# Estimation of Constant Stress Partially Accelerated Life Test for Fréchet Distribution with Type-I Censoring

Authors: Abdullah Ali H. Ahmadini; Wali Khan Mashwani; Rehman Ahmad Khan Sherwani; Shokrya S. Alshqaq; Farrukh Jamal; Miftahuddin Miftahuddin; Kamran Abbas; Faiza Razaq; Mohammed Elgarhy; Sanaa Al-Marzouki
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Modern reliability engineering accelerated life tests (ALT) and partially accelerated life tests (PALT) are widely used to obtain the timely information on the reliability of objects, products, elements, and materials as well as to save time and cost. The ALTs or PALTs are useful in determining the failed manners of the items at routine conditions by using the information of the data generated from the experiment. PALT is the most sensible method to be used for estimating both ordinary and ALTs. In this research, constant stress PALT design for the Fréchet distribution with type-I censoring has been investigated due to a wide applicability of the Fréchet distribution in engineering problems especially in hydrology. The distribution parameters and acceleration factor are obtained by using the maximum likelihood method. Fisher's information matrix is used to develop the asymptotic confidence interval estimates of the model parameters. A simulation study is conducted to illustrate the statistical properties of the parameters and the confidence intervals by using the R software. The results indicated that the constant stress PALT plan works well. Moreover, a numerical example is given to exemplify the performance of the proposed methods.


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## Body

## 1. Introduction

The continuous progress in manufacturing design makes the products and materials highly reliable [1]. It is becoming more and more difficult to get information under normal use conditions about the lifespan of objects and materials with higher reliability at the testing time, and it becomes very expensive and time-consuming under such conditions [2]. The sample is examined for these objects at more serious circumstances than usual [3]. These situations belong to stresses, which may be in the form of voltage, force, temperature, vibrations, moisture, etc [4–7]. To develop a new life testing method, these problems have motivated the researcher to get information timely on the reliability of components, objects, and products.Particularly, when test items are run at both normal and higher than normal stress conditions, the more suitable test to be applied is the partially accelerated life test (PALT) [8–10]. Furthermore, it is the most important method to be utilized for evaluating the acceleration factor and thus extrapolating the accelerated test data to use condition. The PALT combines both accelerated and ordinary life tests. Thus, PALT is reasonable for estimating the acceleration factor (β > 1), “which is the ratio of the hazard rate at the accelerated condition to that at normal use conditions” [11].The commonly used methods in PALT are constant stress PALT (CSPALT) [4, 12, 13], step stress PALT (SSPALT) [5, 14–16], and progressive stress PALT (PSPALT) [17–19]. In CSPALT, products are examined at either accelerated condition or at normally used condition before the test is being terminated.Additionally, censoring is very ordinary in life testing experiments. It frequently happens that the experiment is censored when the experimenter may not be able to examine the lifespan of all units put on the test because of time restrictions and other limitations on the data collection. It is usually used when a distribution of exact lifetimes is known only for a part of the test items, and the remainder of the lifetimes is known only to exceed certain values under a life test. The two most common types of censoring are type-I censoring (time censoring) and type-II censoring (failure censoring).For a brief overview of PALT, literature is abundant on designing PALT. Bai and Chung [8, 9] discussed both the problem of estimation and optimally designing constant stress PALT for exponential and lognormal distributions. Ismail [20] has applied maximum likelihood (ML) and Bayesian methods for evaluating the parameters of Pareto distribution of the second kind and the acceleration factor. Ismail et al. [21] considered the case of CSPALT when under type-I censoring two stress levels was included. Similarly, Cheng and Wang [22], Ismail [23], Kamal et al. [13], Wang [24, 25], and Hassan et al. [12] applied PALT and acceleration factors using different lifetime distributions; PALT and CSPALT have been studied via Bayesian approach [19, 26–28].In this paper, the CSPALT plan using type-I censoring with the assumption that the lifetimes of the test items at use condition follow Fréchet distribution which is considered. Maurice Fréchet was a French mathematician who had identified one possible limit distribution for the largest order statistic in 1927 [29]. The probability density function (PDF) of Fréchet distribution is(1)ft=αθtθ−α+1exp−tθ−α,t,α,θ>0,where α is the shape parameter and θ is the scale parameter. This distribution has a heavy upper tail and is bounded on the lower side (t > 0). The cumulative distribution function (CDF) is(2)Ft=exp−tθ−α,t,α,θ>0,The reliability function of Fréchet distribution is(3)Rt=1−exp−tθ−α,t,α,θ>0,The Fréchet distribution is applied to extreme events and linked to the modeling of several real-world phenomena, including human lifetimes, flood and seismic analyses, radioactive emissions, and maximum one-day rainfalls.The rest of the study is arranged as follows. In Section2, the assumption and test procedures for Fréchet distribution with constant stress using type-I censoring are discussed. The maximum likelihood estimates (MLEs) of the distribution parameters and acceleration factor are presented in Section 3. Fisher information matrix and asymptotic confidence interval are discussed in Section 4. A simulation study is presented in Section 5. Finally, Section 6 presents the results, conclusion, and recommendations for future work.


## 2. Model Description

The following notations are used for model description:(i)
n = total number of objects in a PALT(ii)
τ = censoring time of CSPALT(iii)
t = lifespan of an object at normal use condition(iv)
X = lifespan of an object at accelerated condition(v)
β = acceleration factor (β > 1) and defined as β = T/X(vi)
ti = observed lifetime of ith object at usual use condition(vii)
xj = observed lifetime of jth unit at accelerated condition(viii)
t1 ≤ … ≤ tnu ≤ τ = ordered failure time at use condition(ix)
x1 ≤ … ≤ xna ≤ τ = ordered failure time at accelerated condition


### 2.1. Test Procedure of CSPALT

In CSPALT, total test units “n” are randomly divided into two samples of sizes “nr” and n(1 − r), respectively, where “r” is the sample proportion. In the first sample, “nr” items are randomly chosen among “n” items, which are assigned to accelerated conditions and the other n(1 − r) are assigned to regular use conditions. Each test object of every sample is run until the censoring time “τ” and the test condition is not changed.


### 2.2. Assumptions

The following assumptions are also made for CSPALT.(i)
Under normal-use condition, lifespan of an object supports Fréchet distribution.(ii)
The lifespan of an object at accelerated condition is attained byX = β−1T, where β > 1, which is an acceleration factor. Therefore, the PDF at accelerated condition is(4)fx,α,β,θ=αβθβxθ−α+1exp−βxθ−α,x,θ,β>0.(iii)
The lifespanTi,i=1,2,…,n1−r of objects allocated to the normal condition are independently and identically distributed (i.i.d) random variables.(iv)
The lifespanXj,j=1,2,…,nr of objects assigned to accelerate are i.i.d. random variables.(v)
The lifetimesTi and Xjare mutually independent.


## 2.1. Test Procedure of CSPALT

In CSPALT, total test units “n” are randomly divided into two samples of sizes “nr” and n(1 − r), respectively, where “r” is the sample proportion. In the first sample, “nr” items are randomly chosen among “n” items, which are assigned to accelerated conditions and the other n(1 − r) are assigned to regular use conditions. Each test object of every sample is run until the censoring time “τ” and the test condition is not changed.


## 2.2. Assumptions

The following assumptions are also made for CSPALT.(i)
Under normal-use condition, lifespan of an object supports Fréchet distribution.(ii)
The lifespan of an object at accelerated condition is attained byX = β−1T, where β > 1, which is an acceleration factor. Therefore, the PDF at accelerated condition is(4)fx,α,β,θ=αβθβxθ−α+1exp−βxθ−α,x,θ,β>0.(iii)
The lifespanTi,i=1,2,…,n1−r of objects allocated to the normal condition are independently and identically distributed (i.i.d) random variables.(iv)
The lifespanXj,j=1,2,…,nr of objects assigned to accelerate are i.i.d. random variables.(v)
The lifetimesTi and Xjare mutually independent.


## 3. Parameter Estimation

The ML method is one of the most important and widely used methods in statistics. The main reasons are that the ML method is very vigorous and provides the estimates of parameters with good statistical properties such as consistency, asymptotic unbiased, asymptotic efficiency, and asymptotic normality.Letδui and δaj be the indicator functions under the use and accelerated conditions, respectively, such that(5)δui=1,ti≤τ,0,otherwise,i=1,2,…,n1−r,δaj=1,xj≤τ,0,otherwise,j=1,2,…,nr.The likelihood function forti,δuiandxj,δaj can be expressed as(6)Luiti,δui|α,θ=∏i=1n1−rαθtiθ−α+1exp−tiθ−αδui.1−exp−τθ−αδ¯ui,(7)Lajxj,δaj|α,β,θ=∏j=1nrαβθβxjθ−α+1exp−βxjθ−αδaj×1−exp−βτθ−αδ¯aj,where δ¯ui=1−δui and δ¯aj=1−δaj.The total likelihood fort1;δu1,…,tn1−r;δun1−r,x1;δa1,…,xnr;δanr is(8)Lt,x|α,β,θ=Luiti,δui|α,θ∗Lajxj,δaj|α,β,θ,Lt,x|α,β,θ=∏i=1n1−rαθtiθ−α+1exp−tiθ−αδui1−exp−τθ−αδ¯ui×∏j=1nrαβθβxjθ−α+1exp−βxjθ−αδaj×1−exp−βτθ−αδ¯aj.It is more convenient to work with the log-likelihood function. The log-likelihood function of (8) is(9)ℓ=lnL=∑i=1n1−rδuilnα−lnθ−α+1lntiθ−tiθ−α+∑i=1n1−rδ¯uiln1−exp−τθ−α+∑j=1nrδajlnα+lnβ−lnθ−α+1lnβxjθ−βxjθ−α+∑j=1nrδ¯aj.ln1−exp−βτθ−α.Letnu and na be the numbers of items that failed at normal and accelerated conditions, respectively. Similarly, let cu and ca be the numbers of items censored at normal and accelerated conditions, respectively:(10)∑i=1n1−rδui=nu,∑j=1nrδaj=na,∑i=1n1−rδ¯ui=cu,∑i=1n1−rδ¯ui=ca,l=nulnα−nulnθ−α+1∑i=1n1−rδuilntiθ−∑i=1n1−rδuitiθ−α+cu.ln1−exp−τθ−α+nalnα+nalnβ−nalnθ−α+1∑j=1nrδajlnβxjθ−∑j=1nrδajβxjθ−α+ca.ln1−exp−βτθ−α,(11)l=nu+nalnα−nu+nalnθ+nalnβ−α+1∑i=1n1−rδuilntiθ−∑i=1n1−rδuitiθ−α+cu∗ln1−exp−τθ−α−α+1∑j=1nrδajlnβxjθ−∑j=1nrδajβxjθ−α+caln1−exp−βτθ−α.MLEs ofα, θ, and β are obtained by solving the system of equations ∂l/∂α=0,∂l/∂θ=0and∂l/∂β=0, respectively:(12)∂l∂α=nu+naα−∑i=1n1−rδuilntiθ+∑i=1n1−rδuitiθ−αlntiθ−cuτ/θ−αlnτ/θe−τ/θ−α1−e−τ/θ−α−∑j=1nrδajlnβxjθ+∑j=1nrδajβxjθ−αlnβxjθ−caβτ/θ−αlnβτ/θe−βτ/θ−α1−e−βτ/θ−α,(13)∂l∂θ=nu+naαθ+cuαθα−1τ−αe−τ/θ−α1−e−τ/θ−α+caαθα−1βτ−αe−βτ/θ−α1−e−βτ/θ−α−αθα−1∑i=1n1−rδuiti−α+β−α∑j=1nrδajxj−α,(14)∂l∂β=−naαβ+αθαβ−α+1∑j=1nrδajxj−α−caαβτ/θ−αe−βτ/θ−αβ1−e−βτ/θ−α.The above equations cannot be written in the closed form. So, an iterative procedure can be used to get MLEs. Here, we use the BFGS Quasi-Newton Optimization method which is available in the R software. Furthermore, for interval estimates of model parameters, we need the Fisher information matrix.


### 3.1. Fisher Information Matrix

Here, it is hard to obtain an explicit solution to nonlinear equations, so an iterative method such as the Newton Raphson method [30] is used to find the solution of equations to obtain MLEs. Also, we get the asymptotic confidence intervals. Hence, the asymptotic variance of the MLE can be attained by the inverse of the observed Fisher information matrix:(15)V=AVarα^ACovα^θ^ACovα^β^ACovθ^α^AVarθ^ACovθ^β^ACovβ^α^ACovβ^θ^AVarβ^.The 100 (1 −α) percent asymptotic confidence interval for α, θ, and β can be written as(16)α^±Z1−αAVarα^,θ^±Z1−αAVarθ^,β^±Z1−αAVarβ^,respectively.


## 3.1. Fisher Information Matrix

Here, it is hard to obtain an explicit solution to nonlinear equations, so an iterative method such as the Newton Raphson method [30] is used to find the solution of equations to obtain MLEs. Also, we get the asymptotic confidence intervals. Hence, the asymptotic variance of the MLE can be attained by the inverse of the observed Fisher information matrix:(15)V=AVarα^ACovα^θ^ACovα^β^ACovθ^α^AVarθ^ACovθ^β^ACovβ^α^ACovβ^θ^AVarβ^.The 100 (1 −α) percent asymptotic confidence interval for α, θ, and β can be written as(16)α^±Z1−αAVarα^,θ^±Z1−αAVarθ^,β^±Z1−αAVarβ^,respectively.


## 4. Simulation Study

A simulation study is carried out to evaluate the performance of proposed estimators. It is performed by using the R software (3.2.3) for demonstrating the theoretical outcomes of the estimation problems. The performances of MLEs are evaluated through mean square error (MSE) and absolute bias of estimates. Also, 90% and 95% confidence limits are constructed for parameters and acceleration factors. For this purpose, several data sets are generated from two-parameter Fréchet distribution with sizesn=50,100,150,200,300,and500. Samples are generated by inverse transformation of the Fréchet distribution and the process is replicated 1000 times.The simulation steps are summarized below:(1)
For estimation, the sample size is varied to see the effect of large and small samples are considered. The random samples are generated from Fréchet distribution by using transformationx=θ−logu−1/α, where u is uniformly distributed, i.e., U ∼ uniform (0, 1).(2)
The values for true parameters are taken as (α = 0.75, β = 1.2, θ = 1), (α = 1, β = 1.1, θ = 1), and (α = 0.8, β = 1.2, θ = 0.8), respectively. The selected censoring level is 0.7 ∗ n(1 − r) and 0.3 ∗ nr.(3)
The “n” test item is split with sample proportion r = 0.3, such that “nr” items are assigned at the accelerated condition and n(1 − r) are allotted to normal use condition.(4)
The estimates of the model parameters and their corresponding summary statistics are obtained by the CSPALT model.(5)
For the different combination of stresses, the MLEs forα, β, and θ, MSE, and absolute bias of the estimates and ninety percent and ninety-five percent CIs are derived.(6)
Tables1–3 summarize the results of the ML estimates for α, β, and θ. The numerical results presented in Table 4 are based on 1000 simulation replications.(7)
Figures1–3 present the MSEs against different sample sizes for the MLEs of α, β, and θ.Table 1
Average estimates of CSPLAT withα = 0.75, β = 1.2, θ = 1, and r = 0.3.
ηParametersEstimatesMSEAbsolute bias95% CI50α1.58700.81090.8370−0.8226, 3.9966β1.18300.12290.01690.49494, 1.8711θ0.41590.34850.5840−1.2117, 2.0436100α1.47550.58360.7352−0.5724, 3.5234β1.26350.07150.05310.7797, 1.7473θ0.41470.34650.5858−1.2112, 2.0408150α1.46030.532740.7103−0.5569, 3.5019β1.29140.048930.07760.8076, 1.7464θ0.41360.346110.5864−1.2172, 2.0429200α1.43860.492540.68863−0.5043, 3.3997β1.29920.041600.099180.8576, 1.7275θ0.41260.346630.58743−1.2125, 2.0402300α1.42210.463640.67210−0.4592, 3.3088β1.30890.036550.108860.8753, 1.7414θ0.41280.345920.58724−1.2159, 2.0416Table 2
Average estimates of CSPLAT withα = 1, β = 1.1, θ = 1, and r = 0.3
ηParametersEstimatesMSEAbsolute bias95% CI50α2.14661.46011.12465−1.1840, 5.5040β0.71120.464570.38876−0.9333, 2.2885θ0.50800.248110.49196−0.8467, 1.8718100α1.98241.044110.98242−0.8141, 4.7996β0.75130.283920.34868−0.4538, 1.9765θ0.51610.237070.48393−0.8336, 1.8634150α1.94430.941510.94430−0.7090, 4.5968β0.80990.233700.29013−0.2890, 1.9258θ0.51560.236480.48444−0.8350, 1.8635200α1.93180.902380.93182−0.6499, 4.4828β0.80660.218640.29344−0.2114, 1.9076θ0.51600.235740.48405−0.8345, 1.8628300α1.90430.840350.90430−0.6121, 4.4129β0.83150.182930.26852−0.1070, 1.8217θ0.51660.234640.48344−0.8333, 1.8619500α1.89910.822070.89914−0.5948, 4.3840β0.84880.169800.25116−0.0358, 1.7549θ0.51450.236280.48553−0.8374, 1.8635Table 3
Average estimates of CSPLAT withα = 0.8, β = 1.2, θ = 0.8, and r = 0.3
ηParametersEstimatesMSEAbsolute bias95% CI50α1.70920.95420.9092−0.9064, 4.3248β1.20320.20570.00320.3143, 2.0921θ0.34490.21110.4551−0.9225, 1.6123100α1.58060.65790.7806−0.6258, 3.7871β1.22560.11980.02570.5454, 1.9059θ0.35190.20280.4480−0.8932, 1.5971150α1.55920.60600.7592−0.5721, 3.6905β1.25630.10700.05630.6059, 1.9067θ0.34650.20700.4535−0.9124, 1.6055200α1.54070.57230.7407−0.5344, 3.6158β1.27710.08290.07720.6928, 1.8614θ0.34980.20580.4502−0.8998, 1.5994300α1.52040.53240.7203−0.4893, 3.5301β1.29060.06820.09050.7487, 1.8324θ0.34810.20490.4519−0.9058, 1.6019500α1.50810.50920.7081−0.4623, 3.4784β1.29250.05340.09260.8045, 1.7806θ0.34670.20400.4502−0.9103, 1.6038Table 4
Interval estimates of the parameters.
Parameters90% CI95% CIα−0.6142, 3.7657−0.2623, 3.4137β0.26319, 2.57640.4491, 2.3905θ−0.5521, 0.9953−0.4277, 0.8710Figure 1
Graph of MSE versus sample size whenα = 0.75, β = 1.2, and θ = 1.Figure 2
Graph of MSE versus sample size whenα = 1, β = 1.1, and θ = 1.Figure 3
Graph of MSE versus sample size whenα = 0.8, β = 1.2, and θ = 0.8.


## 5. Numerical Example

For illustration, we assume that the values of Fréchet constant stress model parameters are known and use the simulated data based onn = 100, α = 0.8, β = 1.3, θ = 0.5, and r = 0.3, and at accelerated condition, na=10. The MLEs of the model parameters and acceleration factor obtained by using the R software are α^=1.5757,θ^=0.2216,andβ^=1.4198. The inverse of the estimated Fisher information matrix is(17)F=138.4549154.526224.7334154.526260.14885.716924.733405.716920.6185.The 90% and 95% interval estimates for the model parameters and acceleration factor are provided in Table4.The asymptotic variance-covariance matrix ofα, θ, and β are(18)V=0.00720.00650.04040.00650.01660.17490.04040.17490.0485.The 95% asymptotic confidence interval forα, θ, and β are presented, respectively:(19)1.4093≤α≤1.7420,−0.0309≤θ≤0.4741,0.9881≤β≤1.8514.From the results, we observe that the point estimates are stable. We also examine that the results support theoretical findings of CSPALT for Fréchet distribution.


## 6. Conclusion

From Tables1–3, it is observed that the MLEs converge to the true values of the parameters with an increase in sample size “n.” Also, we find that the MLEs have good statistical properties for a fixed value of parameters, as we increase sample size, the MSE and bias of the estimators are decreased. Moreover, from Tables 1–3, the MSEs of α decrease as we increase the sample size. It can be seen from results in Tables 1–3 that when there is an increment in sample size, the interval of the estimates decreases. The outcomes of simulation suggest that PALT is an appropriate model using Fréchet distribution which enables us to save cost and time without using a higher level of stress to all test items. From simulation results, it is concluded that the MLEs have good statistical properties.


## 7. Recommendation

Some pertinent suggestions are also given for future research which is tantamount to provide a pathway for future researchers in the field of PALT. This work can be extended for type-II censoring of CSPALT using Fréchet distribution.



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