Abstract
Inference on R = P(X < Y ) has been considered when X and Y belong to independent exponentiated family of distributions. Maximum Likelihood Estimator (MLE), Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes Estimator of R has been derived and compared through simulation study. Exact and approximate confidence intervals and Bayesian credible intervals have also been derived.
Highlights
R = P( X < Y ) is used in various applications e.g. stress-strength reliability, statistical tolerancing, measuring demand-supply system performance, measuring heritability of a genetic trait, bio-equivalence study etc
We report the estimates of R, R1, R2 and R4 using the Maximum Likelihood Estimator (MLE), UMVUE and empirical Bayes procedure assuming conjugate priors, and the average biases and mean squared errors (MSEs) of R in tables 1-4 over 1000 replications
We have discussed inference problem of R = P( X < Y ) for exponentiated family of distributions. This family is obtained by adding a parameter to the exponent of a distribution function to make resulting distribution richer and more flexible for modeling data
Summary
V) A certain unit - be it a receptor in a human eye or ear or any other organ operates only if it is stimulated by the source of random magnitude Y and the stimulus exceeds a lower threshold X specific for that unit In this case, R is the probability that the unit functions. Birnbaum used Mann-Whitney statistic to estimate R and found the confidence interval of R in nonparametric set up following the Hodges-Lehmann approach. We look into the problem in more general set up under any known baseline distribution not necessarily restricted to the location-scale family. An outline is given for inference about R in general case i.e. when the baseline distributions for X and Y are different through parameter.
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