## Abstract

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 H_{0} : Σ = Σ_{0}, where Σ_{0} is a given matrix, and obtain the null and non-null 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.

Original language | English (US) |
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Pages (from-to) | 603-620 |

Number of pages | 18 |

Journal | Journal of Multivariate Analysis |

Volume | 101 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2010 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty