematical simplifications, we can define θi=θ1hib, where hi=u1/ui.Assume the test begins withn units, and the stress level in each level is ordered as follows: τ0<τ1t<⋯<τkt, and τ0 is the utilization-stress. Additionally, assume that ni units are assessed at each progressive-stress level τit, i=1,2,3,……,k. The progressive censorship procedure is defined as follows: when really the first failure occuring at time ti1:mi:ni is reached, Ri1 of the residual units is randomly withdrawn, from the ni−1 surviving units. When the second malfunction occurring at ti2:mi:ni is hit, Ri2 units are automatically selected from the existing ni−2−Ri1 living units. The test is halted at the specified stage i at the times when mi-th failures happened at timi:mi:ni; at the moment, all that remaining Rimi=ni−mi−∑j=1mi−1Rij items are withdrawn. The observed life times by using the progressive-stress τit are ti1:mi:ni<ti2:mi:ni<⋯<timi:mi:ni,i=1,2,…..,k.The CDF for progressive-stressτit lifetimes is calculated using the cumulative exposure model’s assumptions as(4)Git=Fi△t,i=1,2,…,k,where △t=∫0tdu/θiu=t/θ1hibb+1 and Fi. is introduced in (2) with scale parameter equal to 1. Therefore,(5)Git=1−e−et/σi−1α,t>0,α,σi>0,where σi=θ1hibb+1. The PDF of (5) is given by(6)git=αet/σi−et/σi−1αet/σi−1α−1σi,t>0,α,σi,θ>0.Let us assume that an individual’s removal from the test is uncorrelated with the others, but just as likely, with probabilityP. Here, the number of items that fail per time has a binomial distribution. That is, R1∼binomialni−mi,P, Rij∼binomialni−mi−∑i=1mi−1Rij,P,i=2,…,mi, and Rimi=ni−mi−∑j=1mi−1Rij.


## 2.1. MKEx Distribution

Al-Babtain et al. [36] introduced a new lifetime distribution, and they are called MKEx distribution. This distribution has a lot of merits. It has a bathtub shape, increasing and decreasing failure rate. Furthermore, it has the ability to model negatively and positively skewed data. Additionally, it has a closed formula for the cumulative distribution function (CDF) and is extremely simple to handle, and the distribution is well-suited for use in a variety of fields, including life testing, quality, medical research, and statistical techniques. Aljohani et al. [37] discussed ranked set sampling with application of MKEx distribution. In this context, Al-Babtain et al. [36] used two different types of real data to show that this distribution may be a good alternative to many popular distributions such as Weibull, Marshall–Olkin exponential, Kumaraswamy exponential, beta exponential, gamma, and general exponential distribution. The CDF of the MKEx distribution is(1)Fx;α,σ=1−e−eσx−1α,x>0,α,σ>0.The corresponding probability density function (PDF) of (1) is given by(2)fx;α,σ=ασeασx1−e−σxα−1e−eσx−1α,x>0,α,σ>0,where σ is the scale parameter and α is a shape parameter.Figure1 shows a graphical representation of the PDF of the proposed model for various values of α and σ. This representation shows that the PDF of the proposed model can be right-skewed, left-skewed, unimodal, symmetric, increasing, or decreasing density function.Figure 1
Plots of the proposed model PDF for different parametric values.


## 2.2. PSALT Assumptions and Acceleration Model

In this part of the paper, we identify the assumptions used for PSALT design:(1)
A unit’s lifetime is determined by Modified Kies distributionσ,α.(2)
The relation between the progressive-stress at leveli and the time is controlled by the law, i.e., τit=uit,0<u1<u2<⋯<uk. τit increases by increasing the time ui, where ui is the ramp value, may be Volt or temperature.(3)
As shown in the following, the life characteristicθi (the scale parameter) and the stress loading τit are controlled by the following law, namely, the inverse power law:(3)θit=1aτitb,in whicha and b are unknown parameters; see the experimental section on acceleration (for more details about these acceleration ways, we can refer to Cha. 2 in [1]).(4)
The cumulative exposure models are used to determine how stress affects failure time as it progresses through one level to the next (see, e.g., Nelson [1]).(5)
Regarding the relation in (3) and by making some mathematical simplifications, we can define θi=θ1hib, where hi=u1/ui.Assume the test begins withn units, and the stress level in each level is ordered as follows: τ0<τ1t<⋯<τkt, and τ0 is the utilization-stress. Additionally, assume that ni units are assessed at each progressive-stress level τit, i=1,2,3,……,k. The progressive censorship procedure is defined as follows: when really the first failure occuring at time ti1:mi:ni is reached, Ri1 of the residual units is randomly withdrawn, from the ni−1 surviving units. When the second malfunction occurring at ti2:mi:ni is hit, Ri2 units are automatically selected from the existing ni−2−Ri1 living units. The test is halted at the specified stage i at the times when mi-th failures happened at timi:mi:ni; at the moment, all that remaining Rimi=ni−mi−∑j=1mi−1Rij items are withdrawn. The observed life times by using the progressive-stress τit are ti1:mi:ni<ti2:mi:ni<⋯<timi:mi:ni,i=1,2,…..,k.The CDF for progressive-stressτit lifetimes is calculated using the cumulative exposure model’s assumptions as(4)Git=Fi△t,i=1,2,…,k,where △t=∫0tdu/θiu=t/θ1hibb+1 and Fi. is introduced in (2) with scale parameter equal to 1. Therefore,(5)Git=1−e−et/σi−1α,t>0,α,σi>0,where σi=θ1hibb+1. The PDF of (5) is given by(6)git=αet/σi−et/σi−1αet/σi−1α−1σi,t>0,α,σi,θ>0.Let us assume that an individual’s removal from the test is uncorrelated with the others, but just as likely, with probabilityP. Here, the number of items that fail per time has a binomial distribution. That is, R1∼binomialni−mi,P, Rij∼binomialni−mi−∑i=1mi−1Rij,P,i=2,…,mi, and Rimi=ni−mi−∑j=1mi−1Rij.


## 3. Maximum Likelihood Estimation

In this part, we employed the famous classical method, namely, the MLEs method to evaluate the parametersα, θ1, and b of the distribution such that we can use these values in the simulation section. For simplification, let tij:mi:ni=tij, that also comprises the lifetimes of progressive censorship under progressive-stress conditions at each stress levels τit, i=1,2,3,…..,k and j=1,2,….,mi with binomial removal obtained, then the PDF of Rij, i=1,2,…,k, is given as follows:(7)PrRi1=ri1=ni−miri1Pri11−Pni−mi−ri1,while, for j=2,3,…,mi−1,(8)PrRij=rij|Ri−1j=ni−mi−∑j=1i−1rijrijPrij1−Pni−mi−∑j=1irij,where 0≤rij≤ni−mi−∑j=1i−1rij. Furthermore, we suppose that Rij is independent of Xij:mi:ni for all i, i=1,2,…,k. Therefore, the likelihood functions α, θ1, and b under PT-II censoring with binomial removal are given by(9)L∗=∏i=1kCi∏j=1miftij;α,θ1,b1−Ftij;α,θ1,bRijPrRij=rij.Sincetij:mi:ni and Rij, for all i=1,2,…,k, are independent, then the MLE of P can be derived by maximizing PrRij=rij directly. Hence, the MLE of P is given by(10)atP=∑i=1i∑i=1mi−1rijmi−1ni−mi−∑i=1mi−1mi−i−1rij.After estimating the value ofP, we can write the likelihood function as shown in the following, such that the CDF in (5) and the corresponding PDF in (6):(11)Lα,θ1,b=∏i=1kCi∏j=1migitij1−GitijRij,where Ci is a constant and can be written as Ci=nini−1−Ri1ni−2−Ri1−Ri2…ni−mi+1−∑j=1mi−1Rij. For more details, see [23].From (5) and (6) in (11), we get(12)Gt;α,θ1,b=1−e−etij/θ1b+1hib−1α,x>0,α,θ1>0.The corresponding probability density function (PDF) of (12) is defined by(13)gt;α,θ1,b=α1θ1b+1hibeαtij/θ1b+1hibetij/θ1b+1hib−1α−1×exp−etij/θ1b+1hib−1α;x>0,α,σ>0.(14)Lα,θ1,b=∏i=1kCi∏j=1miα1θ1b+1hibeαtij/θ1b+1hibetij/θ1b+1hib−1α−1exp−∑i=1k∑j=1mietij/θ1b+1hib−1αRij+1.We can write the log-likelihood function denoted by(15)ℓα,θ1,b=∑i=1klnCi+∑i=1kmilnα−lnθ1−lnb+1+α∑i=1k∑j=1mitijθ1b+1hib−b∑i=1kmilnhi+α−1∑i=1k∑j=1milnetij/θ1b+1hib−1−∑i=1k∑j=1mietij/θ1b+1hib−1αRij+1,and the normal equation for the distribution parameter α, θ1, and b is as follows, respectively:(16)∂ℓ∂α=∑i=1kmiα+∑i=1k∑j=1mitijθ1b+1hib+∑i=1k∑j=1milnetij/θ1b+1hib−1−∑i=1k∑j=1miRij+1etij/θ1b+1hib−1αRij+1lnetij/θ1b+1hib−1.(17)∂ℓ∂θ1=−∑i=1kmiθ1−α∑i=1k∑j=1mitijθ12b+1hib+α−1∑i=1k∑j=1mitij/θ12b+1hibetij/θ1b+1hibetij/θ1b+1hib−1−α∑i=1k∑j=1mitijθ12b+1hibRij+1etij/θ1b+1hib−1αRij+1−1etij/θ1b+1hib.(18)∂ℓ∂b=∑i=1kmi−1b+1−hi−α∑i=1k∑j=1mitij1+b+1lnhiθ1b+12hib−α−1θ1∑i=1k∑j=1mitij1+b+1lnhi/θ1b+12hibetij/θ1b+1hibetij/θ1b+1hib−1+θ1∑i=1k∑j=1miαRij+1tij1+b+1lnhiθ1b+12hibetij/θ1b+1hibetij/θ1b+1hib−1αRij+1−1.By solving the system of three nonlinear equations (16)–(18), in three unknowns α,θ1, and b, the MLE of the parameters is achieved. That can be done by using the Newton–Raphson method.


## 4. The Bayesian Approach

The BEs for the parameters are calculated in this section using square error (SE) function and linear exponential (LINEX) loss functions; the BEs corresponding to the parametersα, θ1, and b will be obtained. We will use gamma prior for the distribution parameters α, θ1, and b such that(19)π1α∝αμ1−1e−α/λ1,α>0,μ1,λ1>0,π2θ1∝θ1μ2−1e−θ1/λ2,θ1>0,μ2,λ2>0, (20)π3b∝bμ3−1e−b/λ3,b>0,μ3,λ3>0.Assume that now the model’s parameters are independent, where the joint PDF of the prior is provided by(21)πα,θ1,b∝bμ3−1θ1μ2−1αμ1−1e−α/λ1+θ1/λ2+b/λ3.The parameter’s posterior density function ofα,θ1, and b is provided by the aid of (14) and (21) which is presented as follows:(22)π∗Θ|t=π∗α,θ1,b|t∝Lα,θ1,bπα,θ1,b∝bμ3−1∏i=1kCi∏j=1miαμ1θ1μ2−21b+1hibeαtij/θ1b+1hibetij/θ1b+1hib−1α−1e−α/λ1+θ1/λ2+b/λ3exp−∑i=1k∑j=1mietij/θ1b+1hib−1αRij+1.The BEs of parameters functionU=UΘ, Θ=α,θ1,b under the SE loss function are defined by(23)ÛSE=∫ΘUπ∗Θ|tdΘ,while by depending on the LINEX loss function, the BEs of U=UΘ are defined by(24)ÛLINEX=−1clog∫Θe−cUπ∗Θ|tdΘ,where the shape parameter of the LINEX loss function c≠0.It is self-evident that the integrals in (23) and (24) are difficult. As a result, we must employ the Markov Chain Monte Carlo (MCMC) approach to determine the values of integrals. The MH algorithm is a critical component of the MCMC technique; it can often be referred to as the random walk algorithm. This algorithm is similar to acceptance-rejection sampling; the MH algorithms can consider how candidate values can be generated from the normal proposals distributions for each iteration of the algorithm. The MH algorithm is discussed as follows (Algorithm 1).Algorithm 1: The steps followed to get the estimates of the parameters using the MH algorithm.
(1)
Use the initial for the parameters as the MLE values for the MH algorithm as follows:α0=α^, θ10=θ^1,b0=b^.(2)
Consider this iteration as first iteration withi=1.(3)
Now, we will generateα∗ from proposal distribution from Nαi−1,/varαi−1, where N is considered as the normal distribution.(4)
In order to decide which value will accept or even reject, we must estimate the acceptance probabilityAαi−1|α∗=min1,π∗α∗|θ1i−1,bi−1/π∗αi−1|θ1i−1,bi−1.(5)
Now, in order to complete the steps, we must generate a value from zero to one so we will use the uniform distribution asU0,1.(6)
IfU≤Aαi−1|α∗, then put αi=α∗; else put αi=αi−1.(7)
There are two rest parameters needed to be estimated, so we will repeat the above steps from ((2)–(6)) for the other two parameters,θ1 and b.(8)
Puti=i+1.(9)
Repeat steps ((3)–(8)),N=10000 times to obtain α1,θ11,b1, …, αN,θ1N,bN.Then, the BEs ofuα,θ1,b using MCMC under SE and LINEX loss functions are defined, respectively, as follows: u˜SE=1/N−M∑i=M+1Nuαi,θ1i,bi,u˜LINEX=−1/clog1/N−M∑i=M+1Nexp−cuαi,θ1i,bi.The conditional posterior distributions used in the MH algorithm are given as follows:(25)π∗α|θ1,b,t∝bμ3−1∏i=1k∏j=1miαμ1eαtij/θ1b+1hibeαtij/θ1b+1hib−1α−1e−α/λ1exp−∑i=1k∑j=1mieαtij/θ1b+1hib−1αRij+1,(26)π∗θ1|α,b,t∝∏i=1k∏j=1miθ1μ2−2eαtij/θ1b+1hibetij/θ1b+1hib−1α−1e−θ1/λ2exp−∑i=1k∑j=1mietij/θ1b+1hib−1αRij+1,(27)π∗b|α,θ1,t∝bμ3−1∏i=1k∏j=1mi1b+1hibeαtij/θ1bhibetij/θ1b+1hib−1α−1e−b/λ3exp−∑i=1k∑j=1mietij/θ1b+1hib−1αRij+1.


## 5. Interval Estimation

The approximate and Bootstrap confidence intervals (CIs) for the parametersα, θ1, and b are discussed in this section of the paper.


### 5.1. Approximate CIs

On the basis of the asymptotic distributions of the unknown parameters’ MLEsα, θ1, and b, the parameters’ normal approximation confidence intervals (CIs) are determined where the MLEs’ asymptotic distribution of α,θ1,b is given by Miller [38].(28)θ^1−θ1,α^−α,b^−b∼N0,η,and we mut consider η=κij,i,j=1,2,3, is the parameters’ variance-covariance matrix.ϑi CIs are calculated by the following formula:(29)ϑ^iL,ϑ^iU=ϑ^i±Zζ/2κii,i=1,2,3,where ϑ1=α, ϑ2=θ1, and ϑ3=b. For more information and reading, we can refer to Miller [38] distribution.


### 5.2. Bootstrap CIs

For the unknown parameters, we get percentile Bootstrap (Boot-p) confidence intervals (CIs). For more reading on Bootstrap (Boot-p) CIs, see Efron et al. [39] (Algorithms 2 and 3).Algorithm 2: Bootstrap CIs.
(1)
Determine the parameters’ MLE estimates by finding a solution numerically to (16)–(18).(2)
The approximation in step 1 can be used to produce a bootstrap sampleX¯∗ utilising the algorithm developed by [39]. Let us assume that the estimates of the distribution’s parameters are α^∗, θ1^∗, and b^∗.(3)
In order to produce a bootstrap, we will iterate the first two steps 1000 iteration.(4)
Ascend the list of estimates such thatϕt1,ϕt2,…,ϕtN,t=1,2,3, where ϕ1=α^∗,ϕ2=θ1^∗,andϕ3=b^∗. Let Gx=Pϕt≤x be the CDF of ϕt.(5)
Defineϕtboot=G−1x for given x. As a result, the upper bounds and lower bounds are defined as follows: ϕtbootβ/2,ϕtboot1−β/2.Algorithm 3: The steps for the Bootstrap-t CIs.
(1)
Repeat the first two steps in the percentile Algorithm 2(2)
EvaluateT=ϑ^b−ϑ^/Vϑ^b, where ϑ is α,θ1, or b and Vϑ^b is the asymptotic variances of ϑ^b.(3)
Repeat the above two stepsℬ times and rearrange T1,T2,…,Tℬ in ascending order.(4)
A two-side1001−γ% bootstrap-t CI for α,θ1, and θ is given by α−Tαbℬγ/2Vαb,α+VαbTαbℬ1−γ/2, θ1−Tθ1bℬγ/2Vθ1b,θ1+Vθ1bTθ1bℬ1−γ/2, and b−Tbbℬγ/2Vbb,b+VbbTbbℬ1−γ/2.


### 5.3. Highest Posterior Density (HPD) Interval Algorithm

Here, we introduced the algorithm for finding1−γ% HPD interval for α,θ1,andb (credible intervals), where the algorithm was proposed by Chen and Shao [40] (Algorithm 4).Algorithm 4: Credible interval algorithm.
(1)
Use the initial for the parameters as the MLEs values for the MH algorithm as follows:α0=α^, θ10=θ^1, and b0=b^.(2)
Consider this iteration as first iteration withi=1.(3)
Now, we will generateα∗ from proposals distribution from Nαi−1,/Nαi−1,varαi−1/, where N is considered as the normal distribution.(4)
In order to decide which value will accept or even reject, we must estimate the acceptance probabilityAαi−1|α∗=min1,π∗α∗θ1i−1,bi−1/π∗αi−1≥|θ1i−1,bi−1.(5)
Now, in order to complete the steps, we must generate a value from zero to one so we will use the uniform distribution asU0,1.(6)
IfU≤Aαi−1|α∗, then put αi=α∗; else put αi=αi−1.(7)
There are two rest parameters needed to be estimated, so we will repeat the above steps from ((2)–(6)) for the other two parameters,θ1 and b.(8)
Puti=i+1.(9)
Repeat the steps ((3)–(8))N=10000 times to get α1,θ11,b1, …, αN,θ1N,bN.(10)
After obtaining the estimates for theN=10,000 parameters, arrange each estimate in ascending order as φ^iSE1,φ^iSE2,…,φ^iSEN, i=1,2,3, where φ^1SE≡α^SE, φ^2SE≡θ^1SE, and φ^3SE≡b^SE.(11)
Then, the1001−γ% credible CIs for φi are given by φ^iSEγN/2,φ^iSE1−γ/2N,i=1,2,3.


## 5.1. Approximate CIs

On the basis of the asymptotic distributions of the unknown parameters’ MLEsα, θ1, and b, the parameters’ normal approximation confidence intervals (CIs) are determined where the MLEs’ asymptotic distribution of α,θ1,b is given by Miller [38].(28)θ^1−θ1,α^−α,b^−b∼N0,η,and we mut consider η=κij,i,j=1,2,3, is the parameters’ variance-covariance matrix.ϑi CIs are calculated by the following formula:(29)ϑ^iL,ϑ^iU=ϑ^i±Zζ/2κii,i=1,2,3,where ϑ1=α, ϑ2=θ1, and ϑ3=b. For more information and reading, we can refer to Miller [38] distribution.


## 5.2. Bootstrap CIs

For the unknown parameters, we get percentile Bootstrap (Boot-p) confidence intervals (CIs). For more reading on Bootstrap (Boot-p) CIs, see Efron et al. [39] (Algorithms 2 and 3).Algorithm 2: Bootstrap CIs.
(1)
Determine the parameters’ MLE estimates by finding a solution numerically to (16)–(18).(2)
The approximation in step 1 can be used to produce a bootstrap sampleX¯∗ utilising the algorithm developed by [39]. Let us assume that the estimates of the distribution’s parameters are α^∗, θ1^∗, and b^∗.(3)
In order to produce a bootstrap, we will iterate the first two steps 1000 iteration.(4)
Ascend the list of estimates such thatϕt1,ϕt2,…,ϕtN,t=1,2,3, where ϕ1=α^∗,ϕ2=θ1^∗,andϕ3=b^∗. Let Gx=Pϕt≤x be the CDF of ϕt.(5)
Defineϕtboot=G−1x for given x. As a result, the upper bounds and lower bounds are defined as follows: ϕtbootβ/2,ϕtboot1−β/2.Algorithm 3: The steps for the Bootstrap-t CIs.
(1)
Repeat the first two steps in the percentile Algorithm 2(2)
EvaluateT=ϑ^b−ϑ^/Vϑ^b, where ϑ is α,θ1, or b and Vϑ^b is the asymptotic variances of ϑ^b.(3)
Repeat the above two stepsℬ times and rearrange T1,T2,…,Tℬ in ascending order.(4)
A two-side1001−γ% bootstrap-t CI for α,θ1, and θ is given by α−Tαbℬγ/2Vαb,α+VαbTαbℬ1−γ/2, θ1−Tθ1bℬγ/2Vθ1b,θ1+Vθ1bTθ1bℬ1−γ/2, and b−Tbbℬγ/2Vbb,b+VbbTbbℬ1−γ/2.


## 5.3. Highest Posterior Density (HPD) Interval Algorithm

Here, we introduced the algorithm for finding1−γ% HPD interval for α,θ1,andb (credible intervals), where the algorithm was proposed by Chen and Shao [40] (Algorithm 4).Algorithm 4: Credible interval algorithm.
(1)
Use the initial for the parameters as the MLEs values for the MH algorithm as follows:α0=α^, θ10=θ^1, and b0=b^.(2)
Consider this iteration as first iteration withi=1.(3)
Now, we will generateα∗ from proposals distribution from Nαi−1,/Nαi−1,varαi−1/, where N is considered as the normal distribution.(4)
In order to decide which value will accept or even reject, we must estimate the acceptance probabilityAαi−1|α∗=min1,π∗α∗θ1i−1,bi−1/π∗αi−1≥|θ1i−1,bi−1.(5)
Now, in order to complete the steps, we must generate a value from zero to one so we will use the uniform distribution asU0,1.(6)
IfU≤Aαi−1|α∗, then put αi=α∗; else put αi=αi−1.(7)
There are two rest parameters needed to be estimated, so we will repeat the above steps from ((2)–(6)) for the other two parameters,θ1 and b.(8)
Puti=i+1.(9)
Repeat the steps ((3)–(8))N=10000 times to get α1,θ11,b1, …, αN,θ1N,bN.(10)
After obtaining the estimates for theN=10,000 parameters, arrange each estimate in ascending order as φ^iSE1,φ^iSE2,…,φ^iSEN, i=1,2,3, where φ^1SE≡α^SE, φ^2SE≡θ^1SE, and φ^3SE≡b^SE.(11)
Then, the1001−γ% credible CIs for φi are given by φ^iSEγN/2,φ^iSE1−γ/2N,i=1,2,3.


## 6. Simulation Studies

We made the simulation study to assess the performance of the distribution when using ALT under PT-II censored sample. We estimate the points and interval estimations of the distribution parameters. After estimating the parameters, we evaluated the Bias, mean square errors (MSEs), and length of CI. We used various sizes for samples in both designs whereni and mi, i=1,2,…,k, are the sample size at level i, i=1,2,…,k, and fixed number of failures, respectively. Also, we used different values for the binomial removal parameter P for the censoring schemes (Rij, j=1,2,…,mi). Additionally, a simulation study is conducted on two different PSALT designs where the first is a simple ramp-stress with two stress levels k=2, where k=2 is the number of levels, and their ramp values are h1=50 and h2=70. The second is a multiple ramp-stress test, which includes four different levels of stress with ramp values k=4h1=50, h2=70, h3=90, and h4=120. Tables 1–4 illustrate the values for the distribution parameters conducted from simulation using Bayesian and classical methods under SE, and their associated CIs are tabulated in Tables 5–8. We used the true values for the parameters as follows:Case 1:α=0.7;θ1=1.3;b=0.5Case 2:α=1.7;θ1=1.3;b=2.5Table 1
Bias and MSE for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 1,k = 2.
α=0.7,θ1=1.3,b=0.5h1=50,h2=70MLEBayesainSELINEXc=−0.5LINEXc=0.5n1,n2Pm1,m2BiasMSEBiasMSEBiasMSEBiasMSE17, 130.2512, 9α0.44500.29800.23470.09980.24420.10730.22530.0928θ10.89942.2620.52510.50150.58520.62580.47140.4076b0.04890.84050.11950.16000.15090.18760.08880.135615, 12α0.39390.21920.20820.06920.21480.07320.20160.0654θ10.47440.98870.31240.21280.34560.25040.28170.1822b0.18270.71860.12410.13280.15580.15660.09360.11300.8512, 9α0.48680.33560.25840.11070.26740.11780.24940.1038θ10.68671.64410.34870.24940.39210.31100.30980.2038b0.11471.01580.10000.13630.13120.16160.06990.114815, 12α0.40330.22930.21560.07480.22230.07910.20900.0707θ10.49170.99960.29350.18260.32770.21810.26240.1542b0.11770.66880.10050.11290.12940.12980.07280.098833, 270.2523, 19α0.36950.16780.20230.05410.20600.05600.19860.0523θ10.51140.80930.31040.17340.33530.19740.28720.1534b0.06030.45390.11160.11810.13840.13560.08590.103330, 24α0.33100.13610.18160.04390.18450.04510.17880.0426θ10.37340.51500.24510.11420.26450.12830.22690.1020b0.08500.37520.08020.07920.10170.08830.05940.07150.8523, 19α0.37070.16760.20550.05550.20920.05740.20180.0537θ10.45770.70670.26550.12710.28800.14410.24440.1123b0.05820.43270.09280.09940.11810.11390.06830.086830, 24α0.32650.12930.17920.04150.18210.04270.17640.0403θ10.41130.53030.25150.11460.27080.12790.23310.1030b0.01880.32360.06980.08260.09280.09360.04760.073555, 650.2538, 46α0.32710.12030.18330.03910.18500.03980.18160.0384θ10.38020.38650.23190.08760.24450.09510.21990.0807b0.02560.23240.08370.06300.10190.07010.06590.056950, 58α0.29920.09910.16700.03180.16830.03230.16570.0314θ10.36550.32360.20940.07130.22000.07700.19920.0661b−0.01110.16350.07930.04740.09560.05280.06330.04270.8538, 46α0.31050.11040.17350.03580.17520.03650.17180.0352θ10.33150.33080.21220.08370.22450.09070.20040.0773b0.06310.24020.10680.08190.12650.09170.08760.073250, 58α0.29140.09570.16230.03080.16360.03120.16100.0303θ10.37360.34770.21070.07100.22120.07650.20060.0659b−0.01390.18680.06970.05200.08560.05740.05410.0474Table 2
Bias and MSE for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 2,k = 2.
α=1.7,θ1=1.3,b=2.5h1=50,h2=70MLEBayesainSELINEXc=−0.5LINEXc=0.5n1,n2Pm1,m2BiasMSEBiasMSEBiasMSEBiasMSE17, 130.2512, 9α1.67783.43281.36292.62751.46263.07571.25392.1626θ10.19220.07730.08290.01390.08470.01430.08110.0135b0.01580.30560.05050.03230.06530.03460.03600.030615, 12α1.60323.08741.25042.10551.32062.37191.17341.8149θ10.15000.04600.05190.00650.05340.00660.05050.0063b−0.01160.25060.03080.03000.04290.03130.01890.02900.8512, 9α1.50692.84721.40912.70381.51003.15151.29872.2347θ10.14180.05370.05220.00780.05390.00810.05040.0076b0.00540.33950.02770.03390.04210.03520.01350.033215, 12α1.43702.76181.28672.25201.36012.54611.20451.9261θ10.15120.05110.04920.00650.05060.00670.04770.0064b−0.03070.25100.02010.02600.03200.02670.00830.025633, 270.2523, 19α1.31472.56191.21471.83071.25631.97321.16871.6726θ10.15110.03910.05370.00570.05460.00590.05280.0056b0.00320.13310.03270.01810.04030.01870.02530.017630, 24α1.29042.20731.10831.47681.13891.56491.07521.3809θ10.13080.02920.04070.00370.04150.00380.04000.0037b0.00760.12080.03010.01740.03640.01780.02380.01700.8523, 19α1.21032.37181.20071.79241.24191.92481.15591.6490θ10.13820.03620.04500.00480.04600.00490.04410.0047b0.00370.15680.02990.01960.03770.02020.02220.019230, 24α1.14872.12081.08661.43181.11661.51551.05441.3419θ10.11950.02640.03630.00330.03710.00330.03560.0032b0.03680.12810.03730.01800.04380.01860.03080.017555, 650.2538, 46α1.17331.31621.01441.15311.03181.19490.99561.1079θ10.12590.02240.03610.00260.03660.00260.03570.0025b0.00790.07220.03020.01130.03440.01160.02590.011150, 58α1.06281.28750.95410.99930.96771.02850.93960.9684θ10.12350.02030.03370.00220.03410.00220.03340.0021b−0.00440.05680.02250.00880.02590.00900.01900.00860.8538, 46α1.17011.21880.99801.12611.01561.16850.97901.0804θ10.12250.02160.03400.00240.03440.00250.03350.0024b−0.00030.07460.02310.01110.02750.01130.01880.010950, 58α1.05621.18470.94660.98040.96001.00930.93230.9497θ10.12770.02160.03490.00220.03530.00230.03450.0022b−0.02320.05450.01450.00790.01800.00800.01110.0078Table 3
Bias and MSE for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 2,k = 4.
α=1.7,θ1=1.3,b=2.5h1=50,h2=70,h3=90,h4=120MLEBayesainSELINEXc = −0.5LINEXc = 0.5n1,n2,n3,n4Pm1,m2,m3,m4BiasMSEBiasMSEBiasMSEBiasMSE9, 8, 7, 60.256, 6, 5, 4α1.58862.31901.11711.76281.19822.04081.03021.4780θ10.22820.07640.09590.01390.09750.01420.09440.0135b−0.00090.0991−0.02660.0136−0.02120.0135−0.03190.01398, 7, 6, 5α1.36802.15801.14471.78741.21172.01961.07151.5392θ10.15890.04320.05780.00640.05910.00650.05660.0062b−0.03020.0802−0.02420.0105−0.01980.0104−0.02860.01080.856, 6, 5, 4α1.56812.27031.37552.46461.46912.85311.27212.0628θ10.14850.04250.05470.00630.05600.00650.05340.0062b−0.01750.0873−0.01950.0110−0.01440.0110−0.02460.01118, 7, 6, 5α1.34892.03211.20341.90731.27072.14561.12971.6540θ10.13280.03640.04280.00520.04400.00530.04160.0050b0.01480.0687−0.00300.00890.00150.0090−0.00750.008920, 18, 15, 70.2514, 13, 11, 5α1.49971.79651.02781.30911.06281.40460.9901.2059θ10.17130.03870.06280.00580.06350.00590.06200.0057b−0.04820.0592−0.03340.0095−0.03030.0093−0.03640.009618, 16, 14, 6α1.08991.36931.03451.26011.06301.33461.00381.1804θ10.13250.02550.03870.00300.03930.00310.03810.0030b−0.01920.0408−0.01360.0060−0.01120.0059−0.01600.00600.8514, 13, 11, 5α1.30401.69901.12151.54261.16021.66451.07891.4101θ10.13450.02820.04210.00370.04280.00380.04130.0037b0.00190.0531−0.00970.0078−0.00670.0077−0.01270.007818, 16, 14, 6α1.18591.49891.03821.27261.06651.34841.00731.1902θ10.12150.02240.03290.00250.03350.00260.03230.0025b−0.00450.0382−0.00650.0056−0.00400.0057−0.00890.005745, 35, 25, 150.2531, 24, 18, 10α1.00771.24580.98911.09181.00721.13430.9691.0457θ10.13760.02440.04180.00290.04210.00290.04140.0029b−0.03630.0236−0.02380.0041−0.02240.0040−0.02520.004140, 32, 22, 14α0.96460.98280.93570.95700.94970.98710.92040.9242θ10.12150.01890.03080.00180.03110.00180.03050.0018b0.00150.0192−0.00380.0030−0.00270.0030−0.00500.00300.8531, 24, 18, 10α1.00191.20600.97751.06990.99511.11100.95841.0255θ10.12350.02020.03300.00210.03340.00210.03260.0021b0.00210.0236−0.00520.0036−0.00370.0036−0.00670.003640, 32, 22, 14α0.95630.97880.93110.95640.94530.98710.91570.9227θ10.11730.01780.02870.00160.02900.00170.02830.0016b0.00390.0174−0.00240.0027−0.00120.0027−0.00360.0027Table 4
Bias and MSE for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 1,k = 4.
α=0.7,θ1=1.3,b=0.5h1=50,h2=70,h3=90,h4=120MLEBayesainSELINEXc = −0.5LINEXc = 0.5n1,n2,n3,n4Pm1,m2,m3,m4BiasMSEBiasMSEBiasMSEBiasMSE9, 8, 7, 60.256, 6, 5, 4α0.39790.27950.19330.07750.20220.08370.18460.0717θ10.91291.39100.65200.62330.69980.72890.60820.5353b−0.04870.42120.11550.11120.14250.12790.08960.09738, 7, 6, 5α0.40320.23380.20460.06890.21150.07320.19770.0649θ10.56370.67560.35540.20150.38050.23100.33260.1776b−0.05580.29930.08000.06200.10180.07110.05910.05450.856, 6, 5, 4α0.51470.41120.26800.11990.27750.12810.25860.1120θ10.53200.61920.35000.21770.37880.26330.32390.1824b−0.06270.33160.07550.07520.09870.08660.05310.06568, 7, 6, 5α0.40790.23270.21240.07120.21890.07510.20600.0676θ10.36620.35580.22980.09420.24740.10670.21330.0836b0.01490.31500.10830.09500.13170.10860.08560.083220, 18, 15, 70.2514, 13, 11, 5α0.34360.14990.18040.04470.18400.04640.1760.0431θ10.69070.77890.41610.24510.43640.27060.39740.2234b−0.18100.25970.04390.04980.06070.05490.02760.045518, 16, 14, 6α0.32970.13470.17710.04170.17990.04290.17430.0405θ10.39340.30050.22850.08070.23800.08650.21930.0754b−0.06670.15810.05670.03930.07120.04340.04270.03590.8514, 13, 11, 5α0.37960.17660.20680.05600.21050.05790.20320.0542θ10.38630.30610.23690.09040.24850.09800.22580.0836b−0.01570.20360.07320.05220.08980.05830.05690.047018, 16, 14, 6α0.34200.14050.18470.04330.18750.04450.18190.0421θ10.30020.17290.18970.05580.19790.05980.18180.0521b0.04250.15240.09540.05100.11090.05720.08030.045545, 35, 25, 150.2531, 24, 18, 10α0.33430.12790.18480.04070.18660.04140.1830.0400θ10.43400.27330.25170.08390.25840.08800.24510.0801b−0.12010.10830.02670.02260.03620.02420.01740.021340, 32, 22, 14α0.29990.10080.16270.03060.16390.03110.16140.0302θ10.28440.12210.16280.03770.16690.03940.15860.0362b−0.01230.07010.06030.02290.06820.02460.05250.02140.8531, 24, 18, 10α0.33950.13060.18820.04160.19000.04240.18650.0409θ10.31930.15420.18730.04780.19280.05020.18200.0455b−0.01350.09350.06360.02710.07320.02930.05410.025140, 32, 22, 14α0.29450.09650.15940.02920.16070.02960.15820.0287θ10.27390.10930.15770.03440.16180.03590.15370.0330b0.00770.06840.07040.02230.07830.02400.06250.0208Table 5
Length of confidence interval for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 1,k = 2.
α=0.7,θ1=1.3,b=0.5h1=50,h2=70MLEBayesianSELINEXc=−0.5LINEXc=0.5n1,n2Pm1,m2L.CIL.BPL.BtCP (%)L.CCIL.BPL.BtL.CCIL.BPL.BtL.CCIL.BPL.Bt17, 130.2512, 9α1.24120.05690.055794.800.71840.03850.03920.74270.03720.03610.70010.03650.0371θ14.67140.20790.206594.801.60770.08300.08301.71340.09340.09081.50100.07700.0756b3.59230.16460.163496.401.23390.06690.06711.30030.06930.06911.17470.06240.062015, 12α0.99340.04320.043394.800.59080.02900.02890.59940.02880.02860.58230.02700.0268θ13.42900.15790.158295.181.22000.06090.06131.27830.06160.06171.17800.05560.0553b3.24810.15650.151897.581.22320.05970.06001.28250.06470.06411.13600.05780.05550.8512, 9α1.23260.05630.056994.600.72350.03520.03530.73910.03810.03850.70960.03520.0351θ14.24910.19060.189794.601.31500.06470.06441.38460.06690.06781.21570.05950.0582b3.92910.18400.184196.401.13300.06710.06701.18790.06500.06601.07850.05560.055515, 12α1.01300.04650.046695.600.57520.03090.03030.58600.03070.03100.56670.02820.0277θ13.41600.14800.145995.001.10470.05610.05621.15190.06050.05951.04600.05120.0512b3.17560.14130.142096.481.11630.05950.05951.19210.06260.06251.02800.05260.052533, 270.2523, 19α0.69350.03110.031294.600.40420.02000.02000.40920.02110.02080.39910.01920.0192θ12.90430.13100.131094.000.96520.04830.04831.02540.05120.05130.93100.04940.0494b2.63290.12030.120196.201.04410.05410.05351.09820.06060.06090.97630.05410.054630, 24α0.63860.02940.029494.960.38580.01860.01880.38820.01900.01890.38210.01830.0181θ12.40460.10500.104894.200.89220.04100.04180.91530.04250.04240.86340.04020.0404b2.38020.10420.102997.000.91780.04640.04570.96300.05000.05000.87140.04460.04560.8523, 19α0.68160.02980.029894.400.44740.02070.02070.45240.02110.02120.44350.02030.0202θ12.76680.12090.121193.400.87110.04170.04210.90100.04190.04170.84100.03870.0389b2.57110.11600.115993.720.99360.05260.05371.03990.05600.05580.92620.05170.051730, 24α0.59160.02640.026194.800.37570.01640.01640.37840.01720.01710.37320.01680.0168θ12.35820.10100.101294.600.80280.03900.03910.82710.04110.04100.78620.03950.0394b2.23100.10370.104695.400.98210.04570.04621.02200.04900.04870.92860.04650.046455, 650.2538, 46α0.45350.02010.020395.800.27310.01300.01300.27490.01270.01260.27100.01350.0133θ11.93000.08830.090695.000.68160.03400.03370.69860.03360.03370.66820.03140.0315b1.88900.08620.086695.200.86430.04250.04230.89560.04240.04280.84460.04040.040050, 58α0.38460.01740.017396.520.24160.01090.01120.24280.01090.01080.24040.01170.0117θ11.71050.07810.076895.480.60850.03060.03020.61760.02900.02910.59910.02870.0285b1.58630.07100.070895.900.73890.03570.03570.76200.03620.03630.71740.03570.03510.8538, 46α0.46380.02090.020894.200.29570.01270.01260.29730.01300.01310.29400.01270.0128θ11.84430.08000.082194.600.72180.03620.03570.74390.03440.03440.70540.03540.0351b1.90710.07970.080696.000.89520.04760.04730.92440.05180.05180.87180.04590.044950, 58α0.40750.01750.017095.800.24470.01130.01120.24620.01170.01160.24290.01180.0118θ11.79030.08330.081595.600.61450.02780.02850.62170.02880.02840.60420.02810.0275b1.69490.07690.077796.540.77900.03950.03910.79380.04050.04010.75260.03710.0377Table 6
Length of confidence interval for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 2,k = 2.
α=1.7,θ1=1.3,b=2.5h1=50,h2=70MLEBayesianSELINEXc=−0.5LINEXc=0.5n1,n2Pm1,m2L.CIL.BPL.BtCP (%)L.CCIL.BPL.BtL.CCIL.BPL.BtL.CCIL.BPL.Bt17, 130.2512, 9α5.56170.35250.342396.203.27630.15650.15553.53990.17290.17212.89290.12760.1279θ10.78790.03470.034794.800.30940.01520.01520.31050.01420.01410.30880.01470.0146b2.16820.09300.094594.600.65010.02970.02960.65980.03100.03090.64960.03060.030815, 12α4.15790.23270.231995.802.47750.12890.12892.63450.13850.13632.29780.11140.1136θ10.60220.02650.027394.800.23610.01010.01020.23670.01080.01080.23580.01050.0105b1.96380.09050.090394.400.64710.02930.02920.64730.03040.02950.64840.02990.03010.8512, 9α5.07860.32030.316595.803.00380.15230.15383.24510.16220.16292.65920.12590.1271θ10.71930.03320.033295.000.28020.01220.01220.28120.01220.01220.27840.01260.0126b2.28610.10090.100894.600.69030.03180.03220.69620.03160.03160.67760.03190.031515, 12α4.87460.27840.268895.602.68760.13580.13432.85280.14380.14422.45370.11960.1191θ10.65970.02980.029895.400.24970.01140.01140.25050.01070.01070.24950.01080.0108b1.96210.09000.091993.400.59430.02790.02800.59660.02820.02820.60070.02890.028833, 270.2523, 19α3.82980.16560.166494.802.14520.09570.09502.24640.11230.11272.01940.10080.0995θ10.50140.02240.022394.200.20970.00900.00910.21010.00970.00950.20920.00980.0100b1.43140.06340.063394.400.50080.02310.02310.50500.02300.02300.49440.02160.021930, 24α2.99090.13380.136395.801.81310.09040.09041.87230.09070.09071.74430.08050.0785θ10.43130.01950.019495.000.17610.00830.00820.17670.00820.00820.17610.00770.0077b1.36350.06270.063095.800.47020.02310.02320.47070.02090.02110.47000.02190.02200.8523, 19α3.82010.16460.164695.002.14780.10060.10062.23450.11080.11102.04580.09820.0962θ10.51240.02270.022694.600.19630.00900.00900.19680.00950.00950.19580.00930.0093b1.55370.07000.071495.400.50920.02450.02490.50960.02520.02520.50960.02390.023430, 24α3.07200.14180.140995.001.87480.08650.08651.93080.09500.09351.79220.08440.0832θ10.43160.01900.018394.800.16560.00820.00830.16610.00780.00790.16540.00800.0080b1.39720.06480.065595.400.47570.02220.02240.47560.02370.02410.47550.02280.022755, 650.2538, 46α2.19220.09930.102296.601.30470.06050.06021.33420.06280.06261.27140.06050.0602θ10.31730.01360.013894.400.13900.00600.00600.13900.00590.00590.13890.00630.0063b1.05370.04700.046895.200.38200.01850.01880.38220.01780.01780.38190.01920.019250, 58α1.86350.08400.083795.801.12690.05120.05081.14070.05520.05501.10610.05300.0530θ10.27940.01240.012695.800.12160.00560.00570.12160.00560.00550.12150.00540.0054b0.93470.04260.042695.000.35930.01660.01670.35790.01610.01600.35840.01650.01650.8538, 46α2.12930.09960.096197.001.29950.06800.06801.31500.06830.06801.26330.06210.0611θ10.31830.01370.013396.000.13370.00640.00630.13390.00630.00630.13360.00600.0060b1.07150.04750.047695.200.38460.01870.01830.38380.01800.01800.38220.01730.017350, 58α1.79770.07970.078695.601.09760.05030.05061.11290.05320.05251.06500.05030.0504θ10.28660.01300.013095.400.11900.00550.00550.11920.00520.00510.11870.00570.0056b0.91170.04240.042494.800.34390.01590.01610.34410.01520.01520.34430.01550.0153Table 7
Length of confidence interval for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 2,k = 4.
α=1.7,θ1=1.3,b=2.5h1=50,h2=70,h3=90,h4=120MLEBayesianSELINEXc=−0.5LINEXc=0.5n1,n2,n3,n4Pm1,m2,m3,m4L.CIL.BPL.BtCP (%)L.CCIL.BPL.BtL.CCIL.BPL.BtL.CCIL.BPL.Bt9, 8, 7, 60.256, 6, 5, 4α5.34600.21510.210994.702.57710.08680.08692.76450.09950.09902.34780.07830.0788θ10.61140.01950.019595.400.25750.00810.00810.25890.00840.00830.25500.00870.0086b1.23430.03920.038194.500.44580.01400.01380.44750.01370.01330.44260.01410.01418, 7, 6, 5α4.50860.17240.170594.602.49160.08560.08572.65960.10130.10082.26260.07630.0748θ10.52580.01650.016595.700.21220.00680.00680.21260.00660.00660.21220.00690.0069b1.10460.03570.036194.900.37560.01220.01200.37180.01210.01230.37340.01280.01280.856, 6, 5, 4α5.25750.20490.207696.102.74300.09290.09412.99170.10630.10632.42580.08260.0829θ10.56070.01710.017495.000.22230.00690.00690.22260.00740.00740.22180.00730.0073b1.15690.03430.034395.400.39090.01360.01370.39350.01320.01300.38980.01250.01258, 7, 6, 5α4.26640.15860.155195.302.42610.07760.07742.61750.08940.08922.23620.07480.0759θ10.53760.01740.017394.300.22190.00710.00690.22150.00760.00760.22140.00780.0077b1.02620.03240.031995.000.35880.01170.01190.36440.01200.01170.35660.01170.011820, 18, 15, 70.2514, 13, 11, 5α3.52820.10690.106694.801.85130.06490.06491.93040.06510.06511.76330.06040.0606θ10.38030.01260.012394.700.17510.00530.00530.17560.00540.00540.17460.00530.0053b0.93560.02990.029994.900.35690.01120.01110.35720.01120.01150.35670.01110.011118, 16, 14, 6α2.94540.09370.093895.401.66390.05520.05501.72550.05710.05611.59990.05090.0510θ10.34920.01050.010595.500.14510.00470.00470.14530.00490.00490.14500.00460.0046b0.78850.02570.025494.800.28880.00970.00970.29020.00980.00980.28800.00940.00960.8514, 13, 11, 5α3.56730.11060.111795.201.90790.06340.06452.00950.06850.06881.77950.06270.0627θ10.39500.01260.012694.500.17440.00600.00590.17470.00550.00560.17400.00560.0056b0.90370.02830.028495.400.32590.01150.01140.32870.01100.01100.32510.01100.011118, 16, 14, 6α2.86750.09330.094895.951.61170.05520.05581.65860.05800.05771.53130.05400.0522θ10.34370.01100.011094.900.14740.00490.00500.14790.00480.00480.14750.00450.0045b0.76600.02560.025895.790.28010.00970.00970.28000.00910.00900.28030.00930.009345, 35, 25, 150.2531, 24, 18, 10α2.22020.06850.069195.901.26790.04290.04301.30240.04250.04231.23590.04240.0422θ10.29120.00950.009494.900.13020.00410.00420.13030.00440.00440.13010.00440.0044b0.58550.01720.017394.800.23120.00720.00720.23160.00730.00740.23130.00720.007140, 32, 22, 14α1.82980.05800.058295.201.07260.03470.03451.09480.03730.03711.04370.03380.0337θ10.25150.00760.007795.100.11220.00360.00360.11240.00370.00370.11210.00370.0036b0.54330.01670.016894.900.21010.00670.00660.20990.00660.00660.20980.00660.00670.8531, 24, 18, 10α2.16200.06870.068796.101.27590.04280.04291.30110.04290.04301.23240.03910.0391θ10.27520.00890.008795.600.12120.00390.00400.12130.00400.00400.12110.00400.0040b0.60240.01980.019895.100.23140.00750.00750.23120.00760.00760.23160.00710.007140, 32, 22, 14α1.88310.05830.058596.851.11470.03740.03711.13810.03820.03781.08800.03630.0373θ10.24820.00750.007495.950.10970.00370.00380.10970.00370.00370.10960.00360.0036b0.51680.01640.016695.200.20370.00640.00630.20420.00670.00660.20350.00660.0067Table 8
Length of confidence interval for MLE and Bayesian estimates of MKEx based on ALT parameters: Case 1,k = 4.
α=0.7,θ1=1.3,b=0.5h1=50,h2=70,h3=90,h4=120MLEBayesianSELINEXc=−0.5LINEXc=0.5n1,n2,n3,n4Pm1,m2,m3,m4L.CIL.BPL.BtCPL.CCIL.BPL.BtL.CCIL.BPL.BtL.CCIL.BPL.Bt9, 8, 7, 60.256, 6, 5, 4α1.36620.06020.05840.95600.69430.03410.03450.71550.03660.03680.67480.03450.0343θ12.93000.12760.12840.94601.53480.08210.08061.64080.09690.09571.43970.06900.0685b2.53940.11260.11280.96801.07900.05340.05351.13990.06140.06061.04470.05120.05018, 7, 6, 5α1.04740.04500.04510.94800.61600.02880.02880.63400.02930.02980.60050.02740.0275θ12.34720.10180.10150.94001.00680.04950.04831.04300.05170.05130.94860.04570.0458b2.13540.09610.09420.96200.86520.04160.04230.89560.04720.04620.81730.04040.04030.856, 6, 5, 4α1.50060.06770.06710.95800.72950.03880.03860.74200.03940.03940.71100.03780.0382θ12.27530.10150.10150.95400.96880.05340.05341.03360.06350.06370.92480.04900.0475b2.24610.10640.10650.97000.94010.04740.04760.95450.05090.05130.88970.04320.04268, 7, 6, 5α1.01070.04460.04390.94600.59960.03090.03090.61060.02910.02860.59040.02910.0291θ11.84760.08390.08490.94400.74600.03760.03760.78240.03610.03670.72170.03410.0341b2.20150.10040.09980.96801.02930.05150.05111.06750.05540.05510.99370.04740.047920, 18, 15, 70.2514, 13, 11, 5α0.69980.03160.03180.95200.39740.01990.01970.40250.01860.01870.39210.01920.0191θ12.15550.09640.09780.94400.99730.04680.04691.03740.04930.04980.95710.04520.0461b1.86920.08450.08490.96600.80650.03770.03770.83480.04110.04030.77010.03540.035918, 16, 14, 6α0.63350.02890.02810.94600.38670.01830.01830.38930.01760.01750.38410.01730.0171θ11.49800.06910.06950.95800.59610.02890.02900.60490.03160.03130.58230.02920.0294b1.53790.06590.06690.94200.73300.03230.03210.75550.03450.03450.70680.03210.03200.8514, 13, 11, 5α0.70760.03120.03120.94200.44760.02000.02030.45370.01970.02000.44150.01920.0190θ11.55420.06840.06870.94200.65810.03210.03210.67210.03430.03330.64570.03150.0316b1.76960.08270.08250.95400.72780.03860.03890.76540.04080.04070.69180.03930.039118, 16, 14, 6α0.60180.02720.02680.95400.34920.01750.01760.35160.01760.01720.34650.01670.0169θ11.12930.04970.05010.95200.51880.02420.02380.53080.02650.02630.50840.02340.0235b1.52300.06610.06660.94400.71480.03700.03810.73230.03640.03610.70580.03260.032445, 35, 25, 150.2531, 24, 18, 10α0.49910.02310.02310.95000.30300.01390.01390.30440.01410.01400.30110.01400.0141θ11.14320.05380.05340.96200.52260.02500.02500.52550.02570.02540.51540.02500.0248b1.20210.05290.05310.94400.52280.02640.02670.53340.02790.02790.51260.02520.025340, 32, 22, 14α0.40860.01830.01820.95000.24300.01150.01150.24400.01170.01150.24250.01100.0110θ10.79630.03660.03630.94600.40390.01870.01910.40930.01820.01820.40130.01780.0180b1.03770.04490.04500.95400.55020.02610.02550.55930.02560.02530.54430.02390.02360.8531, 24, 18, 10α0.48660.02150.02190.95200.29800.01400.01400.29880.01400.01420.29590.01380.0139θ10.89710.04290.04260.95800.40950.02070.02060.41460.02000.01920.40470.01920.0191b1.19870.05500.05440.94800.55660.02640.02630.57010.02680.02670.55160.02690.026940, 32, 22, 14α0.38720.01750.01720.95400.23420.01120.01110.23500.01130.01110.23340.01110.0111θ10.72600.03200.03160.95000.36120.01740.01690.36490.01760.01760.35800.01750.0173b1.02580.04740.04840.95400.47410.02320.02330.48490.02420.02420.46930.02270.0224The MCMC iteration plots and the kernel histograms of the posterior samples of the parametersα, θ1, and b are shown in Figure 2.Figure2 shows that the MCMC samples are mixed well and achieved stationary. Also, it indicates that posterior distributions of the three parameters are symmetric. The simulation study is carried out using Algorithm 5.Figure 2
MCMC iterations plots and the kernel histograms of the posterior samples with true values for each parameters: in case 1,γ=0.7;σ1=1.3;θ=0.5,n1=17;n2=13,m1=12;m2=9; and P=0.8.Algorithm 5: The steps followed in the whole simulation process in the simulation section.
(1)
Choose appropriate initial values of the distribution parameters, the number of level in each experimentk, the ramp value ui, the number of items in each level ni, and the number at which the experiment will be stopped, mi, where i=1,2,3,…,k.(2)
Create random samples of appropriate sizemi based on the number of levels in the test, from uniform distribution 0,1; this random sample is ordered as follows: Ui1,Ui2,…,Uimi, i=1,2,3,…,k.(3)
We will binomially generate the CS andRij values with parameter P, i=1,2,3,…,k and j=1,2,…,mi, with consideration of ∑j=1miRij=ni−mi.(4)
SetEij=Uij1/j+∑d=mi−j+1miRid, j=1,2,…,mi and i=1,2,3,…,k.(5)
ObtainUi1∗,Ui2∗,…,Uimi∗, where Uij∗=1−∏d=mi−j+1miEid, j=1,2,…,mi and i=1,2,3,…,k.(6)
Use the mathematica 12 program to produce and generate different random samplesti1,ti2,…,timi, i=1,2,…,k, from (5) as follows: tij=θ1b+1u1/uiblog−log1−Uij∗1α+1,j=1,2,…,miandi=1,2,…,k.(7)
In step 6, the samples generated are used to determine the parameters’ MLEs by solving the nonlinear system in (16)–(19).(8)
Calculate the BEs through using the MH algorithm under SE and LINEX conditions specified in Algorithm 1.(9)
Calculate the 95% confidence intervals (CIs) for the parameters in equation (29).(10)
Determine the bootstrap confidence intervals (CIs) for the parameters by using Algorithms 2 and 3 and evaluate the credible interval using Algorithm 4.(11)
Repeat the preceding steps from2−10, a total of 1000 times.(12)
Calculate the mean values of the Bias, MSE, and the length of the confidence intervals for the model’s parameters.


### 6.1. Concluded Remarks from the Results in Tables1–8 in the Simulation Section

In this part of the paper, we illustrate the observations concluded from the simulation table to make the readability of the paper very high. According to MSEs and Bias, BEs of model parameters provide more accurate results than MLE. We can say so because all the Bayesian estimates have lower MSE and Bias than the MLE. All the estimates satisfy the consistence property such that the Bias, MSE, and all CIs lengths decreased as the total sample sizen increases for the parameters of MK distributions. For fixed values of the parameters fixed and sample size, the Bias, MSE, and L.CI decrease as P increases. Bayesian estimation is a better estimation method than classical methods regarding the values of the MSE and Bias. Bayesian estimation under LINEX loss function is better than that under SE loss function in most cases. We can say so because all the Bayesian estimates using the LINEX loss function have lower MSE and Bias than the SE loss function. Multiple ramp designs are better than simple ramp designs for the same sample size in both designs as MSE for the parameters is smaller in multiple ramps than that in simple ramp design. The bootstrap CIs are shorter in length than the corresponding approximate CIs. The bootstrap-t CIs are the shortest one compared with the other CIs such as BP and the approximate CIs. At last, LINEX loss function with parameter c=0.5 is better than that with c=−0.5; we can say so because all the Bayesian estimates using the LINEX with c=0.5 have lower MSE and Bias than the other one.


## 6.1. Concluded Remarks from the Results in Tables1–8 in the Simulation Section

In this part of the paper, we illustrate the observations concluded from the simulation table to make the readability of the paper very high. According to MSEs and Bias, BEs of model parameters provide more accurate results than MLE. We can say so because all the Bayesian estimates have lower MSE and Bias than the MLE. All the estimates satisfy the consistence property such that the Bias, MSE, and all CIs lengths decreased as the total sample sizen increases for the parameters of MK distributions. For fixed values of the parameters fixed and sample size, the Bias, MSE, and L.CI decrease as P increases. Bayesian estimation is a better estimation method than classical methods regarding the values of the MSE and Bias. Bayesian estimation under LINEX loss function is better than that under SE loss function in most cases. We can say so because all the Bayesian estimates using the LINEX loss function have lower MSE and Bias than the SE loss function. Multiple ramp designs are better than simple ramp designs for the same sample size in both designs as MSE for the parameters is smaller in multiple ramps than that in simple ramp design. The bootstrap CIs are shorter in length than the corresponding approximate CIs. The bootstrap-t CIs are the shortest one compared with the other CIs such as BP and the approximate CIs. At last, LINEX loss function with parameter c=0.5 is better than that with c=−0.5; we can say so because all the Bayesian estimates using the LINEX with c=0.5 have lower MSE and Bias than the other one.


## 7. Application

We applied a practical experiment in this section to check the superiority of the MKEx distribution against its competitors. So, we used a reliability experiment conducted previously by Zhu [41] where the data in Table 9 are from ramp-voltage experiments with miniature light bulbs. The ramp-voltage experiments used 62 light bulbs in the first level of stress and 61 light bulbs in the second level of stress, with ramp rates of 2.01 V/h and 2.015 V/h, which are defined, respectively, as well as experimental design stress of 2 V. The data presented in Table 9 represents a complete sample and not a censored sample.Table 9
Practical failure times (FTs) for the items under experiment, for both stress levels.
First level under progressive stress 2.01V/hFirst level under progressive stress 2.015V/hItemFTItemFTItemFTItemFTItemFTItemFT113.572272.334342.06119.32249.654331.00219.922372.604447.88223.282351.424434.81323.32475.434554.21323.502451.274536.03427.812575.854654.55426.502553.254643.08531.162676.204755.85527.422654.254745.63631.562777.784856.43628.322755.474846.03734.002879.134958.86728.622856.834946.33846.262980.655060.60830.622956.175049.62946.413082.655162.48934.42308.855149.861050.603190.335262.811035.303111.315250.661156.763214.515363.411135.483211.835350.931256.853315.615463.761238.303314.505451.031360.133415.855564.181340.523414.835551.731465.003517.735666.151443.833517.735651.951565.863619.655766.411543.003619.355752.361666.203721.055869.911643.003725.505854.781766.403821.205971.731743.123826.155955.581866.803924.216072.461844.433927.456055.831966.934024.856173.781945.324027.616157.132068.254131.186278.912047.584128.052170.234235.082147.654230.96We use the Kolmogorov–Smirnov (K–S) analysis to check whether the real data set fits the Modified Kies distribution. The K–S statistic and the correspondingP values are computed and shown in Table 10 using the MLEs for the parameters provided in Table 10. Based on the current P values in Table 11, we deduce that the Modified Kies distribution fits the given data well.Table 10
MLEs estimates for the distribution parameter.
DistributionAICα^θ^1b^γ^Pareto IV1085.336.097 336.590 11.614 30.363 1MKEx1043.082.330 50.623 96140.66—Gamma1071.745.285 9123.3290.080 1—Weibull1048.843.083 1131.8890.448 6—General exponential1082.416.094 5118.330.181 3—Table 11
K–S statistics andP values for each stress levels.
DistributionStress 2.01V/hStress 2.015V/hPareto IVKS value0.225 80.125The corresponding value (P value)0.084 60.786 5MKExKS value0.1290.147 6The corresponding value (P value)0.6860.522 1GammaKS value0.161 20.163 9The corresponding value (P value)0.398 00.398 0WeibullKS value0.129 00.180 3The corresponding value (P value)0.684 60.274 7General exponentialKS value0.177 40.213 1The corresponding value (P value)0.284 90.125 4Table10 includes the MLES for the failure times recorded in Table 9, for the MKEx distribution and its competitors. Results of MLEs, KS, and P values for the gamma distribution, Weibull distribution, and general exponential in Tables 10 and 11 are taken from reference [30], and also results for the Pareto IV distribution are taken from reference [42].


### 7.1. Model Selection Criteria

For some specific data, the selection of model is one of the main basic tasks of the scientific study in choosing the predictive model from a group of proposed models. There are several statistical methods to determine the fitness of competing distributions, and the most widely used criteria are the Akaike information criteria (AIC). A model with the lowest values could be chosen as the best model for the real data set. The AIC criteria are given by(30)AIC=2k−2ℓ,where ℓ defines the values of the log-likelihood function under the MLE and also k refers to the number of parameters in the proposed model, while n is the samples size.


### 7.2. Graph for Reliability Function and Important Results Conducted from Real Data

The following conclusions illustrate the major findings from the results that we have found in this real data set example. According to the results in Tables10 and 11, we found the following.First, we must test for fitting of the data, so we used a K–S method to test whether if our data were good fit for our distribution or not. According to theP value in Table 11, we concluded that the used distribution made a good fit for the failure times of the experiment. By using the estimated parameters of the distribution θ^1,α^,and atb, we deduced θ0, where it is considered as the scale parameter under normal use. From equation θi=θ1hib, we can evaluate the MLE of the scale parameters under the normal conditions θ0=θ1h0b=1.2584, and also we can calculate θ2=θ1h2b=0.43936 which is the scale parameter under the second stress level. By using equation σ_i=θ1hibb+1 and (4), we can evaluate the meantime to failure (MTTF) under normal conditions and first and second stress levels; some times, we called it the expected time to failure which is the result of the integration of the expectation using the PDF of the specific level. MTTF = 109.04, 54.0634, 38.1181 for the normal and first and second levels, respectively. In order to make sure that our calculations are right, we refer to Tables 12 and 13, as we can see the actual average time to failure of the first level and the second level is near as the expected MTTF calculated from the accelerated PDF. The reliability function under normal conditions, stress level one, and level two conditions can be computed by setting i = 1, 2, 3 and j = 0, 1, 2, respectively:(31)Git=e−ex/σj−1α,x>0,α,σj>0,i=1,2,3,j=0,1,2.Table 12
Descriptive statistics for the data in the first level (2.01 V/h).
nMinQ1MedianMeanQ3MaxSkewnessKurtosis6213.5731.5660.36553.295 269.9190.33−0.499 8121.945 82Table 13
Descriptive statistics for the second level (2.015 V/h) data.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis618.8527.574338.62850.95557.13−0.460 7992.048 58By plotting the reliability function, we concluded with these outcomes that the reliability function under normal condition use at time is equal to zero; then, the reliability function is equal to one (see Figure3). Under stress conditions, we found that as time increases, the reliability function decreases (see Figure 4). As the stress increases once more, it approaches zero (see Figure 5). By referring to the values of AIC in Table 10, we can see that the MKEx distribution outperforms all its competitors’ distributions.Figure 3
The graph depicts the reliability function during ordinary situations; we can realize that it reaches one at time equals zero and then reaches zero as time increases, which is very reasonable under standard usages for just any product.Figure 4
The graph demonstrated the reliability function underS = 2.01-volt working use. We could indeed conclude from the diagram that it quickly declines as the given period increases, which is very reasonable under stress use for any items because it was exposed to a rate higher than the use environments, resulting in early failure.Figure 5
The plot shows the reliability function under level of stressS = 2.015 volt conditions. From the plot we concluded with that the reliability function decreases very rapidly by increasing both of the time and stress, and this is very rational outcome under stress use for any item as it if any item was subjected to high level of voltage greater than the normal use conditions, and that leads to early failure.


### 7.3. Existence and Uniqueness

Figure6 discusses that the estimated parameters make maximization for the Log-likelihood function. We estimate the roots using Mathematica 12, with the aid of NMaximize Function, which always finds global maximum, not local maximum. Also, we assured these results by graphing the log-likelihood function. As shown, the red dot denotes that the estimate is at max point thought the whole curve.Figure 6
Plots of the profile-likelihood functions for the three parameters for the real data set.Figure7 shows that the estimates are not only maximum but also unique. As we can see, the graph of the derivative equation is decreasing function, and the curve intersects the X-axis in one point, which is the estimate, so we can conclude that these roots (estimates) are unique.Figure 7
Plots of the existence and uniqueness of model parameters for the real data set.


## 7.1. Model Selection Criteria

For some specific data, the selection of model is one of the main basic tasks of the scientific study in choosing the predictive model from a group of proposed models. There are several statistical methods to determine the fitness of competing distributions, and the most widely used criteria are the Akaike information criteria (AIC). A model with the lowest values could be chosen as the best model for the real data set. The AIC criteria are given by(30)AIC=2k−2ℓ,where ℓ defines the values of the log-likelihood function under the MLE and also k refers to the number of parameters in the proposed model, while n is the samples size.


## 7.2. Graph for Reliability Function and Important Results Conducted from Real Data

The following conclusions illustrate the major findings from the results that we have found in this real data set example. According to the results in Tables10 and 11, we found the following.First, we must test for fitting of the data, so we used a K–S method to test whether if our data were good fit for our distribution or not. According to theP value in Table 11, we concluded that the used distribution made a good fit for the failure times of the experiment. By using the estimated parameters of the distribution θ^1,α^,and atb, we deduced θ0, where it is considered as the scale parameter under normal use. From equation θi=θ1hib, we can evaluate the MLE of the scale parameters under the normal conditions θ0=θ1h0b=1.2584, and also we can calculate θ2=θ1h2b=0.43936 which is the scale parameter under the second stress level. By using equation σ_i=θ1hibb+1 and (4), we can evaluate the meantime to failure (MTTF) under normal conditions and first and second stress levels; some times, we called it the expected time to failure which is the result of the integration of the expectation using the PDF of the specific level. MTTF = 109.04, 54.0634, 38.1181 for the normal and first and second levels, respectively. In order to make sure that our calculations are right, we refer to Tables 12 and 13, as we can see the actual average time to failure of the first level and the second level is near as the expected MTTF calculated from the accelerated PDF. The reliability function under normal conditions, stress level one, and level two conditions can be computed by setting i = 1, 2, 3 and j = 0, 1, 2, respectively:(31)Git=e−ex/σj−1α,x>0,α,σj>0,i=1,2,3,j=0,1,2.Table 12
Descriptive statistics for the data in the first level (2.01 V/h).
nMinQ1MedianMeanQ3MaxSkewnessKurtosis6213.5731.5660.36553.295 269.9190.33−0.499 8121.945 82Table 13
Descriptive statistics for the second level (2.015 V/h) data.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis618.8527.574338.62850.95557.13−0.460 7992.048 58By plotting the reliability function, we concluded with these outcomes that the reliability function under normal condition use at time is equal to zero; then, the reliability function is equal to one (see Figure3). Under stress conditions, we found that as time increases, the reliability function decreases (see Figure 4). As the stress increases once more, it approaches zero (see Figure 5). By referring to the values of AIC in Table 10, we can see that the MKEx distribution outperforms all its competitors’ distributions.Figure 3
The graph depicts the reliability function during ordinary situations; we can realize that it reaches one at time equals zero and then reaches zero as time increases, which is very reasonable under standard usages for just any product.Figure 4
The graph demonstrated the reliability function underS = 2.01-volt working use. We could indeed conclude from the diagram that it quickly declines as the given period increases, which is very reasonable under stress use for any items because it was exposed to a rate higher than the use environments, resulting in early failure.Figure 5
The plot shows the reliability function under level of stressS = 2.015 volt conditions. From the plot we concluded with that the reliability function decreases very rapidly by increasing both of the time and stress, and this is very rational outcome under stress use for any item as it if any item was subjected to high level of voltage greater than the normal use conditions, and that leads to early failure.


## 7.3. Existence and Uniqueness

Figure6 discusses that the estimated parameters make maximization for the Log-likelihood function. We estimate the roots using Mathematica 12, with the aid of NMaximize Function, which always finds global maximum, not local maximum. Also, we assured these results by graphing the log-likelihood function. As shown, the red dot denotes that the estimate is at max point thought the whole curve.Figure 6
Plots of the profile-likelihood functions for the three parameters for the real data set.Figure7 shows that the estimates are not only maximum but also unique. As we can see, the graph of the derivative equation is decreasing function, and the curve intersects the X-axis in one point, which is the estimate, so we can conclude that these roots (estimates) are unique.Figure 7
Plots of the existence and uniqueness of model parameters for the real data set.


## 8. Conclusion

The purpose of this study was to examine statistical inference using progressively censored type-II data when the failure times follow from the Modified Kies distribution under progressive-stress accelerated life testing. We made a simulation study to assess the performance of the distribution and usability in modeling accelerated failure times. We address statistical analysis by estimating the unknown parameters of the Modified Kies distribution using a variety of methods, such as the classical method and Bayesian methods. And, we deduced that the BEs method is better than the classical estimation method as it always has lower values regarding the MSE. Also, we made three types of interval estimation as bootstrap, asymptotic, and credible, and we deduce that the smallest interval is the bootstrap-t. Also, we used a real data practical example, and we fitted these data and compared the MKEx distribution with its competitors. And we deduced that the MKEx distribution outperforms them by referring to the AIC values, and we concluded that the MKEx distribution is the best replacement for them. We made a graphical plot for the reliability functions under normal and stress use, and we deduced that the MTTF decreases by increasing the stress, and that is very rational. Also, we estimated the parameters under stress level with the aid of the estimated parameters and by using the cumulative exposure model. At last, we made modeling for accelerated data by using the MKEx distribution and proved its superiority. Also, we strengthened our application by making sure that the estimates are the global maximum and unique solution from Figures6 and 7. For more details about this subject, see the following references: [43–46].


## 9. Future Work

Neutrosophic statistics is the extension of classical statistics, and it can be used for the data from a complex process or from an uncertain environment. Our current work can be extended using neutrosophic statistics as future work, and we will take the following papers as references and guide in our future work (see [47, 48]).



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*Source: 9558650-2022-04-28.xml*