A study of large sample test criteria through properties of efficient estimates. I. Tests for goodness of fit and contingency tables

Abstract

The existence of m.l. and m.l. equation estimates and their efficiency, in the case of sampling from a finite multinomial distribution, have been established under conditions weaker than those assumed by earlier writers. For instance, the existence of the second derivatives of the cell probabilities as functions of the parameters is not necessary. The asymptotic distributions of the chi-square goodness of fit test and test criteria for examining composite hypotheses in contingency tables have been shown to depend mainly on the efficiency (in the new sense) of estimates used in the construction of the test criteria. The m.l. estimate satisfies the efficiency condition and is, in the sense of second order efficiency, more suitable for the construction of test criteria than other efficient estimates. A new test has been proposed for examining the expected frequencies computed under two different hypotheses of a special nature.