Wednesday, 8 January 2014


Estimation of Stress-Strength Model for Generalized Inverted Exponential Distribution using Ranked Set Sampling
M. A. Hussian
Department of Mathematical Statistics
Institute of Statistical Studies and Research (ISSR), Cairo University, Cairo, Egypt

In this paper, the estimation of R=P(Y < X), when X and Y are two generalized inverted exponential random variables with different parameters is considered. This problem arises naturally in the area of reliability for a system with strength X and stress Y. The estimation is made using simple random sampling (SRS) and ranked set sampling (RSS) approaches. The maximum likelihood estimator (MLE) of R is derived using both approaches. Assuming that the common scale parameter is known, MLEs of R are obtained. Monte Carlo simulations are performed to compare the estimators obtained using both approaches. . The properties of these estimators are investigated and compared with known estimators based on simple random sample (SRS) data. The comparison is based on biases, mean squared errors (MSEs) and the efficiency of the estimators of R based on RSS with respect to those based on SRS. The estimators based on RSS is found to dominate those based on SRS.
Keywords: generalized exponential distribution; reliability; stress-strength; ranked set sampling, simple random sampling; maximum likelihood estimators.
      I.            Introduction
The estimation of reliability is a very common problem in statistical literature. The most widely approach applied for reliability estimation is the well-known stress-strength model. This model is used in many applications of physics and engineering such as strength failure and the system collapse. In the stress-strength modeling, R=P(Y < X) is a measure of component reliability when it is subjected to random stress Y and has strength X. In this context, R can be considered as a measure of system performance and it is naturally arise in electrical and electronic systems. Another interpretation can be that, the reliability of the system is the probability that the system is strong enough to overcome the stress imposed on it. It may be mentioned that R is of greater interest than just reliability since it provides a general measure of the difference between two populations and has applications in many areas. For example, if X is the response for a control group, and Y refers to a treatment group, R is a measure of the effect of the treatment. In addition, it may be mentioned that R equals the area under the receiver operating characteristic (ROC) curve for diagnostic test or biomarkers with continuous outcome, see; Bamber, [1]. The ROC curve is widely used, in biological, medical and health service research, to evaluate the ability of diagnostic tests or biomarkers and to distinguish between two groups of subjects, usually non-diseased and diseased subjects. For complete review and more applications of R; see [2-14].
Ranked set sampling (RSS) is a sampling protocol that can often be used to improve the cost and efficiency for experiments [15]. It is often used when a ranking of the sampling units can be obtained cheaply without having to actually measure the characteristics of interest, which may be time consuming or costly [16,17]. Such a technique is well received and widely applicable in environmental applications, reliability and quality control experiments [18-20]. A modification of ranked set sampling (RSS) called moving extremes ranked set sampling (MERSS) was considered for the estimation of the scale parameter of scale distributions [21] and an improved RSS estimator for the population mean was obtained [22]. On the other hand, Ozturk has developed two sampling designs to create artificially stratified samples using RSS [23]
Recently, many authors have been interested in estimating R using RSS. For example, Sengupta and Mukhuti [24], considered an unbiased estimation of R using RSS for exponential populations. Muttlak and co-authors [25], proposed three estimators of R using RSS when X and Y independent one-parameter exponential populations. In a RSS procedure, m independent sets of SRS each of size m are drawn from the distribution under consideration. these samples are ranked by some auxiliary criterion that does not require actual measurements and only the ith smallest observation is quantified from the ith set, i = 1,2,…,m. This completes a cycle of the sampling. Then, the cycle is repeated k times to obtain a ranked set sample of size n = m k .

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