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
ABSTRACT
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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