We 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 of over(Σ, ̂) and of the estimated regression matrix, over(Σ, ̂)12 over(Σ, ̂)22- 1. We represent over(Σ, ̂) 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 of over(Σ, ̂) and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing H0 : Σ = Σ0, where Σ0 is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing H0 : (μ, Σ) = (μ0, Σ0), where μ0 and Σ0 are given, we prove that the likelihood ratio criterion is unbiased and obtain its null and non-null distributions. For the sphericity test, H0 : Σ ∝ Ip + q, we obtain the null distribution of the likelihood ratio criterion. In testing H0 : Σ12 = 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.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty