## Mathematics & Statistics Theses & Dissertations

Summer 2003

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

Dayanand N. Naik

N. Rao Chaganty

Larry Filer

Edward Markowski

#### Abstract

Most of the inferential statistical methods for multivariate data are developed under the fundamental assumption that the data are from a multivariate normal distribution. Unfortunately, one can never be sure a set of data is really from a multivariate normal distribution. There are numerous methods for checking (testing) multivariate normality, but based on many published and our own simulation studies, provided in the first chapter of this dissertation, we observe that these tests are generally not very powerful, especially for smaller sample sizes. Hence it is always beneficial to have alternative multivariate distributions available along with the methodology for using them.

In this dissertation, we focus on a probability distribution, called the Kotz type distribution, which has fatter tail regions than that of multivariate normal distribution and has its probability density function (pdf ) in the form [special characters omitted]where μ ∈ [special characters omitted], Σ is a positive definite matrix and c = [special characters omitted]. Using this distribution as the basis we have developed statistical methods for performing various statistical inferences for multivariate data. Our main contributions in this dissertation are the following: (i) Various characteristics of this distribution, such as, its moments, the marginal, and conditional distributions in specific forms, and a simulation algorithm for simulating samples from this distribution are provided. (ii) Estimation of the parameters of this distribution using the maximum likelihood method under different assumptions of one and more populations, and under different covariance is performed. An interesting and important observation is that the maximum likelihood estimators derived under this distribution are the generalized spatial median (GSM) estimators of C. R. Rao (1988). (iii) Using the asymptotic distribution of the estimates, multivariate analysis of variance (MANOVA) is performed and simultaneous confidence intervals for contrasts are constructed and illustrated on data sets. (iv) Finally, discrimination and classification rules under a Kotz type distribution are derived and compared with the rules based on a multivariate normal distribution using estimated expected error of misclassification. It is concluded that the expected error of misclassification can be reduced by using the methods developed here when the underlying distributions are not multivariate normal, but are of Kotz type distributions.

#### DOI

10.25777/6rcj-ch14

9780496554034

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