Inference for Some Incompletely Specified Models Involving Normal Approximations to Discrete Data

Abstract

This paper considers the problem of pooling means of two independent random samples from discrete distributions (in particular Poisson and binomial) which can be approximated by normal distributions after the appropriate transformations. We first develop the theory for two samples from N(iL , U2), i = 1, 2, o-2 assumed known, and the parameter of interest being ji, . Using a preliminary test of significance (PTS) at level a to test ju = A2, a new estimator x is proposed, both to estimate g I, and to test the hypothesis ,ui = ,io. The bias and mean squared error of x are studied and the regions in the parameter space in which x has smaller mean squared error than x, the mean of the first sample, are determined. Similarly, the size and power of the overall hypothesis testing procedure are studied. It is recommended that in order to control the mean squared error and the size of the overall test procedure based on x*, the level of significance of PTS should be about .25. These results are then applied to corresponding problems for the data from Poisson and binomial distributions.