Unbiased Estimation of P(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling

Abstract

This paper addresses the problem of unbiased estimation of P[X > Y] = θ for two independent exponentially distributed random variables X and Y. We present (unique) unbiased estimator of θ based on a single pair of order statistics obtained from two independent random samples from the two populations. We also indicate how this estimator can be utilized to obtain unbiased estimators of θ when only a few selected order statistics are available from the two random samples as well as when the samples are selected by an alternative procedure known as ranked set sampling. It is proved that for ranked set samples of size two, the proposed estimator is uniformly better than the conventional non-parametric unbiased estimator and further, a modified ranked set sampling procedure provides an unbiased estimator even better than the proposed estimator.

Keywords:

• Exponential population
• Order statistic
• Ranked set sampling
• Simple random sampling
• Unbiased estimation

Mathematics Subject Classification:

• Primary 62D05
• Secondary 62F10

Acknowledgment

Research of the second author is supported by Council of Scientific and Industrial Research, India (Sanction no. 9/28(566)/2002/EMR-I). We also thank the referee for his helpful comments.