Convergence rates and powers of six power-divergence statistics for testing independence in 2 by 2 contingency table

Abstract

This thesis investigates the convergence rates to the limiting null distribution and the powers of six test statistics of the power-divergence family (Cressie and Read, 1984) for testing independence in the 2 by 2 Contingency Table. This family of statistics can be expressed by -- [special characters omitted] -- which is indexed by the parameter ⋋, x₁.j are observed cell frequencies and m₁.j are expected cell frequencies. It can easily be seen that the Pearson's X² (⋋ = 1), the log likelihood ratio statistic G² (⋋ = 0), the Freeman-Tukey statistic T² (⋋ = -1/2), the modified log likelihood ratio statistic MG² (⋋ = -1), the Neyman modified chi-square statistic MX² (⋋ = —2) and the Cressie-Read statistic (⋋ = 2/3), are all special cases. -- For calculating the convergence rates and the powers of these six statistics, an iterative procedure for obtaining the minimum power-divergence estimates for the unkown parameters will be presented. It is found that among these six statistics, the convergence rate of Pearson's X² (⋋ = 1) to the limiting null distribution is the best. For the power of the test, for different alternatives, each of X², G² and MX² is the most powerful. It is also found that the power of the test depends not only on the noncentrality parameter but on the location of alternative hypothesis. The working rules for deciding which statistic is to be used will also be presented for the practitioner. -- Key Words: Convergence Rate; Power; Power-Divergent Family; Independence Model; Minimun-Distance Estimator; Asymptotic Distribution; Non-Central Chi-Squared Distribution.