For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.

As a typical non-normal case, we consider a family of elliptically symmetric distributions. Then, the moment parameter and its consistent estimator are presented. Also, the asymptotic expectation and the asymptotic variance of the consistent estimator of the general moment parameter are given. Besides, the numerical results obtained by Monte Carlo simulation for some selected parameters are provided.

Tn this paper we obtain asymptotic approximations for the expected probability of misclassification(RPMC) of a classification statistic including the quadratic discriminant function as well as the linear discriminant function when both the sample sizes and the dimension are large. We also make a comparison for the accuracy of our asymptotic approximations with other asymptotic expansions when the sample sizes are large. It is shown that the approximations proposed in this paper are considerably good for low-dimensional cases as well as high-dimensional cases. Further, among them, we propose to use an extension of Raudys [8] as a better approximation.

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