#### Date of Award

1984

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics and Statistics

#### Keywords

Mathematics.

#### Rights

CC BY-NC-ND 4.0

#### Abstract

In classical tests of hypotheses, assumptions concerning normality and homogeneity of variances are needed. Often practical data do not meet some of these assumptions and the idea of robustness is advanced. To avoid making assumptions underlying a tests, Mielke, Berry and Johnson introduced the MRPP (Multi-response Permutation Procedures) test. The test statistic (delta) is simply a weighted average of some distance measure between pairs of observations within a group. It tests H(,0): Classification of data into g groups is random against H(,1): Classification is done according to some a priori scheme. Special cases of (delta) are equivalent to some well-known test statistics. When the distance measure is the Euclidean distance between ranks of observations and the weights are proportional to the size of groups, in the 2-sample case, the MRPP statistic, called (delta)(,1), performs better than the Wilcoxon test for some underlying distributions. The null distribution of (delta) is often highly negatively skewed, and is, in general, asymptotically non-normal. To account for the skewness, Mielke and others have recommended the use of the Pearson Type III approximation, determined by the first three moments of (delta). We define 23 symmetric functions in order to obtain the fourth moment of (delta). In the case of two equal samples, an explicit result for the fourth moment of (delta)(,1) is obtained. We obtain empirical powers of (delta)(,1), considering 10,000 samples for both small and large samples, using a Pearson type approximation based on four moments, as well as the Type III approximation. These powers are compared with those of the Wilcoxon test against various shifts in location for several underlying distributions, viz., uniform, normal, logistic, 10% 3N, 10% 10N, Laplace, U-shaped, Cauchy, and exponential. We conclude the dissertation by discussing the scope for further work with the use of the fourth moment.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1984 .T358. Source: Dissertation Abstracts International, Volume: 46-02, Section: B, page: 0549. Thesis (Ph.D.)--University of Windsor (Canada), 1984.

#### Recommended Citation

TAJUDDIN, ISLAMUDDIN H., "FOURTH MOMENT AND SIMULATED POWERS OF MRPP STATISTICS." (1984). *Electronic Theses and Dissertations*. 2151.

http://scholar.uwindsor.ca/etd/2151