Optimum Step Stress Accelerated Life Testing For Rayleigh Distribution

Accelerated testing is needed when testing even large sample sizes at use stress would yield few or no failures within a reasonable time. The step-stress accelerated life test is used to increase the stress levels at fixed times during the experiment. This paper deals with the problem of designing an optimum step stress accelerated life test for Rayleigh distribution. The scale parameter of the distribution is assumed to be a log linear function of stress. The maximum likelihood estimates of the parameters under consideration are obtained. Interval estimation that generates narrow intervals to the unknown parameters of the distribution with high probability is obtained. Optimization criterion is also discussed and simulation results are obtained to explain the techniques used in the paper.


Introduction
With today's high technology, some life tests result in none or very few failures, by the end of the test. In such cases, an approach is to do life test at higher-than-usual stress conditions, in order to obtain failures quickly. This can be achieved by using accelerated life test (ALT). ALT is achieved by subjecting units and components to test conditions such that failure occurs sooner. Thus, prediction of the long-term reliability can be made within a short period of time. Results from the A LT are used to extrapolate the unit characteristics at any future time and given at normal operating conditions. ALT can be applied only when a model relating life length to stress is known. One way to accelerate failure is step-stress which increases the stress applied to test product in a specified discrete sequence. Generally, as indicated by Xiong and Ji [1], a test unit starts at a certain low stress. If the unit does not fail at a particular time, stress on it is raised and held at a specified time. Stress is repeatedly increased and held, until the test unit fails or a censoring point is reached.
Miller and Nelson [2] described an optimu m simp le stepstress plan for A LT when failure t imes are exponentially d is trib u ted . A rrh en iu s in v ers e-p o wer law p lay ed a prominent role in modeling in the paper by Schatzoff and Lane [3]. Bai, Kim, and Lee [4] extend ed the resu lts of Miller and Nelson [2] to the case of exponential censoring. Nelson [5] has provided a very good survey on the SSALT. Zarrin et al. [6] obtained a 2 step step-stress test plan for exponentiated Weibull distribution. Tang et al. [7] have used a linear cu mu lative exposure model to analy ze data fro m a SSA LT using 3-parameter Weibull distribution. In the paper by Khamis and Higgins [8] an optimu m 3 step step-stress test plan has been proposed when the failure time distribution is exponential and the life stress relationship is linear or quadratic. Bhattacharyya and Soejoeti [9] developed a tampered failure-rate model. Bhattacharyya [10] also derived an approach using a Gaussian stochastic process which was later modified and extended by Doksum and Hoyland [11] and Lu and Storer [12] The Ray leigh d istribution has played an impo rtant role in modeling the lifetime o f random phenomena. It arises in many areas of applications, including reliability, life testing and survival analysis. Ray leigh d istribution is a special case of Weibull distribution (shape parameter=2) and is frequently used to model wave heights in oceanography, and in co mmunication theory to describe hourly median and instantaneous peak power of received radio signals. It has been used to model the frequency of different wind speeds over a year and a wind turbine sites. The distance from one individual to its nearest neighbor when the spatial pattern is generated by Poisson distribution follows a Ray leigh distribution. In co mmun ication theory, Ray leigh distributio n is used to model scattered signals that reach a receiver by mu ltip le paths. Depending on the density of scatter, the signal will display different fading characteristics. Rayleigh distribution is used to model dense scatter.
Other than this introductory section, this paper includes six more sections. Section 2 deals with the model and assumptions. In section 3 likelihood estimates of parameters and fisher information matrix is evaluated. Confidence intervals for the model parameters based on the asymptotic normality of the M LE are obtained in section 4. Optimizatio n criterion is discussed in section 5. For illustration of the article a simu lation study is given in section 6, follo wed by some concluding remarks in section 7.

The Model
The life time of a product at any level of stress is assumed to fo llo w Rayleigh distribution. The scale parameter of the distribution is assumed to be a log linear function of stress. The probability density function (p.d.f.) of Ray leigh distribution is given by is the scale parameter. The corresponding survival function is The hazard function of , denoted as The log likelihood function is (2) By using the relation the log likelihood function (2) beco mes (3) where Differentiating eq. (3) partially with respect to and we get (4) Where MLEs and for the model parameters and are the value which maximize the likelihood function (1). (4) and (5) are very co mplex non-linear equations to be solved. Therefore, an iterative procedure is required to solve these equations numerically. Newton Raphson method is used to obtain the M LEs of and . Once the value of is obtained, can easily be calcu lated by using (5).
But the exact mathematical exp ression for the expectation is too difficult to find. So it can be appro ximated by numerically inverting the asymptotic fisher information mat rix. It is composed of the negative second and mixed derivatives of the normal logarith m of the likelihood function evaluated at the MLE. So, asymptotic fisher information matrix can be written as where,

Confidence Intervals for Model Parameters
The most common method to set confidence bounds for the parameters is to use asymptotic normal distribution of maximu m likelihood estimators, see Vander Wiel and Meeker [13].
To construct a confidence interval fo r a population parameter ; assume that and are functions of the sample data such that where the interval is called a two sided confidence interval for . and are the lower and upper confidence limits for , respectively. The random limits and enclose with probability . Asymptotically, the maximu m likelihood estimators, under appropriate regularity conditions are consistent and normally distributed. Therefore, the two sided appro ximate confidence limits for a population parameter can be constructed such that: where is the standard normal percentile. Therefore, the two sided appro ximate confidence limits for and are given respectively as follows:

Optimization Criterion
The optimal test plan is to determine the duration of the lower stress level. According to Al-Haj Ebrahem and Al-Masri [14], an optimu m test plan can be determined with respect to the change time, the asymptotic variance (A V) of the MLE o f a given percentile at the design stress . The log of the percentile of the lifetime the design stress is given by Generally, the optimization criterion is defined to minimize the asymptotic variance of the percentile estimate at the design stress. The MLE is used for the percentile estimate. Then, the A V of the percentile estimate at the design stress can be obtained as follows: where F is the Fisher information matrix g iven in Section (3), and Then

Simulation Study
In optimu m test plan the duration of the lower stress level is obtained by min imizing the A V o f the M LE. But it is very difficult to study the complex derived equations. So a simu lat ion study is performed using Newton-Raphson meth od to explain the results obtained in the study.    5. The resulting estimates of parameters are used to obtain the variance covariance matrix.
The results obtained in the above simulat ion study are summarized in Table 1 Table 1 shows different measures for stress combination and . Similarly table 2 and 3 show the various results for stress combination , and , respectively. Fro m these tables the following observations can be made on the performance of SSALT parameter estimation of Rayleigh distribution 1. For the first set of stresses ( , ) the maximu m likelihood estimators have good statistical properties than the second ( , ) and third ( , ) set for all sample sizes. 2. As the value of stresses increase the estimates have smaller MSE and RE. As the sample size increases the RABs and MSEs of the estimated parameters decrease on an average. This indicates that the maximu m likelihood estimates provide asymptotically normally d istributed and consistent estimator for the parameters.
3. The asymptotic variances of the estimators are decreasing when the sample size increasing.

Discussion and conclusions
ALTs are used to estimate the lifetime of highly reliable products within a reasonable testing time. The test units are run at higher than usual levels of stress to induce early failures. The test data obtained at the accelerated conditions are analyzed in terms of a model and then extrapolated to design stress to estimate the life time distribution. This study considered the inference on the parameters and optimally designing simple step stress plan for the Ray leigh distribution. The MLEs of the model parameters were obtained. Performance of step stress testing plans and model assumptions are usually evaluated by the properties of the MLEs of model parameters.
In this study the parameters have good statistical properties for the first set of stresses than the other two sets for all samp le sizes. Maximu m likelihood estimators are consistent and asymptotically normally d istributed. As the sample sizes increase the asymptotic variance and covariance of estimators decrease.
Here, some final discussions about the results in the study are given. First, a statistical methodology is presented to analyze the data obtained from a step stress accelerated life test. This methodology will be especially useful when intermittent inspection is the only feasible way of checking the status of test units during a step stress test. Second, although the cumulative exposure model has been a popular choice for the analysis of step stress life test data, its validity in various situations remain to be tested. Third, for a simp le step stress test, we have derived the optimu m choice for the stress change time wh ich minimizes AV under a Rayleigh distribution. This optimu m design is particularly important at the designing stage of the test as it gives a guide to experimenters about when the intermittent inspection and the stress change should be carried out during the test. The results on the optimu m design are based on the assumption of Rayleigh distribution but its sensitivity is not addressed in this study. These questions provide the direction of future research in this area associated with the data obtained from SSA LT is to study the statistical analysis and the optimu m design when the stress change time is random variab le: such as the order statistics at the current stress levels and when only these order statistics are observed during testing.