On Bayesian estimation of step stress accelerated life testing for exponentiated Lomax distribution based on censored samples

In reliability analysis and life-testing experiments, the researcher is often interested in the effects of changing stress factors such as “temperature”, “voltage” and “load” on the lifetimes of the units. Step-stress (SS) test, which is a special class from the well-known accelerated life-tests, allows the experimenter to increase the stress levels at some constant times to obtain information on the unknown parameters of the life models more speedily than under usual operating conditions. In this paper, a simple SS model from the exponentiated Lomax (ExpLx) distribution when there is time limitation on the duration of the experiment is considered. Bayesian estimates of the parameters assuming a cumulative exposure model with lifetimes being ExpLx distribution are resultant using Markov chain Monte Carlo (M.C.M.C) procedures. Also, the credible intervals and predicted values of the scale parameter, reliability and hazard are derived. Finally, the numerical study and real data are presented to illustrate the proposed study.


Introduction and motivation
Generally, there are two well-known types of SS loadings which are concerned in accelerating life tests ALTs: The 1 st is the linearly increasing stress and the 2nd is the stable stress. There are two types of real data which are found from ALT. The 1st type is the whole data set in which failure time (FT) of each unit is seen and the 2 nd is the censored data in which FT of every sample unit mayn't be observed yet. Miller and Nelsen (1983) studied finest test plans which is employed to minimize the asymptotic variance (Asy-Var.) of the MLE of the mean life at a plan (use) stress for the two-step ALT (2S-ALT) when every units are gone to death. Bai et al. (1989) further studied optimum 2S-ALT where a pre-determined censoring time is considered. Rene et al. The major aim and motivation of this article is to provide a model for 2-SS ALTs based on the ExpLx model. We reflect the Bayesian estimation of the scale parameter, reliability, hazard rate of the distribution of FTs under regular operating conditions and the SS ALT model based on cumulative break that helps a log linear model. We find the probability density and cumulative distribution functions (P-D-F & C-D-F) for lifetimes from the ExpLx model. Finally, the performance of these methods is illustrated by a M.C.M.C simulation study and application under censoring type-I.

Basic assumption SS ALT model
The following basic assumptions are taken in the experiment of SS testing For any stress 1 ≤ 2 ≤ ⋯ ≤ , the lifetime model is ExpLx ( , , ). The P-D-F is written as are constants with respect to stress , and the scale parameter is an affected by stress , = 1,2, … , through the log linear model in the form where both a & b are unknown parameters. Suppose that for a certain pattern of stress , units run at stress starting at time ( −1) and going to time The conduct of these units is as follows: At step 1, the population portion 1 ( ) of units failing by time under steady stress 1 is Let ( ) be the population C-D-F of units failing (dying) under SS, then in the 1 st step let ( ) = 1 ( ),0 < < 1 , where 1 is the time when the stress is grown from 1 to be 2 .
When the 2 nd step starts, units which have same age 1 created the alike fraction unsuccessful seen at the termination of the step.
In other meaning, the stayers at the time 1 will be converted to a certain stress 2 stating at 2 , which can be controlled as the result of 2 ( 1 ) = 2 ( 1 ) 1 ( 1 ), where ∆ 0 = τ 1 − τ 0 , u 0 = 0 and ∆ j−2 = τ j−1 − τ j−2 , j=2,3,…,k, by taking the root and taking the root -β to two sides, the cumulative exposure model for j steps is written as follows where −1 = − exp ( ( −1 − )) (∆ −2 + −2 ). It seen that F(t), for a SS pattern next F(t), can be printed in the form: and the associated P-D-F f(t), is presented as the following form: In this work, the ML process is consumed for the SS model. The P-D-F for each test is shown in (3) and it is the time derivative of the C-D-F given in (2). The sample likelihood is the product of such P-D-F's evaluated at FTs if the uncensored data is used or the seen P-D-F's of such survival function (SF) evaluated at censoring time when censoring is put on. It is indicated that F(t), differs for units with unlike SS forms.
The MLE on Type I when there are 2 SS as special case. A special circumstance, we let k=2, 1 is the time at which the stress will change from 1 to be 2 and is the time at which the experiment is finished (censoring time). In time SS, the stress −1 is grown to j at τ j−1 , j = 2, … , k. It is implicit that the test is stayed till all components die or until time . The likelihood function has the identical form as The log likelihood function ℓ can be presented as ℓ = ∑ [ 1 ( + 1 ) + log( + ) + 1 (− − 1) (1 + exp( + 1 ) 1 )

Bayesian analysis
In this section, we introduce Bayesian estimation of the unknown parameters, reliability and hazard rate functions (H-R-Fs) of the SS ExpLx model based squared error loss (SEL) function and censoring from the type I.
Due to Sinha (1998), the Jeffrey's rule for picking the non-informative prior P-D-F for the unrelated random variables (RVs) a, b, , are studied as well-known uniform distribution.
The joint non-informative prior density function of the random parameters is found as The informative prior P-D-F of a, b, , have a lognormal (LogN) distribution which are totally uncorrelated with the location and scale parameters 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , respectively. It has the following P-D-F The joint informative prior P-D-F of the random parameters is given as and hence the posterior P-D-F with non-informative prior is provided as follows and hence the posterior P-D-F with informative prior is presented as follows It is obvious that from the posterior P-D-F the usual Bayesian methods of parameter estimation with integration is hard so one of the M.C.M.C methods are spent to simulate or make direct draws from some the complex model. Then, we shall study the SEL function to get the a consistent estimators from the marginal posterior P-D-Fs.

Bayesian estimators under the squared loss function
It is well known that the Bayesian estimate based on SEL function, is the posterior mean. The SEL function is a symmetric loss function and reads as where denotes a constant and ̂ is an estimator. The Bayesian estimators of a, b, , of SS ExpLx(a, b, , ) under SEL can be got when the parameters are unknown, respectively as follows

Credible intervals
The confidence intervals (CIs) estimation in Bayesian technique is straighter than the non-Bayesian method which build on CIs. Once the marginal P-D-F of has been derived, a symmetric 100(1 − )% two-sided Bayesian probability interval estimate of , denoted by [ , ], can be built easily. The Bayesian equivalent to the CIs is called a credibility interval. Generally, we have for the limits and . Another time, these equations are not in closed forms, so we concern proper numerical techniques to find a solution of these non-linear equations.

Numerical illustration
Under Bayesian estimation, we shall assume three M.C.M.C chains with dissimilar initial values for many levels of integration necessary to find the standardizing constant and the marginal posterior P-D-Fs.

The simulation Algorithm
Accelerated life data from ExpLx model are generated using R language at different samples of size (n=40, n=60), λ = 2, = 2, = 3. Two accelerated stress stages If the experiment is ended once wholly the objects fail or when a fixed censoring time is reached (Type I censoring). When 1 = 2 = 20, = 0.90, 0.80 and 1 = 2 = 30, = 0.90, 0.80, respectively. The analogous likelihood function is found in (4). where P-I=the non-informative prior, U= uniform distribution and P-II= informative prior, lnor =LogN. The posterior mean and variance of , given t can be designed, respectively, as follows and where and p is the p th percentile.

An application
A real data set of FTs of "84 aircrafts windshield" is used to compare the fits of the Exp-Lx model. We consider the data on FTs for a specified model windshield given in Murthy et al. (2004). These data were analyzed by Ramos et al.