Asymptotic expansions of the distributions of some test statistics for elliptical populations.

*(English)*Zbl 0955.62016
Ghosh, Subir (ed.), Multivariate analysis, design of experiments, and survey sampling. A tribute to Jagdish N. Srivastava. New York, NY: Marcel Dekker. Stat., Textb. Monogr. 159, 433-467 (1999).

For samples of multivariate observations from a multivariate normal population the distributional properties of the sample covariance matrix and related quantities, such as its latent roots and latent vectors, have been examined at the past. The present paper aims at reviewing and describing similar results for the case when the normality assumption is relaxed and the data come from the more general family of elliptical populations.

Asymptotic expansions for the distribution of the sample mean and the sample covariance matrix are presented. However, since standard techniques used for the derivation of asymptotic distributions in the case of normal populations are not applicable, the author presents alternative asymptotic expansions for the case of elliptical populations. The asymptotic behavior of latent roots of a Wishart matrix and of likelihood ratio criteria and related statistics for several hypotheses concerning a covariance matrix, latent roots and latent vectors of it are considered. These test statistics allow for several hypothesis testing situations that are common in the multivariate analysis practice. These expansions verify preexisting results for the multivariate normal population case.

For the entire collection see [Zbl 0927.00053].

Asymptotic expansions for the distribution of the sample mean and the sample covariance matrix are presented. However, since standard techniques used for the derivation of asymptotic distributions in the case of normal populations are not applicable, the author presents alternative asymptotic expansions for the case of elliptical populations. The asymptotic behavior of latent roots of a Wishart matrix and of likelihood ratio criteria and related statistics for several hypotheses concerning a covariance matrix, latent roots and latent vectors of it are considered. These test statistics allow for several hypothesis testing situations that are common in the multivariate analysis practice. These expansions verify preexisting results for the multivariate normal population case.

For the entire collection see [Zbl 0927.00053].

Reviewer: Dimitris Karlis (Athens)