Exponential Families and Conditioning on Statistics which are not Minimal Sufficient

Abstract

SUMMARY The likelihood function for each of k independent sets of data is assumed to belong to the two-parameter exponential family, the two parameters for the ith data set being si, which is a nuisance parameter, and 0, which is common to all the data sets and is the parameter of interest. The theory of similar tests suggests that the appropriate method for testing 0 is to condition on the statistic which is minimal sufficient for * = (b1, ..., a/4) when 0 is known. This minimal sufficient statistic will be different depending on whether or not the .s's are known to be equal. The effect of assuming that the as's are not equal when in fact they are (and vice versa) is investigated and some examples are given. The broad conclusion is that, provided the k samples are balanced in a certain sense, no great harm is done by conditioning on the incorrect statistic.