TEXT SIZE

search for



CrossRef (0)
Inference for exponentiated Weibull distribution under constant stress partially accelerated life tests with multiple censored
Communications for Statistical Applications and Methods 2019;26:131-148
Published online March 31, 2019
© 2019 Korean Statistical Society.

Said G. Nassr1,a, Neema M. Elharouna

aDepartment of Quantitative Methods, Sinai University, Egypt
Correspondence to: 1Faculty of Business Administration and International Marketing, Sinai University, Egypt. E-mail: dr.saidstat@gmail.com
Received August 13, 2018; Revised December 21, 2018; Accepted January 7, 2019.
 Abstract

Constant stress partially accelerated life tests are studied according to exponentiated Weibull distribution. Grounded on multiple censoring, the maximum likelihood estimators are determined in connection with unknown distribution parameters and accelerated factor. The confidence intervals of the unknown parameters and acceleration factor are constructed for large sample size. However, it is not possible to obtain the Bayes estimates in plain form, so we apply a Markov chain Monte Carlo method to deal with this issue, which permits us to create a credible interval of the associated parameters. Finally, based on constant stress partially accelerated life tests scheme with exponentiated Weibull distribution under multiple censoring, the illustrative example and the simulation results are used to investigate the maximum likelihood, and Bayesian estimates of the unknown parameters.

Keywords : constant stress partially accelerated life tests, exponentiated Weibull distribution, multiple censored scheme, maximum likelihood estimation, Bayesian estimation, Markov chain Monte Carlo
1. Introduction

Under reliability development and continuous quality of products make it difficult to grasp failure information under normal condition. A lifetime test under such conditions therefore becomes sluggish and futile over a long time span. An accelerated life test (ALT) is being used to quickly acquire information about the lifetime distribution of a product or materials. Accelerated life testing could be realized by submitting the test units to conditions that are more intense than what the normal ones are in imposing higher levels of pressure, temperature, vibration, voltage, load, and cycling rate. Research work on ALT began in the 1950’s to advance a more operative testing procedure. Chernoff (1962) and Bessler et al. (1962) innovated and considered the concept of ALTs. In accelerated life testing, the experiment is to be initiated both at higher stresses than standard ones and sustained under the stated conditions or it may be under normal conditions. Hence, there are two types of accelerated life testing. The first is the ordinary accelerated life test (OALT), while the second one is the partially accelerated life testing (PALT). The chief supposition in OALT is that the mathematical model linking the lifetime of the unit to the stress must be recognized or can be anticipated; therefore, these models do not occur or are very difficult to assume in certain cases. PALT is practical to execute the life test. The chief supposition in PALT is that the mathematical pattern associating the mean lifetime and the stress are not known and cannot be suspected. PALT is applied for difficulties where it is a suitable test only at a particularized accelerated state, and then we extrapolated the data to a normal state. In such cases, PALT is the reasonable structure to implement and estimate the acceleration factor, which is the rate of the hazard level at the accelerated state to a normal use state (β > 1). There are two different types of PALTs. The first is a step-stress partially accelerated life testing (SS-PALT) that lets the test be modified from a normal state to accelerated state at a pre-defined time. The second is the constant-stress partially accelerated life testing (CS-PALT) where every item happens only in either normal condition or accelerated conditions.

The stress imposed in a PALT can be conceived in different methods. They comprise step stress, constant stress, and random stress. One approach to accelerate failures is constant stress in which each test item is only run at both a normal use state or accelerated state, i.e., each item is developed at a perpetual stress level until the test is completed. Nelson (1990) explained that constant stress testing has many benefits: First, it is simpler to conserve a constant stress level in most tests. Second, accelerated test patterns for constant stress are better to develop for some elements and products. Thirdly, data investigation for reliability estimation is well developed. The stress imposed in a PALT can be conceived in different methods. They comprise step stress, constant stress, and random stress. One approach to accelerated failures is constant stress in which each test item is run at both normal use state or accelerated state only, i.e., each item is developed at a perpetual stress level until the test is completed. Nelson (1990) explained that constant stress testing has many benefits: First, it is simpler to conserve a constant stress level in most tests. Second, accelerated test patterns for constant stress are better to develop some elements and products. Thirdly, data investigation for reliability estimation is well developed.

In life test experiments, sometimes the experiment could not get under control fully because items may break by mistake. Even though, Type I and Type II censoring schemes do not tolerate items to be eliminated from the test during the life testing duration. Progressive censoring schemes allow for items to be neglected only under control conditions. However, multiple censoring schemes are a suitable in this situation. Multiple censoring allows for items to be extracted from the test at any time during the life test duration. Multiple censoring may also; occur when the testing component fails for more than one reason. Hence Type I and II are exceptional cases of numerous censoring (Tobias and Trindade, 1995).

There is an aggregate of collected works on designing CS-PALT. Bai and Chung (1992) estimated the scale parameter and acceleration factor for exponential distribution under two kinds of PALT which are a step and constant stresses in case of Type I censored sample utilizing maximum likelihood method. Abdel-Ghani (1998) reflected that the estimating problems of Weibull distribution parameters and accelerated factor under CS-PALT. Ismail (2006) provided the optimum design of CS-PALT under Type II censoring supposing that the lifetime design stress has a Weibull distribution. Hassan (2007) estimated accelerated factor and the unknown parameters for generalized exponential distribution under CS-PALT using a maximum likelihood method. Ismail et al. (2011) used maximum likelihood methodology to estimate the accelerated factor and parameters for Pareto distribution with Type I censored data. Cheng and Wang (2012) used two maximum likelihood estimation methods, which accompany observed data and complete data likelihood function respectively, to estimate the accelerated factor and the unknown parameters for Burr XII distribution under CS-PALT. Zarrin et al. (2012) manipulated the maximum likelihood approach to estimate the accelerated factor and the parameters of Rayleigh’s distribution. This work was done the CS-PALT under Type I censored data.

Kamal et al. (2013) manipulated the maximum likelihood approach to obtain the point and interval estimate for the accelerated factor and unknown parameters for inverted Weibull distribution under CS-PALT in case type I censoring. Srivastava and Mittal (2013a, 2013b) used an optimal design of the CS-PALT plan under a failure constant and time constant for truncated logistic distribution based on type I censoring data. They determined the optimal sample proportion allocated to both normal and accelerated use conditions in the generalized asymptotic variance of maximum likelihood estimators (MLEs) for the accelerated factor and the unknown parameters. Jaheen et al. (2014) estimated the unknown parameters and accelerated factor for generalized exponential distribution under constant stress partially accelerated life testing with progressive Type II censored data using maximum likelihood and Bayesian approaches. Abushal and Soliman (2015) applied constant stress partially accelerated life testing with Type II progressively censored data to estimate unknown parameters and the accelerated factor for Pareto distribution using maximum likelihood and Bayesian methods. Hassan et al. (2015) used the maximum likelihood estimation to estimate the accelerated factor and the unknown parameters for inverted Weibull distribution under CS-PALT established on multiple censored data.

This paper is concerned with the estimation problem in the case of an exponentiated Weibull (EW) distribution under CS-PALT using multiple censored data. The rest of this paper constructed as follows. Section 2 introduces the EW distribution as the lifetime model, and the assumptions of the CS-PALT are presented. Section 3 displays the estimates of the unknown parameters and acceleration factor for the EW distribution with multiple censored samples. In addition, the confidence intervals of the unknown parameters are developed. The Bayesian estimators of the unknown parameters and acceleration factor for the EW distribution and Markov chain Monte Carlo (MCMC) technique are presented in Section 4. Section 5 presents a numerical example to illustrate all methods of inference developed in this article. Section 6 gives the simulation studies and the results. The conclusions are presented in Section 7.

2. The model and test procedure

This section presents the assumption pattern for product life test. The details of the test method of how to manipulate the experiment under multiple censored data using the EW model are illuminated.

2.1. The exponentiated Weibull distribution: as a failure time pattern

Mudholkar and Srivastava (1993) introduced an expansion distribution of the renowned Weibull distribution called EW distribution. The EW family includes distribution with monotone failure ratios aside from a broader class of monotone failure ratios. In practice, various lifetime data are of bathtub shape or upside-down bathtub shape failure ratios, and so the EW distribution as a failure pattern is more practical than monotone failure ratios and performs an important part to describe such data. The test procedure of CS-PALT based on multiple censored data assuming the life time item has EW distribution is described as


f1(ti)=αθtiα-1e-tiα(1-e-tiα)θ-1;α,θ>0,i=1,2,,n1

and


F1(ti)=(1-e-tiα)θ,

where ti is the ith observed life time of test item at normal condition. Under accelerated condition, the probability density function (pdf) and cumulative distribution function of lifetime X = β−1T, where β is accelerated factor, β > 1, take the following forms:


f2(xj)=αβθ(βxj)α-1e-(βxj)α(1-e-(βxj)α)θ-1;α,θ>0,β>1,j=1,2,3,,n2

and


F2(xj)=(1-e-(βxj)α)θ,

where xj is the jth observed life time of test item at accelerated condition.

2.2. Basic assumptions

The basic assumptions are:

  • The lifetimes of items Ti, i = 1, 2, …, n1 devoted to normal condition are independent and identically distributed random variables with pdf (2.1).

  • The lifetimes of items Xj, j = 1, 2, …, n2 devoted to accelerated condition are independent and identically distributed random variables with pdf (2.3).

  • The lifetimes Ti and Xj are mutually independent.

3. Maximum likelihood estimators

Suppose that the observed values of the total lifetime T at normal condition are t(1) < 떙 < t(n1), and the observed values of the total lifetime X at accelerated condition are x(1) < 떙 < x(n2). The likelihood function for the unknown parameters φ = (α, θ, β) of the EW distribution with multiple censored data is given as


L(x)=i=1n[f1(ti)]γi,(1,f)[1-F1(ti)]γi,(1,c)[f2(xi)]γi,(2,f)[1-F2(xi)]γi,(2,c),

where ti = t(i), xi = x(i) for simplicity of notation, and γi,(1, f ), γi,(1,c) , γi,(2, f ), γi,(2,c) be the indicator functions, i = 1, 2, …, n, such that,

γi,(1,f)={1,the unit fails at normal condition,0,otherwise,γi,(1,c)={1,the unit censoring at normal condition,0,otherwise,γi,(2,f)={1,the unit fails at stress condition,0,otherwise,γi,(2,c)={1,the unit censoring at stress condition,0,otherwise,

Moreover, the equations are obtained as:

i=1nγi,(1,f)=n1f,i=1nγi,(1,c)=n1c,i=1nγi,(2,f)=n2f,i=1nγi,(2,c)=n2c,nf=n1f+n2f,n1=n1f+n1c,n2=n2f+n2c,n=n1+n2.

Substituting the probability density and cumulative distribution functions (2.1), (2.2), (2.3), and (2.4) in likelihood function (3.1), then:


L(x)=i=1n[αθtiα-1e-tiα(1-e-tiα)θ-1]γi,(1,f)[1-(1-e-tiα)θ]γi,(1,c)×[αβθ(βxi)α-1e-(βxi)α(1-e-(βxi)α)θ-1]γi,(2,f)[1-(1-e-(βxi)α)θ]γi,(2,c).

Hence, it is usually easier to maximize the natural logarithm of the likelihood function instead of the likelihood function itself, the logarithm of likelihood function is considered as


ln L=nfln α+nfln θ+n2f(ln β-βα)-i=1nγi,(1,f)tiα-i=1nγi,(2,f)xiα+(α-1)[i=1nγi,(1,f)ln ti+i=1nγi,(2,f)ln(βxi)]+i=1nγi,(1,c)ln [1-u1iθ]+(θ-1)[i=1nγi,(1,f)ln(u1i)+i=1nγi,(2,f)ln(u2i)]+i=1nγi,(2,c)ln [1-u2iθ],

where u1i=(1-e-tiα) and u2i = (1 − e−(βxi)α).

Maximum likelihood estimators of φ = (α, θ, β) are a solution to the system of equations achieved by addressing the first partial derivatives of the total log-likelihood to be zero concerning α, θ, and β respectively. Thus, the system of equations is considered as:


ln Lα=nfα-n2fβαln β-i=1nγi,(1,f)tiαln ti-i=1nγi,(2,f)xiαln xi+i=1nγi,(1,f)ln ti+i=1nγi,(2,f)ln(βxi)+(θ-1)[i=1nγi,(1,f)1i+i=1nγi,(2,f)2i]-θ[i=1nγi,(1,c)3i+i=1nγi,(2,c)4i],ln Lθ=nfθ+i=1nγi,(1,f)ln(u1i)+i=1nγi,(2,f)ln(u2i)-i=1nγi,(1,f)ψ1i-i=1nγi,(2,c)ψ2i,ln Lβ=n2fβ-1(1-αβα)+(α-1)i=1nγi,(2,f)xi(βxi)-1+α(θ-1)i=1nγi,(2,c)ξ1i-αθi=1nγi,(2,c)ξ2i,

where

(φ1iφ2iφ3iφ4iψ1iψ2iξ1iξ2i)=(e-tiαtiαln tiu1i-1e-(βxi)α(βxi)αln(βxi)u2i-1φ1iu1iθ[1-u1iθ]-1φ2iu2iθ[1-u2iθ]-1u1iθln(u1i)[1-u1iθ]-1u2iθln(u2i)[1-u2iθ]-1xi(βxi)α-1e-(βxi)αu2i-1xi(βxi)α-1e-(βxi)αu2iθ-1[1-u2iθ]-1).

Since, the closed form solutions to the nonlinear Equations (3.4), (3.5), and (3.6) are difficult to reach. The numerical method is then used to solve these simultaneous equations for obtaining α, θ, and β, the second partial derivative of the matrix likelihood function are give as:

2ln Lα2=-nfα2-n2fβα(ln β)-i=1nγi,(1,f)tiα(ln ti)2-i=1nγi,(2,f)xiα(ln xi)2+(θ-1)[i=1nγi,(1,f)φ1iϖ1i+i=1nγi,(2,f)φ2iϖ2i]-θ{i=1nγi,(1,c)φ3i[ϖ1i+θ(φ1i+φ3i)]+i=1nγi,(2,c)φ4i[ϖ2i+θ(φ2i+φ4i)]},2ln Lθ2=-nfα2-i=1nγi,(1,c)ψ1i[ln(u1i)+ψ1i]-i=1nγi,(2,c)ψ2i[ln(u2i)+u2iθψ2i],2ln Lβ2=-n2fβ-2[1+α(α-1)βα]-(α-1)i=1nγi,(2,f)xi2(βxi)-2+α(θ-1)i=1nγi,(2,f)ξ1iϖ3i-αθi=1nγi,(2,c)ξ2i[ϖ3i+θα(ξ1i+ξ2i)],2ln Lαθ=i=1nγi,(1,f)φ1i+i=1nγi,(2,f)φ2i-i=1nγi,(1,c)φ3i[1+θln(u1i)+θψ1i]-i=1nγi,(2,c)φ4i[1+θln(u2i)+θψ2i],2ln Lαβ=-n2fβα[ln β+β-1]+i=1nγi,(2,f)xi(βxi)-1+(θ-1)i=1nγi,(2,f)ξ1iln(βxi)ϖ4i-θi=1nγi,(2,c)ξ2iln(βxi)[ϖ4i+θαxi-1(βxi)(ξ1i+ξ2i)],2ln Lθβ=αi=1nγi,(2,f)ξ1i-αi=1nγi,(2,c)ξ2i[θln(u2i)+1+ψ2i],

where

(ϖ1iϖ2iϖ3iϖ4i)=(ln ti-tiαln ti-φ1iln(βxi)-(βxi)αln(βxi)-φ2i(α-1)xi(βxi)-1-αxi(βxi)α-1-αξ1i[ln(βxi)]-1-α(βxi)α+α-αxi-1(βxi)ξ1i).

That is, the interval estimation of the model parameters that occur in 3 × 3 observed information matrix I (φ) = [Ur,s] for (r, s = α, θ and β ).

Frequently, following standard regularity conditions, the multivariate normal N3[0, I−1(φ)] distribution is applied to form approximate confidence intervals for the parameters. The approximate 100(1 − ϒ)% two-sided confidence intervals for α, θ, and β are then given, respectively, by:

α^±Zϒ2var (α^), 듼 듼 듼θ^±Zϒ2var (θ^), 듼 듼 듼β^±Zϒ2var (β^).

Here, Zϒ/2 is the upper (ϒ/2)th percentile of the standard normal distribution and var(·)’s indicates the diagonal elements of I−1(φ) corresponding to the model parameters.

4. Bayesian estimation

In this section, the Bayesian estimator using a squared error loss function is considered. Following the assumption of gamma distribution to be a prior of two unknown parameters of the EW distribution will be concern based on multiple censored scheme. The Bayesian estimator under the assumption that the random variables α and θ are independently distributed gamma prior distribution (with known shape and scale parameters ϒ, b, c, and d) considered as

g(α)αυ-1e-αb,υ,b>0and g(θ)θc-1e-θd,c,d>0.

While the random variable β distributed as the following non-informative (NIP) type of prior

g(β)β-1,β>1.

Then, the joint prior density of unknown parameters = α, θ, and β can be written as


π()β-1αυ-1θc-1e-(αb+θd),α,θ>0,β>1.

Combining (3.2) and (4.1) to get the posterior density of as follows


π*(x)=L(x)π()100L(x)π()dαdθdβ,

thus, the Bayesian estimators of function of the parameters, say = (α, θ, β) based on multiple censored scheme under squared error loss function; denoted by (BESL)() can be determined by the following equations


˜(BESL)=E(x_)=100L(x)π()dαdθdβ100L(x)π()dαdθdβ.

The ratio of four integrals provided by Equation (4.3) and the three integrals are not achieved in a closed form. It is often instructive to apply the MCMC method to generate samples from the posterior distributions and then calculate the Bayes estimates of the particular parameters.

This section focuses on the Bayesian approach to estimate the unknown parameters of the EW model under multiple censored scheme considering the Gibbs sampler method for approximation of integrals. Often integrals occurring in the Bayesian estimation never admit a closed form, as a result, the analytical method is not attractable and numerical integration is needed. Gibbs sampler give an alternative method to approximate these integrals, this procedure intends to obtain a Markov chain that has a purpose posterior as a limiting distribution, and then the simulation samples or chain can be utilized to estimate any desired feature. The simulation Markov chain (after some burn-in) will have achievements that are observed as simulated from the posterior distribution rather than the straight computation of the posterior. We then utilized the generated samples to estimate the parameters of interest and function.

The MCMC scheme is a good method for parameter estimation. Several schemes of MCMC are available, one significant sub-class of MCMC methods is Gibbs sampling and Metropolis within-Gibbs samplers. MCMC has an advantage over the maximum likelihood method by structuring the probability intervals based on the empirical posterior distribution, we get an appropriate interval estimate of the parameters. The MCMC samples utilized to summarize the posterior uncertainty about the parameters = (α, θ, β), within the joint posterior density function of α, θ and β can be addressed as


π*(x)αnf+υ-1θnf+c-1βn2fα-1e-[αb+θd+i=1nγi,(1,f)tiα+i=1nγi,(2,f)(βxi)α][i=1nti(α-1)(γi,(1,f))]×[i=1nxi(α-1)(γi,(2,f))][i=1nu1i(θ-1)(γi,(1,f))][i=1n(1-u1iθ)γi(1,c)][i=1nu2i(θ-1)(γi,(2,f))]×[i=1n(1-u2iθ)γi(2,c)],α,λ>0,β>1.

The conditional posterior densities of α, θ and β can be computed by:

π1*(αx)αnf+υ-1βn2fα-1e-[αb+i=1nγi,(1,f)tiα+i=1nγi,(2,f)(βxi)α][i=1nti(α-1)(γi,(1,f))][i=1nxi(α-1)(γi,(2,f))]×[i=1nu1i(θ-1)(γi,(1,f))][i=1nu2i(θ-1)(γi,(2,f))][i=1n(1-u1iθ)γi,(1,c)][i=1n(1-u2iθ)γi,(2,c)],π2*(θx)θnf+c-1e-θd[i=1nu1i(θ-1)(γi,(1,f))][i=1nu2i(θ-1)(γi,(2,f))][i=1n(1-u1iθ)γi,(1,c)][i=1n(1-u2iθ)γi,(2,c)],π3*(βx)βn2fα-1e-i=1nγi,(2,f)(βxi)α[i=1nu2i(θ-1)(γi,(2,f))][i=1n(1-u2iθ)γi,(2,c)].

Consequently, to generate from these distributions, the Metropolis-Hastings method (Metropolis et al., 1953) and Robert and Casella (2004), are utilized for details concerning the implementation of Metropolis-Hastings algorithm.

The procedure of Gibbs sampling can be described as:

  • Step 1: Start with an (α(0) = α, θ(0) = θ) and (β(0) = β) and set I = 1.

  • Step 2: Generate αI from π1*(αx).

  • Step 3: Generate θI from π2*(θx).

  • Step 4: Generate βI from π3*(βx).

  • Step 5: Compute αI ,θI and βI .

  • Step 6: Set I = I + 1.

  • Step 7: Repeat Step 2–5 N times.

  • Step 8: Obtain the Bayes MCMC point estimate of q (1 = α, 2 = θ, 3 = β), q = 1, 2, 3 as

    E(qdata)(i=M+1Nq(i))N-M,

    where, M is the burn-in period (that is, some iterations before the stationary distribution perform) and the posterior variance of φ becomes

    V^(qdata)i=M+1N(q(i)-E(qdata))2N-M.

  • Step 9: To compute the credible intervals of q, we use the quantiles of the sample as the endpoints of the interval. Order q(M+1),q(M+2),,q(N) as q(1), q(2), …, q(NM). Consider the 100(1− 2ϒ)% symmetric credible interval as [q[ϒ(NM)], q[(1–ϒ)(NM)]].

5. Illustrative example

Illustrative maximum likelihood estimation for the EW distribution based on another data set was simulated from distribution in CS-PALT scheme with α = 1.5, θ = 3.5, β = 1.25 the data are represented as follows.

  • Under normal condition:

    0.296, 0.448, 0.449, 0.697, 0.791, 0.852, 0.896*, 0.907, 0.908*, 0.915, 0.918*, 0.922*, 0.942, 0.946*, 0.955, 0.984*, 0.988, 1.101, 1.116, 1.156, 1.222, 1.258, 1.327, 1.329*, 1.363, 1.373, 1.379, 1.412, 1.422*, 1.459*, 1.463, 1.480, 1.492, 1.542, 1.562, 1.566, 1.584, 1.588, 1.604, 1.717, 1.746*, 1.759*, 1.780, 1.829, 1.848, 1.849, 1.865*, 1.929, 1.938, 2.025*, 2.037*, 2.041, 2.084, 2.092, 2.164, 2.214, 2.668, 2.782*, 3.196, 3.658.

  • Under stress condition:

    0.249, 0.465, 0.608*, 0.638, 0.644*, 0.664, 0.666, 0.673, 0.688, 0.695*, 0.715, 0.734*, 0.737*, 0.770, 0.778, 0.786, 0.794, 0.813, 0.834, 0.853, 0.865, 0.897, 0.904, 0.921*, 0.927*, 0.937, 0.938*, 1.043*, 1.045, 1.063, 1.075, 1.075, 1.092, 1.097, 1.098, 1.107*, 1.114, 1.141, 1.148, 1.152, 1.167, 1.184, 1.198, 1.20*, 1.206, 1.21, 1.212, 1.225, 1.234, 1.238*, 1.240, 1.249, 1.266, 1.269, 1.275, 1.295*, 1.301, 1.305, 1.306, 1.323*, 1.346, 1.360*, 1.381, 1.401, 1.402, 1.442, 1.451*, 1.459, 1.469*, 1.471, 1.473, 1.493, 1.553*, 1.593*, 1.596, 1.641*, 1.660, 1.660, 1.672, 1.678, 1.681, 1.690, 1.701*, 1.711, 1.763*, 1.831, 1.895*, 1.985, 2.088, 2.437.

There are 150 samples with censoring level (CL) = 0.20. The π value, which is the proportion of the sample under normal condition is 0.40. The sample of failure and censoring in two groups based on CS-PALT, respectively, are 45 failures in group 1, 15 censoring in group 1, 67 failures in group 2 and 23 censoring in group 2. The symbol “*” denotes censored values. Before computing the MLEs, we plot the profile log-likelihood function in Figure 1. It is noted from Figure 1 that the profile loglikelihood function is unimodel. Then, the maximum likelihood estimates are α = 1.257, θ = 3.672, and β = 1.291, and the corresponding 95% confidence intervals are (1.094, 1.271), (2.887, 4.457), and (1.093, 1.489). To compute the Bayes estimates of α, θ, and β, we assume the gamma prior distribution for (α, θ) and non-informative prior distribution for β. Based on MCMC samples of size 10,000, the Bayes estimates under SE loss function are θ = 1.331, θ = 4.368, and β = 1.708 and the corresponding credible intervals are (1.271, 1.731), (0.604, 5.104), and (1.700, 1.711), respectively.

6. Simulation study

In any estimation problems, it is required to study the properties of the derived estimators. The derived expressions for the estimators are too complicated to study analytically. Consequently, a numerical study will be established and treat the sampling distribution of the estimators separately. A numerical study is achieved to compare the different estimates discussed in the previous section. The performances of the different estimates are compared regarding their mean squared error (MSE), relative bias (RB), relative error (RE), standard deviation (SD) and standard error (SE). The numerical procedures are:

  • Step 1: Divide the total sample size n in two sub samples n1 = and n2 = n(1 − π) where π is the sample proportion in normal condition.

  • Step 2: Generate n1 random samples as normal condition samples t1,1 < 떙 < tn1,1 from EW distribution which has cdf as F1(t) = (1 − etα)θ; α, θ > 0. In addition; generate n2 random samples as a stress condition samples x1,2 < 떙 < xn2,2 from EW distribution which has cdf as F2(x) = (1 − e−(βx)α)θ with accelerated factor, where α, θ > 0, and β > 1.

  • Step 3: Select n1f = n1 × (1 − CL), and n2f = n2 × (1 − CL) to be the number of failure samples in the normal condition and in the stress condition, respectively, where CL is the censoring level. For example, CL = 0.4 indicates 60% failures and 40% censored data.

  • Step 4: Let

    ki,1={1,i=1,,n1f,ui,1,i=n1f+1,,n1,kj,2={1,j=1,,n2f,uj,1,j=n2f+1,,n2,

    where, ui,1 and uj,2 are drawn from uniform distribution U(0, 1).

  • Step 5: Set yi,1 = ki,1×ti,1, i = 1, 2, …, n1 and yj,2 = kj,2×xj,2, j = 1, 2, …, n2. Then, y = [yi,1, yj,2; i = 1, 2, …, n1; j = 1, 2, …, n2] is generated as a multiple censored data for the EW distribution.

  • Step 6: 10,000 random samples of sizes 50, 100, 150, 300, and 500 are generated from EW distribution. The parameters values are chosen as case I ≡ (α = 2, θ = 4, β = 1.2), case II ≡ (α = 1.5, θ = 4, β = 1.2) and case III ≡ (α = 2, θ = 4, β = 1.5). The CL is selected as 0.2, 0.3, and 0.5.

  • Step 7: For each sample and for the selected sets of parameters, the population parameters and the accelerated factor are estimated in CS-PALT under multiple censored samples. An iterative procedure is employed for solving the nonlinear Equations (3.4), (3.5), and (3.6) to obtain the estimates of α, θ, and β.

  • Step 8: The MSE, RB, RE, and SE of the estimators for acceleration factor and two shape parameters for all sample sizes and for three sets of parameters are classified.

  • Step 9: The confidence limit with confidence level and coverage probability (CP) of the acceleration factor and the two shape parameters are constructed at ϒ = 0.95 and ϒ = 0.99.

Tables 15 report the simulation results. Tables 13 represent the MSE, RB, RE, and SE for different cases of parameters and the approximated confidence intervals at 95% and 99% for the unknown parameters. In addition; the Bayes estimate and credible intervals for different cases of parameters are displayed in Table 4 and Table 5. The posterior kernel densities and dynamic trace for the unknown parameters (α, θ, β) are given in Figure 2 and Figure 3.

The approximate marginal posterior density of each parameter can be displayed in Figure 2. Trace Plots of the Parameters sampled from posterior are shown in Figure 3.

The dynamic trace for parameters (α, θ, β) in Figure 3 show that the plots looks like a horizontal band, with no long upward or downward trends and the three chains are well-mixed and is indicative of convergence.

The following conclusions can be observed on the properties of the estimated parameters from EW lifetime distribution under CS-PALT:

  • For all the cases, it is clear that MSEs, RB, RE, and SE decrease as sample size increases (Tables 13).

  • The MSEs of α increase when the CL increases. But, the MSEs of θ and β decrease when CL increases.

  • The RBs and REs of α increase as the CL increases. Whereas, the RBs and REs of θ and β decrease as the CL increases.

  • For fixed values of θ, β and as the value of α decreases, the MSEs, RBs, and REs of α decrease (Table 1 and Table 2).

  • For fixed values of θ, β and as the value of α decreases; the average length of confidence intervals (CIs) (at ϒ = 0.95 and 0.99) of α decrease.

  • When sample size increase, the MSEs, RBs, REs for θ and the length of confidence interval for unknown parameters (θ, β) increases (Table 1 and Table 2).

  • For fixed values of α, θ and as the value of β increases, the MSEs, RBs, REs, and the length of CIs (at ϒ= 0.95 and 0.99) for (α, θ , β) increase as the sample size increases (Table 1 and Table 3).

  • For fixed values of θ and as the value of α and β increases, the MSEs, RBs, REs, and the length of CIs ( at ϒ = 0.95 and 0.99) of α and θ increase. But, the MSEs, RBs, REs, and the length of CIs at ϒ = (0.95 and 0.99) of θ and β decrease when sample size increase (Table 2 and Table 3).

  • In case I, the maximum likelihood estimates have good statistical properties than the corresponding cases of parameters at CL = 0.30 and 0.50 (Tables 13).

  • The length of the interval of all estimates decreases as the sample size increases. In addition; the length of the interval of the estimates at ϒ = 0.95 is smaller than the corresponding at ϒ = 0.99 (Tables 13).

  • The Bayes estimate, SD and MCMC error decrease as sample size increases (Table 4 and Table 5).

  • The Bayes estimates for case II have good statistical properties than case I at CL = 0.50 (Table 4 and Table 5).

7. Conclusion

This paper presents the constant stress partially accelerated life test for EW distribution lifetime distribution and discuss the statistical inference based on multiple censored data. Both maximum likelihood and Bayes estimates are considered for the model parameters and accelerated factor. Additionally, the two-sided confidence limit and coverage probability of the model parameters are also constructed. The problem of Bayesian estimation of unknown parameters and accelerated factor under constant stress partially accelerated life test can be extended to include gamma and non-informative priors. In addition, other techniques such as MCMC technique are used to compute the approximate Bayes estimates and credible confidence intervals.

Figures
Fig. 1. Profile log-likelihood function of θ.
Fig. 2. Posterior kernel density for the parameters (α, θ, β) in different sample size and censored level (CL).
Fig. 3. Posterior trace plots for the parameters (α, θ, β) in different sample size and censoring level (CL).
TABLES

Table 1

Results of simulation study of MSE, RB, RE, SE, and confidence interval (CI) of different values of parameters case I ≡ (α = 2, θ = 4, β = 1.2) for different samples size

n Properties 95% 99%



MSE RB RE SE Lower Upper Length CP Lower Upper Length CP
CL = 0.2 50 α 0.185 0.215 0.215 3.377* 1.524 1.617 0.094 0.948 1.509 1.632 0.123 0.962
θ 0.438 0.047 0.165 0.090 2.946 5.432 2.486 0.956 2.553 5.826 3.273 0.988
β 0.015 0.089 0.103 8.858* 1.184 1.429 0.246 0.951 1.145 1.468 0.323 0.988

100 α 0.182 0.213 0.213 1.557* 1.543 1.604 0.061 0.954 1.534 1.614 0.080 0.967
θ 0.197 0.034 0.111 0.042 3.307 4.963 1.655 0.945 3.045 5.224 2.179 0.985
β 0.012 0.083 0.092 4.585* 1.210 1.390 0.180 0.951 1.182 1.418 0.237 0.991

150 α 0.180 0.212 0.212 6.396* 1.561 1.591 0.031 0.967 1.556 1.596 0.040 0.972
θ 0.139 0.029 0.093 0.029 3.426 4.810 1.384 0.954 3.207 5.029 1.822 0.987
β 0.011 0.082 0.089 3.112* 1.223 1.373 0.149 0.952 1.200 1.396 0.197 0.988

300 α 0.179 0.211 0.212 0.245* 1.569 1.585 0.017 0.974 1.566 1.588 0.022 0.984
θ 0.069 0.026 0.066 0.014 3.633 4.577 0.944 0.950 3.484 4.726 1.243 0.990
β 0.010 0.082 0.085 1.527* 1.246 1.350 0.104 0.950 1.230 1.366 0.136 0.995

500 α 0.179 0.211 0.211 0.007* 1.575 1.580 5.849* 0.961 1.574 1.581 7.699* 0.968
θ 0.046 0.026 0.054 8.397* 3.735 4.471 0.736 0.949 3.619 4.587 0.969 0.990
β 9.821* 0.081 0.083 9.345* 1.256 1.338 0.082 0.947 1.243 1.351 0.108 0.989

CL = 0.3 50 α 0.249 0.246 0.249 0.011 1.350 1.666 0.315 0.940 1.300 1.716 0.415 0.985
θ 0.412 0.029 0.161 0.089 2.876 5.353 2.477 0.959 2.485 5.745 3.261 0.987
β 6.270* 0.029 0.066 0.010 1.096 1.374 0.278 0.943 1.052 1.418 0.366 0.988

100 α 0.180 8.932* 0.106 0.042 3.136 4.793 1.657 0.955 2.874 5.055 2.181 0.986
θ 0.180 8.932* 0.106 0.042 3.136 4.793 1.657 0.955 2.874 5.055 2.181 0.986
β 3.414* 0.026 0.049 4.921* 1.135 1.328 0.193 0.946 1.105 1.358 0.254 0.992

150 α 0.234 0.241 0.242 4.448* 1.412 1.626 0.214 0.954 1.378 1.659 0.281 0.988
θ 0.112 0.014 0.084 0.027 3.298 4.593 1.295 0.958 3.093 4.797 1.705 0.992
β 2.398* 0.022 0.041 3.389* 1.145 1.307 0.163 0.947 1.119 1.333 0.214 0.992

300 α 0.226 0.237 0.238 2.338* 1.447 1.606 0.159 0.959 1.422 1.631 0.209 0.992
θ 0.061 0.020 0.062 0.014 3.461 4.380 0.919 0.945 3.316 4.525 1.210 0.986
β 1.535* 0.022 0.033 1.656* 1.170 1.283 0.112 0.952 1.153 1.301 0.148 0.996

500 α 0.228 0.238 0.239 1.575* 1.455 1.593 0.138 0.969 1.433 1.615 0.182 0.995
θ 0.041 0.025 0.051 7.860* 3.554 4.243 0.689 0.953 3.445 4.352 0.907 0.991
β 1.161* 0.021 0.028 1.099* 1.181 1.270 0.088 0.946 1.167 1.284 0.116 0.987

CL = 0.5 50 α 0.637 0.396 0.399 0.015 1.003 1.415 0.412 0.957 0.937 1.480 0.543 0.988
θ 0.249 0.024 0.125 0.069 3.135 5.055 1.921 0.951 2.831 5.359 2.528 0.986
β 6.503* 0.018 0.067 0.011 1.026 1.332 0.305 0.957 0.978 1.380 0.402 0.989

100 α 0.635 0.397 0.398 7.529* 1.059 1.354 0.295 0.954 1.012 1.401 0.388 0.991
θ 0.111 0.026 0.083 0.032 3.275 4.517 1.242 0.954 3.079 4.713 1.635 0.988
β 3.710* 0.023 0.051 5.432* 1.066 1.279 0.213 0.955 1.032 1.313 0.280 0.986

150 α 0.651 0.402 0.403 4.748* 1.081 1.309 0.228 0.959 1.045 1.345 0.300 0.992
θ 0.071 0.017 0.067 0.021 3.426 4.437 1.011 0.951 3.266 4.597 1.331 0.987
β 2.779* 0.022 0.044 3.709* 1.084 1.262 0.178 0.947 1.056 1.290 0.234 0.986

300 α 0.647 0.402 0.402 2.494* 1.112 1.282 0.169 0.954 1.086 1.308 0.223 0.989
θ 0.047 0.031 0.054 0.010 3.526 4.224 0.698 0.954 3.416 4.335 0.919 0.992
β 1.772* 0.024 0.035 1.791* 1.111 1.232 0.122 0.939 1.092 1.252 0.160 0.992

500 α 0.651 0.403 0.403 1.463* 1.130 1.258 0.128 0.956 1.109 1.278 0.169 0.989
θ 0.035 0.032 0.047 6.113* 3.603 4.139 0.536 0.944 3.518 4.224 0.705 0.985
β 1.349* 0.023 0.031 1.067* 1.125 1.219 0.093 0.950 1.111 1.234 0.123 0.992

*

Indicate that the value multiply 10−3.

MSE = mean squared error; RB = relative bias; RE = relative error; SE = standard error; CP = coverage probability; CL = censoring level.


Table 2

Results of simulation study of MSE, RB, RE, SE, and confidence interval (CI) of different values of parameters case I ≡ (α = 1.5, θ = 4, β = 1.2) for different samples size

n Properties 95% 99%



MSE RB RE SE Lower Upper Length CP Lower Upper Length CP
CL = 0.2 50 α 0.054 0.137 0.154 0.015 1.086 1.502 0.416 0.954 1.020 1.568 0.548 0.984
θ 0.710 0.100 0.211 0.105 2.943 5.853 2.910 0.960 2.483 6.314 3.831 0.986
β 0.027 0.113 0.136 0.013 1.157 1.514 0.357 0.951 1.100 1.571 0.470 0.985

100 α 0.054 0.147 0.155 7.390* 1.134 1.424 0.290 0.954 1.088 1.470 0.381 0.989
θ 0.345 0.072 0.147 0.051 3.286 5.292 2.005 0.957 2.969 5.609 2.640 0.986
β 0.023 0.113 0.125 6.476* 1.209 1.462 0.254 0.942 1.168 1.503 0.334 0.992

150 α 0.057 0.154 0.159 4.903* 1.151 1.386 0.235 0.949 1.114 1.424 0.310 0.987
θ 0.236 0.066 0.121 0.033 3.469 5.062 1.592 0.952 3.218 5.314 2.096 0.985
β 0.021 0.112 0.120 4.290* 1.232 1.438 0.206 0.955 1.199 1.470 0.271 0.990

300 α 0.057 0.156 0.159 2.383* 1.184 1.346 0.162 0.946 1.159 1.372 0.213 0.989
θ 0.123 0.054 0.088 0.016 3.672 4.758 1.086 0.954 3.500 4.929 1.429 0.990
β 0.019 0.109 0.114 2.288* 1.253 1.409 0.155 0.952 1.229 1.433 0.204 0.983

500 α 0.058 0.159 0.161 1.473* 1.197 1.326 0.129 0.952 1.176 1.346 0.170 0.990
θ 0.091 0.054 0.075 9.460* 3.801 4.630 0.829 0.947 3.670 4.761 1.092 0.993
β 0.019 0.111 0.114 1.287* 1.277 1.390 0.113 0.941 1.259 1.408 0.149 0.991

CL = 0.3 50 α 0.127 0.228 0.237 0.014 0.964 1.353 0.389 0.958 0.902 1.414 0.512 0.985
θ 0.479 0.060 0.173 0.092 2.970 5.513 2.543 0.957 2.568 5.916 3.347 0.985
β 0.016 0.066 0.106 0.014 1.085 1.474 0.388 0.959 1.024 1.535 0.511 0.991

100 α 0.127 0.233 0.238 6.942* 1.014 1.286 0.272 0.948 0.971 1.329 0.358 0.992
θ 0.184 0.013 0.107 0.043 3.215 4.887 1.671 0.955 0.951 5.151 2.200 0.985
β 0.011 0.067 0.088 6.921* 1.145 1.416 0.271 0.944 1.102 1.459 0.357 0.987

150 α 0.128 0.236 0.239 4.467* 1.039 1.253 0.214 0.948 1.005 1.287 0.282 0.989
θ 0.120 0.018 0.086 0.028 3.412 4.736 1.324 0.953 0.202 4.945 1.743 0.986
β 9.083* 0.065 0.079 4.511* 1.169 1.386 0.217 0.953 1.135 1.420 0.285 0.992

700 α 0.128 0.238 0.239 2.137* 1.071 1.216 0.145 0.948 1.048 1.239 0.191 0.985
θ 0.061 4.482* 0.062 0.014 3.537 4.499 0.962 0.950 3.385 4.651 1.266 0.991
β 8.074* 0.067 0.075 2.330* 1.201 1.359 0.158 0.947 1.176 1.384 0.208 0.994

500 α 0.129 0.238 0.239 1.351* 1.084 1.202 0.118 0.952 1.065 1.221 0.156 0.988
θ 0.035 0.416* 0.047 8.347* 3.636 4.367 0.732 0.945 3.520 4.483 0.963 0.990
β 6.716* 0.063 0.068 1.363* 1.216 1.336 0.119 0.945 1.197 1.355 0.157 0.992

CL = 0.5 50 α 0.357 0.394 0.398 0.011 0.750 1.066 0.316 0.948 0.700 1.116 0.416 0.993
θ 0.286 0.043 0.134 0.072 3.180 5.165 1.985 0.946 2.866 5.479 2.613 0.982
β 0.011 0.011 0.087 0.015 1.010 1.415 0.406 0.943 0.946 1.480 0.534 0.987

100 α 0.355 0.395 0.397 5.685* 0.796 1.018 0.223 0.950 0.760 1.054 0.293 0.984
θ 0.102 1.077* 0.080 0.032 3.369 4.623 1.254 0.958 3.171 4.821 1.650 0.993
β 5.076* 1.354* 0.059 7.123* 1.062 1.341 0.279 0.955 1.018 1.385 0.368 0.994

150 α 0.364 0.401 0.402 3.682* 0.810 0.987 0.177 0.952 0.782 1.015 0.233 0.989
θ 0.072 4.204* 0.067 0.022 3.492 4.541 1.049 0.954 3.326 4.707 1.381 0.986
β 3.390* 3.498* 0.049 4.742* 1.090 1.318 0.228 0.946 1.054 1.354 0.300 0.987

300 α 0.365 0.402 0.403 1.839* 0.834 0.959 0.125 0.954 0.814 0.979 0.164 0.986
θ 0.036 0.015 0.048 0.010 3.585 4.297 0.711 0.959 3.473 4.409 0.936 0.989
β 1.700* 2.738* 0.034 2.373* 1.116 1.277 0.161 0.955 1.091 1.303 0.212 0.992

500 α 0.366 0.403 0.403 1.097* 0.847 0.944 0.096 0.952 0.832 0.959 0.127 0.991
θ 0.024 0.015 0.038 6.306* 3.662 4.215 0.553 0.954 3.575 4.302 0.728 0.992
β 1.161* 3.909* 0.028 1.510* 1.129 1.261 0.132 0.952 1.108 1.282 0.174 0.986

*

Indicate that the value multiply 10−3.

MSE = mean squared error; RB = relative bias; RE = relative error; SE = standard error; CP = coverage probability; CL = censoring level.


Table 3

Results of simulation study of MSE, RB, RE, SE, and confidence interval (CI) of different values of parameters case I ≡ (α = 2, θ = 4, β = 1.5) for different samples size

n Properties 95% 99%



MSE RB RE SE Lower Upper Length CP Lower Upper Length CP
CL = 0.2 50 α 0.196 0.220 0.221 5.999* 1.476 1.642 0.166 0.939 1.450 1.669 0.219 0.961
θ 0.506 0.128 0.178 0.070 2.519 4.460 1.942 0.955 2.211 4.767 2.556 0.983
β 0.017 0.073 0.086 9.468* 1.259 1.522 0.262 0.942 1.218 1.563 0.345 0.986

100 α 0.187 0.216 0.216 2.389* 1.522 1.615 0.094 0.937 1.507 1.630 0.123 0.956
θ 0.420 0.134 0.162 0.036 2.750 4.178 1.428 0.955 2.524 4.403 1.879 0.987
β 0.016 0.077 0.084 5.138* 1.283 1.485 0.201 0.961 1.252 1.517 0.265 0.988

150 α 0.184 0.214 0.215 1.478* 1.536 1.607 0.071 0.948 1.524 1.618 0.093 0.961
θ 0.415 0.142 0.161 0.025 2.841 4.021 1.180 0.948 2.654 4.208 1.554 0.990
β 0.016 0.080 0.084 3.442* 1.298 1.463 0.165 0.950 1.272 1.489 0.218 0.990

300 α 0.181 0.213 0.213 0.549* 1.556 1.593 0.037 0.956 1.550 1.599 0.049 0.966
θ 0.387 0.147 0.155 0.012 3.007 3.821 0.814 0.948 2.878 3.950 1.072 0.988
β 0.016 0.082 0.085 1.770* 1.316 1.437 0.120 0.955 1.297 1.456 0.158 0.991

500 α 0.180 0.212 0.212 0.297* 1.563 1.589 0.026 0.962 1.559 1.593 0.034 0.977
θ 0.385 0.150 0.155 7.351* 3.079 3.724 0.644 0.949 2.977 3.826 0.848 0.994
β 0.016 0.084 0.085 1.066* 1.328 1.421 0.093 0.954 1.313 1.436 0.123 0.993

CL = 0.3 50 α 0.284 0.262 0.266 0.013 1.293 1.658 0.366 0.955 1.235 1.716 0.481 0.990
θ 0.473 0.118 0.172 0.071 2.546 4.513 1.967 0.956 2.235 4.825 2.590 0.985
β 0.032 0.105 0.119 0.012 1.179 1.506 0.327 0.953 1.127 1.557 0.430 0.989

100 α 0.271 0.258 0.260 7.260* 1.342 1.627 0.285 0.970 1.297 1.672 0.375 0.994
θ 0.494 0.152 0.176 0.036 2.695 4.090 1.395 0.950 2.474 4.310 1.836 0.987
β 0.030 0.108 0.115 5.859* 1.223 1.453 0.230 0.953 1.187 1.489 0.302 0.994

150 α 0.270 0.258 0.260 5.223* 1.359 1.610 0.251 0.973 1.319 1.649 0.330 0.997
θ 0.468 0.156 0.171 0.023 2.822 3.933 1.110 0.955 2.647 4.108 1.461 0.986
β 0.031 0.113 0.118 4.269* 1.228 1.433 0.205 0.952 1.196 1.466 0.270 0.987

700 α 0.263 0.255 0.256 2.874* 1.392 1.587 0.195 0.984 1.361 1.618 0.257 0.996
θ 0.482 0.167 0.174 0.011 2.959 3.705 0.746 0.948 2.841 3.823 0.982 0.987
β 0.031 0.115 0.117 1.990* 1.261 1.396 0.135 0.944 1.239 1.417 0.178 0.992

500 α 0.265 0.256 0.257 1.771* 1.410 1.565 0.155 0.940 1.385 1.589 0.204 0.998
θ 0.476 0.169 0.173 6.436* 3.043 3.607 0.564 0.954 2.954 3.696 0.743 0.993
β 0.030 0.114 0.116 1.191* 1.276 1.380 0.104 0.945 1.260 1.397 0.137 0.992

CL = 0.5 50 α 0.696 0.414 0.417 0.015 0.962 1.383 0.421 0.954 0.896 1.450 0.554 0.989
θ 0.248 0.069 0.125 0.059 2.911 4.533 1.622 0.956 2.654 4.790 2.135 0.985
β 0.039 0.117 0.132 0.013 1.148 1.501 0.353 0.959 1.092 1.556 0.464 0.992

100 α 0.693 0.415 0.416 7.417* 1.025 1.316 0.291 0.953 0.979 1.362 0.383 0.988
θ 0.278 0.112 0.132 0.028 3.007 4.097 1.091 0.951 2.834 4.270 1.436 0.992
β 0.039 0.125 0.132 6.667* 1.182 1.444 0.261 0.953 1.141 1.485 0.344 0.993

150 α 0.714 0.421 0.422 4.883* 1.040 1.275 0.234 0.949 1.003 1.312 0.309 0.986
θ 0.219 0.103 0.117 0.018 3.155 4.020 0.865 0.953 3.018 4.157 1.139 0.988
β 0.036 0.122 0.127 4.247* 1.215 1.419 0.204 0.954 1.183 1.452 0.268 0.989

300 α 0.709 0.421 0.421 2.335* 1.080 1.238 0.159 0.956 1.055 1.263 0.209 0.988
θ 0.243 0.117 0.123 9.146* 3.223 3.844 0.621 0.953 3.125 3.942 0.817 0.988
β 0.036 0.125 0.127 2.121* 1.241 1.385 0.144 0.942 1.218 1.407 0.190 0.990

500 α 0.713 0.422 0.422 1.455* 1.092 1.220 0.128 0.942 1.072 1.240 0.168 0.985
θ 0.238 0.118 0.122 5.497* 3.287 3.769 0.482 0.944 3.211 3.845 0.634 0.992
β 0.036 0.125 0.126 1.315* 1.255 1.371 0.115 0.948 1.237 1.389 0.152 0.988

*

Indicate that the value multiply 10−3.

MSE = mean squared error; RB = relative bias; RE = relative error; SE = standard error; CP = coverage probability; CL = censoring level.


Table 4

The Bayes estimate of the unknown parameters case I ≡ (α = 0.3, θ = 0.45, β = 1.6) for different sized sample with hyper parameter values (ν = 1.5, b = 1.25, c = 2, d = 1.75).

n Properties 95% Start Sample


Mean SD MC error Median Lower Upper Length
CL = 0.4 10 α 0.6549 0.07380 0.001040 0.6547 0.5346 0.7757 0.2411 501 10500
θ 0.2172 0.02105 2.164E-4 0.2164 0.1841 0.2534 0.0693 501 10500
β 1.8290 0.15640 0.003156 1.7790 1.7050 2.1330 0.4280 501 10500

20 α 0.6142 0.02528 2.727E-4 0.6139 0.5729 0.6560 0.0831 501 10500
θ 0.4105 0.02191 2.668E-4 0.4103 0.3752 0.4464 0.0712 501 10500
β 1.7170 0.01729 2.645E-4 1.7120 1.7010 1.7510 0.0500 501 10500

30 α 0.5131 0.01738 2.239E-4 0.5130 0.4850 0.5421 0.0571 501 10500
θ 0.4014 0.01645 2.026E-4 0.4013 0.3753 0.4289 0.0536 501 10500
β 1.7060 0.00642 1.224E-4 1.7040 1.7000 1.7190 0.0190 501 10500

50 α 0.3084 0.00476 5.978E-5 0.3083 0.3007 0.3163 0.0156 501 10500
θ 0.3973 0.00847 1.096E-4 0.3973 0.3832 0.4110 0.0278 501 10500
β 1.7010 8.59E-4 1.504E-5 1.7010 1.7000 1.7030 0.0030 501 10500

100 α 0.2868 0.00225 2.398E-5 0.2867 0.2831 0.2905 0.0074 501 10500
θ 0.3973 0.00452 5.821E-5 0.3973 0.3899 0.4048 0.0149 501 10500
β 1.7000 1.84E-4 3.581E-6 1.7000 1.7000 1.7010 0.0010 501 10500

CL = 0.5 10 α 0.7041 0.07437 7.490E-4 0.7028 0.5815 0.8289 0.2474 501 10500
θ 0.2176 0.02068 1.887E-4 0.2171 0.1848 0.2521 0.0673 501 10500
β 1.7960 0.10540 0.001739 1.7630 1.7040 1.9960 0.2920 501 10500

20 α 0.6019 0.02399 2.669E-4 0.6018 0.5632 0.6416 0.0784 501 10500
θ 0.4566 0.02337 2.233E-4 0.4562 0.4186 0.4962 0.0776 501 10500
β 1.7200 0.02094 3.464E-4 1.7140 1.7010 1.7610 0.0600 501 10500

30 α 0.5270 0.01692 1.684E-4 0.5267 0.4994 0.5550 0.0556 501 10500
θ 0.4542 0.01785 1.903E-4 0.4540 0.4252 0.4840 0.0588 501 10500
β 1.7070 0.00729 1.378E-4 1.7050 1.7000 1.7220 0.0220 501 10500

50 α 0.2993 0.00483 5.848E-5 0.2992 0.2913 0.3072 0.0159 501 10500
θ 0.3736 0.00833 1.061E-4 0.3735 0.3599 0.3874 0.0275 501 10500
β 1.7010 9.11E-4 1.580E-5 1.7010 1.7000 1.7030 0.0030 501 10500

100 α 0.2836 0.00227 2.639E-5 0.2836 0.2799 0.2874 0.0075 501 10500
θ 0.3997 0.00470 5.663E-5 0.3996 0.3920 0.4075 0.0155 501 10500
β 1.7000 1.99E-4 3.401E-6 1.7000 1.7000 1.7010 0.0010 501 10500

CL = censoring level.


Table 5

The Bayes estimate of the unknown parameters case I ≡ (α = 0.4, θ = 0.75, β = 1.6) for different sized sample with hyper parameter values (ν = 1.5, b = 1.25, c = 2, d = 1.75)

n Properties 95% Start Sample


Mean SD MC error Median Lower Upper Length
CL = 0.4 10 α 0.2021 0.02114 2.132E-4 0.2017 0.1682 0.2374 0.0692 501 10500
θ 0.2488 0.02224 2.177E-4 0.2482 0.2135 0.2868 0.0733 501 10500
β 2.6720 0.27920 0.004418 2.7460 2.0920 2.9820 0.8900 501 10500

20 α 0.2228 0.00952 1.014E-4 0.2229 0.2069 0.2388 0.0319 501 10500
θ 0.7208 0.03708 4.030E-4 0.7205 0.6605 0.7832 0.1227 501 10500
β 1.7320 0.03266 6.086E-4 1.7220 1.7020 1.7960 0.0940 501 10500

30 α 0.2810 0.00838 9.386E-5 0.2810 0.2673 0.2950 0.0277 501 10500
θ 0.6216 0.02279 2.720E-4 0.6213 0.5847 0.6593 0.0746 501 10500
β 1.7100 0.00992 1.618E-4 1.7070 1.7000 1.7300 0.0300 501 10500

50 α 0.3599 0.00638 8.779E-5 0.3600 0.3495 0.3704 0.0209 501 10500
θ 0.7683 0.01716 2.335E-4 0.7681 0.7404 0.7965 0.0561 501 10500
β 1.7040 0.00354 5.780E-5 1.7020 1.7000 1.7100 0.0100 501 10500

100 α 0.3283 0.00299 3.803E-5 0.3283 0.3234 0.3332 0.0098 501 10500
θ 0.7716 0.00922 1.228E-4 0.7716 0.7566 0.7866 0.0300 501 10500
β 1.7010 6.21E-4 1.065E-5 1.7000 1.7000 1.7020 0.0020 501 10500

CL = 0.5 10 α 0.2125 0.02164 2.168E-4 0.2121 0.1777 0.2492 0.0715 501 10500
θ 0.2488 0.02191 2.285E-4 0.2480 0.2142 0.2861 0.0719 501 10500
β 2.2740 0.37610 0.004562 2.2370 1.7430 2.9120 1.1690 501 10500

20 α 0.2288 0.00962 1.126E-4 0.2288 0.2132 0.2448 0.0316 501 10500
θ 0.8006 0.03868 4.545E-4 0.8004 0.7369 0.8650 0.1281 501 10500
β 2.3690 0.35980 0.004547 2.3650 1.7910 2.9290 1.1380 501 10500

30 α 0.2967 0.00809 9.804E-4 0.2967 0.2834 0.3103 0.0269 501 10500
θ 0.7012 0.02429 2.923E-4 0.7007 0.6620 0.7423 0.0803 501 10500
β 1.7130 0.01309 2.137E-4 1.7090 1.7010 1.7390 0.0380 501 10500

50 α 0.3539 0.00658 8.562E-5 0.3538 0.3432 0.3648 0.0216 501 10500
θ 0.7939 0.01819 2.421E-4 0.7939 0.7650 0.8240 0.0590 501 10500
β 1.7070 0.00657 1.211E-4 1.7040 1.7000 1.7200 0.0200 501 10500

100 α 0.3228 0.00291 3.645E-5 0.3228 0.3180 0.3276 0.0096 501 10500
θ 0.7793 0.00917 9.365E-5 0.7792 0.7644 0.7944 0.0300 501 10500
β 1.7010 8.25E-4 1.491E-5 1.7010 1.7000 1.7020 0.0020 501 10500

CL = censoring level.


References
  1. Abdel-Ghani MM (1998). Investigation of some lifetime models under partially accelerated life tests (Ph.D. Thesis) , Department of Statistics, Faculty of Economics and Political Science, Cairo University, Egypt.
  2. Abushal TA and Soliman AA (2015). Estimating the Pareto parameters under censoring data for constant partially accelerated life tests. Journal of Statistical Computation and Simulation, 85, 917-934.
    CrossRef
  3. Bai DS and Chung SW (1992). Optimal design of partially accelerated life tests for the exponential distribution under type I censoring. IEEE Transactions on Reliability, 41, 400-406.
    CrossRef
  4. Bessler S, Chernoff H, and Marshall AW (1962). An optimal sequential for step-stress accelerated life test. Technometrics, 4, 367-379.
    CrossRef
  5. Cheng Y and Wang F (2012). Estimating the Burr XII parameters in constant stress partially accelerated life tests under multiple censored data. Communication in Statistics - Simulation and Computation, 41, 1711-1727.
    CrossRef
  6. Chernoff H (1962). Optimal accelerated life designs for estimation. Technometrics, 4, 381-408.
    CrossRef
  7. Hassan AS (2007). Estimation of the generalized exponential distribution parameters under constant stress partially accelerated life testing using type I censoring. The Egyptian Statistical Journal, Institute of Statistical Studies & Research, Cairo University, 52, 48-62.
  8. Hassan AS, Assar SM, and Zaky AN (2015). Constraint stress partially accelerated life tests for inverted Weibull distribution with multiple censored data. International Journal of Advanced Statistics and Probability, 3, 72-82.
    CrossRef
  9. Ismail AA (2006). Optimum constant-stress partially accelerated life test plans with type II censoring: the case of Weibull failure distribution InterStat, Electronic Journal .
  10. Ismail AA, Abdel-Ghaly AA, and El-Khodary EH (2011). Optimum constant stress life test plans for Pareto distribution under type I censoring. Journal of Statistical Computation and Simulation, 81, 1835-1845.
    CrossRef
  11. Jaheen ZF, Moustafa HM, and Abd El-monem GH (2014). Bayes inference in constant partially accelerated life tests for the generalized exponential distribution with progressive censoring. Communication in Statistics - Theory and Methods, 43, 2973-2988.
    CrossRef
  12. Kamal M, Zarrin S, and Islam AU (2013). Constant stress partially accelerated life test design for inverted Weibull distribution with type I censoring. Algorithms Research, 2, 43-49.
  13. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, and Teller E (1953). Equations of state calculations by fast computing machines. Journal Chemical Physics, 21, 1087-1091.
    CrossRef
  14. Mudholkar GS and Srivastava DK (1993). ExponentiatedWeibull family for analyzing bathtub failure rate data. IEEE Transactions on Reliability, 42, 299-302.
    CrossRef
  15. Nelson W (1990). Accelerated Life Testing: Statistical Models, Test Plans and Data Analysis, New York, John Wiley & Sons.
    CrossRef
  16. Robert C and Casella G (2004). Monte Carlo Statistical Methods (2nd ed), New York, Springer-Verlag.
    CrossRef
  17. Srivastava PW and Mittal N (2013a). Failure censored optimum constant stress partially accelerated life tests for the truncated logistic life distribution. International Journal of Business Management, 3, 41-66.
  18. Srivastava PW and Mittal N (2013b). Optimum constant stress partially accelerated life tests for the truncated logistic life distribution under time constraint. International Journal of Operations Research Nepal, 2, 33-47.
  19. Tobias PA and Trindade DC (1995). Applied Reliability (2nd ed), New York, Chapman and Hall/CRC.
  20. Zarrin S, Kamal M, and Saxena S (2012). Estimation in constant stress partially accelerated life tests for Rayleigh distribution using type I censoring. Electronic Journal of International Group on Reliability, 7.