Accelerated Life Testing Design using Geometric Process for Marshall-Olkin Extended Exponential Distribution with Type II censored data

In this paper the geometric process is used for the analysis of accelerated life testing for MarshallOlkin Extended Exponential (MOEE) distribution using Type II censored data. Assuming that the lifetimes under increasing stress levels form a geometric process, The parameters are estimated by using the maximum likelihood method and the original parameters instead of the developing inference for the parameters of the log linear link function are used. The asymptotic interval estimates of the parameters of the distribution using Fisher information matrix are also obtained. The simulation study is conducted to illustrate the statistical properties of the parameters and the confidence intervals. Keywords— Maximum Likelihood Estimation; Type II Censoring; Survival Function; Fisher Information Matrix; Asymptotic Confidence Interval; Simulation Study. 1. Introduct ion Models and methods of accelerated life-testing are useful when technical systems under test tend to have long lifetimes. Under normal operating conditions, as systems usually last long, the corresponding life-tests become too time-consuming and expensive. In these cases, accelerated life tests (ALT) can be applied to reduce the experimental time and hence the cost. There are basically three types of accelerated life tests: constant-stress test, step-stress test and progressive stress. In constant-stress test each experimental unit put at only one of the stress levels and in step-stress the level of stress is increased step by step until all items have failed or the test stops for other reasons. Progressive-stress loading is quite like the step stress testing with the difference that the stress level increases continuously Failure data obtained from ALT can be divided into two categories: complete (all failure data are available) or censored (some of failure data are missing). Due to different types of censoring, censored data can be divided into time-censored (or type I censored) data and failurecensored (or type II censored) data. Time censored (or type I censored) data is usually obtained when censoring time is fixed, and then the number of failures in that fixed time is a random variable. Failure censored (or type II censored) data is obtained when the test is terminated after a specified number of failures, and then time to obtain that fixed number of failures is a random variable. For more details about ALTs one can consult Bagdonavicius and Nikulin [i], Meeker and Escobar [ii], Nelson [iii, iv], Mann and Singpurwalla [v]. Constant stress ALT with different types of data and test planning has been studied by many authors. Watkins and John [vi] considers constant stress accelerated life tests based on Weibull distributions with constant shape and a log-linear link between scale and the stress factor which is terminated by a Type-II censoring regime at one of the stress levels. Ding et al. [vii] dealt with Weibull distribution to obtain accelerated life test sampling plans under type I progressive interval censoring with random removals. Pan et al. [viii] proposed a bivariate constant stress accelerated degradation test model by assuming that the copula parameter is a function of the stress level that can be described by a logistic function. Chen et al. [ix] discuss the optimal design of multiple stresses constant accelerated life test plan on non-rectangle test region. Fan and Yu [x] discuss the reliability analysis of the constant stress accelerated life tests when a parameter in the generalized gamma lifetime distribution is linear in the stress level. Islam and Ahmad [xi] and Ahmad and Islam [xii] discuss the optimal constant stress accelerated life test designs under periodic inspection and Type-I censoring. Geometric process (GP) is first used by Lam [xiii] in the study of repair replacement problem. Kamal [xiv] estimates Weibull parameters in accelerated life testing using geometric process with type-II censored data. This article is to focus on the maximum likelihood method for estimating the acceleration factor and the parameters of Marshall-Olkin Extended distribution. This work was conducted for CSALT with type II censored scheme. The confidence intervals for parameters are also obtained by using the asymptotic properties of normal distribution. In the last, the statistical properties of estimates and confidence intervals are examined through a simulation study. 2. Model Description 2.1 The geometric Process A GP is a stochastic process 1,2,...} = , { n X n such that 1,2,...} = , { 1 n X n n  forms a renewal process where 0 >  is real valued and called the ratio of the GP. It is easy to show that if 1,2,...} = , { n X n is a GP and the probability density function of 1 X is ) (x f with mean  and variance 2  then the probability density function of n X will be ) ( 1 1 x f n n     with mean 1 /    n and variance 1) 2( /    n . It is clear to see that a GP is stochastically increasing if 1 < < 0  and stochastically decreasing if 1 >  . Therefore, GP is a natural approach to analyze the data from a series of events with trend. 2.2 The Marshall-Olkin Extended Exponential distribution (MOEE) Marshall and Olkin [xv] proposed a new method for adding a parameter to a family of distributions. Suppose we have a given distributionwith survival function (SF)   < < ), ( x x F then the Marshall-Olkin extended 539 International Journal of Scientific Engineering and Technology (ISSN : 2277-1581) Volume No.3 Issue No.5, pp : 538-542 1 May 2014 IJSET@2014 Page 539 distribution is defined by the SF           1 = 0, > , < < ) ( 1 ) ( = ) ( x x F x F x G (1) If the survival function of exponential distribution is 0 x , e x F x > , = ) (    , and put it in equation (1), we obtain the SF             1 = 0, > , < < , 1 = ) ( x e x G x (2) The distribution with the survival function (2) is called the MOEE with parameters  and  . The probability density function (pdf) and the cumulative distribution function (cdf) and the hazard rate of the Marshall-Olkin extended exponential with the survival function (2), respectively are given by               1 = 0, > , < < , ] [ = ) , ; ( 2 x e e x g x x (3) ] [ 1 = ) , ; (        x x