A new approach to distribution free tests in contingency tables

We suggest an extremely wide class of asymptotically distribution free goodness of fit tests for testing independence in two-way contingency tables, or equivalently, independence of two discrete random variables. The nature of these tests is that the test statistics can be viewed as definite functions of the transformation of (widehat{T}_n = (widehat{T}_{ij})=Big (frac{nu _{ij}- nhat{a}_ihat{b}_j}{sqrt{nhat{a}_ihat{b}_j}}Big )) where (nu _{ij}) are frequencies and (hat{a}_i, hat{b}_j) are estimated marginal distributions. Our method is also applicable for testing independence of two discrete random vectors. We make some comparisons on statistical powers of the new tests with the conventional chi-square test and suggest some cases in which this class is significantly more powerful.