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

  • Authors:

  • Wan-Ying Chang;Donald St. P. Richards

  • Affiliations:

  • Washington Department of Fish and Wildlife, Olympia, WA 98501, USA;Department of Statistics, Penn State University, University Park, PA 16802, USA and The Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, NC 27709, USA

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2010

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Abstract

We continue our recent work on inference with two-step, monotone incomplete data from a multivariate normal population with mean @m and covariance matrix @S. Under the assumption that @S is block-diagonal when partitioned according to the two-step pattern, we derive the distributions of the diagonal blocks of @S@? and of the estimated regression matrix, @S@?"1"2@S@?"2"2^-^1. We represent @S@? 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 @S@? and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing H"0:@S=@S"0, where @S"0 is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing H"0:(@m,@S)=(@m"0,@S"0), where @m"0 and @S"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:@S@KI"p"+"q, we obtain the null distribution of the likelihood ratio criterion. In testing H"0:@S"1"2=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.