Finite-sample inference with monotone incomplete multivariate normal data, II

Abstract

a b s t r a c tWe continue our recent work on inference with two-step, monotone incomplete data from a multivariate normal population with mean µ and covariance matrix Σ. Under the assumption that Σ is block-diagonal when partitioned according to the two-step pattern, we derive the distributions of the diagonal blocks ofΣ and of the estimated regression matrix,Σ 12Σ −1 22 . We representΣ in terms of independent matrices; derive its exact distribution, thereby generalizing the Wishart distribution to the setting of monotone incomplete data; and obtain saddlepoint approximations for the distributions ofΣ and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing H 0 : Σ = Σ 0 , where Σ 0 is a given matrix, and obtain the null and nonnull distributions of the test statistic. In testing H 0 : (µ, Σ) = (µ 0 , Σ 0 ), where µ 0 and Σ 0 are given, we prove that the likelihood ratio criterion is unbiased and obtain its null and non-null distributions. For the sphericity test, H 0 : Σ ∝ I p+q , we obtain the null distribution of the likelihood ratio criterion. In testing H 0 : Σ 12 = 0 we show that a modified locally most powerful invariant statistic has the same distribution as a Bartlett-Pillai-Nanda trace statistic in multivariate analysis of variance.

Chi-square mixture representations for the distribution of the scalar Schur complement in a noncentral Wishart matrix

“…6]. In particular, the Schur complement is a crucial quantity in multivariate statistical inference with monotone incomplete data [3], [4].…”

Finite-sample inference with monotone incomplete multivariate normal data, III: Hotelling’s T2-statistic

“…In the sequel, we will need to decompose the matrix A 11,N in (2.3) as follows (Chang and Richards, 2010):…”