Improved transformed statistics for the test of independence in rxs contingency tables

We consider a class of statistics C[phi] based on [phi]-divergence for the test of independence in rxs contingency tables. The class of statistics C[phi] includes the statistics Ra based on the power divergence as a special case. Statistic R0 is the log likelihood ratio statistic and R1 is Pearson's X2 statistic. Statistic R2/3 corresponds to the statistic recommended by Cressie and Read [Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. B 46 (1984) 440-464] for the goodness-of-fit test. All members of statistics C[phi] have the same chi-square limiting distribution under the hypothesis of independence. In this paper, we show the derivation of an expression of approximation for the distribution of C[phi] under the hypothesis of independence. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the distribution of C[phi]. Furthermore, on the basis of the approximation, we obtain transformations that improve the speed of convergence to the chi-square limiting distribution of C[phi]. As a competitor of the transformed statistic, we derive a moment-corrected-type statistic. By numerical comparison in the case of Ra, we show that the transformed R1 statistic performs very well.