Estimating the True Probability of Correct Selection: Two Exponential Distributions

Abstract

SYNOPTIC ABSTRACTIndependent random samples of same size are drawn from two exponential populations, having unknown scale parameters. For the goal of selecting the population associated with the larger scale parameter, we consider the natural decision rule, which selects the population corresponding to the larger sample mean, and investigate the problem of estimating the true probability of the correct selection (PCS) using the methods of unbiased estimation and maximum likelihood estimation. The non-existence of the unbiased estimators for the true PCS is established and two approaches to find the maximum likelihood estimators (MLEs) of many-to-one parametric functions are discussed. Using these two approaches, the MLEs for the true PCS are derived and performances of the MLEs are compared, using the Monte Carlo simulations, in terms of the absolute error, the mean squared error and the bias.