A GENERALIZATION OF TESTING INDEPENDENCE OF SETS OF VARIATES

抄録

Assuming that X=(X⁽¹⁾', X⁽²⁾', …, X^(q)')' is distributed according to N_p(μ, Σ) and each X^(j) has p_j components, where p=p₁+…+p_q and p₁≥p₂≥…≥p_q, we consider the testing of the hypothesis that for a=2, …, q, all the smallest p_a-k_a canonical correlations between (X⁽¹⁾', X⁽²⁾', …, X^(a-1)')' and X^(a) are zero. In the case of k₂=k₃=…=k_q=0, this hypothesis is equivalent to the hypothesis of independence of X⁽¹⁾, X⁽²⁾, …, X^(q). This testing problem can then be considered to be a generalization of the problem of testing independence. In this paper we drive the likelihood ratio statistic (LR statistic) and obtain the asymptotic expansion of its null distribution. We also obtain the asymptotic expansions of the null distributions of the other two statistics, which are the statistics corresponding to Lawley-Hotelling's trace and the Bartlett-Nanda-Pillai's trace criterion.