## Mathematics Ph.D. Dissertations

#### Title

Goodness-of-Fit Tests For Dirichlet Distributions With Applications

#### Date of Award

2015

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Statistics

#### First Advisor

Maria Rizzo (Advisor)

#### Second Advisor

Pamela Bechtel (Committee Member)

#### Third Advisor

James Albert (Committee Member)

#### Fourth Advisor

Wei Ning (Committee Member)

#### Abstract

We present five new goodness-of-fit tests for Dirichlet distributions. Since beta, Dirichlet, and generalized Dirichlet distributions are widely used in statistics such as Dirichlet regression, data mining, and machine learning. Thus, goodness-of-fit tests for beta and Dirichlet distribution are important applications. We develop new test procedures for testing beta and Dirichlet distribution. The first test, called energy test, is based on energy statistics. The second test, called distance covariance test, is based on the property of complete neutrality of the Dirichlet distribution and distance covariance tests. We also expand the distance covariance test into testing the mutual in- dependence of all components of arbitrary random vector. The third, fourth and fifth tests, called the triangle tests, are based on the interpoint distances. Simulation studies of power performance for the new beta and Dirichlet goodness-of-fit tests are presented. Results show that distance co- variance goodness-of-fit test have good performance in contaminated Dirichlet distributions which contain the small perturbation in the Dirichlet distribution.

The parameter estimation based on the maximum likelihood is derived for the generalized Dirichlet distribution and the initial value for the iteration of the Newton-Raphson algorithm is also proposed. Applications includes the goodness-of-fit tests of Dirichlet and generalized Dirichlet distributions, model evaluation of Dirichlet regression models, and influence diagnostics of Dirichlet regression models.

#### Recommended Citation

Li, Yi, "Goodness-of-Fit Tests For Dirichlet Distributions With Applications" (2015). *Mathematics Ph.D. Dissertations*. 26.

https://scholarworks.bgsu.edu/math_diss/26