Asymptotic expansions of the non-null distributions of the likelihood ratio criteria for multivariate linear hypothesis and independence

Sugiura & Fujikoshi (1969) — non-null expansions of the LR criterion

Result

the non-null distribution of the LR (determinant-ratio) criterion for the multivariate linear hypothesis, expanded to order $N^{-2}$ under a fixed alternative with no rank assumption on the noncentrality matrix; normal theory.

Use

fixed-alternative asymptotic normality of a determinant ratio and the second-order terms behind O7's coverage table; the fourth-moment version is Fujikoshi (2002), JSPI 108:263–282 (record only).

Verification

abstract read (primary text, OpenAlex), 5 Sep 2026.