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The joint distribution of Studentized residuals under elliptical distributions

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  • Iwashita, Toshiya
  • Klar, Bernhard

Abstract

Scaled and Studentized statistics are encountered frequently, and they often play a decisive role in statistical inference and testing. For instance, taking the sample mean vector X̄=∑j=1NXj/N and the sample covariance matrix S=∑j=1N(Xj−X̄)(Xj−X̄)′/(N−1) for an iid sample {Xj}j=1N, some statistics for testing normality of the underlying distribution consist of the scaled residuals (the Studentized residuals or the transformed samples), Yj=S−1/2(Xj−X̄)(j=1,2,…,N). In this paper, the distribution of the random matrix the columns of which consist of the scaled residuals is derived under elliptical distributions. Also exact distributions of Studentized statistics are discussed as an application of the main result.

Suggested Citation

  • Iwashita, Toshiya & Klar, Bernhard, 2014. "The joint distribution of Studentized residuals under elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 203-209.
    • Handle: RePEc:eee:jmvana:v:128:y:2014:i:c:p:203-209
    DOI: 10.1016/j.jmva.2014.03.015
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    References listed on IDEAS

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    1. Zografos, K., 2008. "On Mardia's and Song's measures of kurtosis in elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 858-879, May.
    2. Iwashita, Toshiya & Kakizawa, Yoshihide & Inoue, Tatsuki & Seo, Takashi, 2009. "An asymptotic expansion of the distribution of Student's t type statistic under spherical distributions," Statistics & Probability Letters, Elsevier, vol. 79(18), pages 1935-1942, September.
    3. Batsidis, Apostolos & Zografos, Konstantinos, 2013. "A necessary test of fit of specific elliptical distributions based on an estimator of Song’s measure," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 91-105.
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    Cited by:

    1. Iwashita, Toshiya & Klar, Bernhard & Amagai, Moe & Hashiguchi, Hiroki, 2017. "A test procedure for uniformity on the Stiefel manifold based on projection," Statistics & Probability Letters, Elsevier, vol. 128(C), pages 89-96.

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