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

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

Abstract

We continue our recent work on inference with two-step, monotone incomplete data from a multivariate normal population with mean and covariance matrix . Under the assumption that is block-diagonal when partitioned according to the two-step pattern, we derive the distributions of the diagonal blocks of and of the estimated regression matrix, . We represent 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 and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing , where is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing , where and are given, we prove that the likelihood ratio criterion is unbiased and obtain its null and non-null distributions. For the sphericity test, , we obtain the null distribution of the likelihood ratio criterion. In testing 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.

Suggested Citation

  • Chang, Wan-Ying & Richards, Donald St. P., 2010. "Finite-sample inference with monotone incomplete multivariate normal data, II," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 603-620, March.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:3:p:603-620
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    References listed on IDEAS

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    1. R. Bhargava, 1975. "Some one-sample hypothesis testing problems when there is a monotone sample from a multivariate normal population," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 27(1), pages 327-339, December.
    2. Hao, Jian & Krishnamoorthy, K., 2001. "Inferences on a Normal Covariance Matrix and Generalized Variance with Monotone Missing Data," Journal of Multivariate Analysis, Elsevier, vol. 78(1), pages 62-82, July.
    3. Chang, Wan-Ying & Richards, Donald St.P., 2009. "Finite-sample inference with monotone incomplete multivariate normal data, I," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1883-1899, October.
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    Cited by:

    1. Tomoya Yamada & Megan Romer & Donald Richards, 2015. "Kurtosis tests for multivariate normality with monotone incomplete data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 532-557, September.
    2. Krishnamoorthy, K., 2013. "Comparison of confidence intervals for correlation coefficients based on incomplete monotone samples and those based on listwise deletion," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 378-388.
    3. Tsukada, Shin-ichi, 2014. "Equivalence testing of mean vector and covariance matrix for multi-populations under a two-step monotone incomplete sample," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 183-196.
    4. Richards, Donald St. P. & Yamada, Tomoya, 2010. "The Stein phenomenon for monotone incomplete multivariate normal data," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 657-678, March.
    5. Shutoh, Nobumichi & Hyodo, Masashi & Seo, Takashi, 2011. "An asymptotic approximation for EPMC in linear discriminant analysis based on two-step monotone missing samples," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 252-263, February.
    6. Romer, Megan M. & Richards, Donald St. P., 2010. "Maximum likelihood estimation of the mean of a multivariate normal population with monotone incomplete data," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1284-1288, September.
    7. Yamada, Tomoya, 2013. "Asymptotic properties of canonical correlation analysis for one group with additional observations," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 389-401.
    8. Tsukada, Shin-ichi, 2014. "Asymptotic expansion for distribution of the trace of a covariance matrix under a two-step monotone incomplete sample," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 206-219.
    9. Keiji Takai & Kenichi Hayashi, 2023. "Model Selection with Missing Data Embedded in Missing-at-Random Data," Stats, MDPI, vol. 6(2), pages 1-11, April.

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