#### Publication Date

Spring 2018

#### Advisor(s) - Committee Chair

Dr. Ngoc Nguyen (Director), Dr. Dominic Lanphier, Dr. John Spraker

#### Degree Program

Department of Mathematics

#### Degree Type

Master of Science

#### Abstract

The main purpose of this thesis is to compare the performance of the score test and the likelihood ratio test by computing type I errors and type II errors when the tests are applied to the geometric distribution and inflated binomial distribution. We first derive test statistics of the score test and the likelihood ratio test for both distributions. We then use the software package R to perform a simulation to study the behavior of the two tests. We derive the R codes to calculate the two types of error for each distribution. We create lots of samples to approximate the likelihood of type I error and type II error by changing the values of parameters.

In the first chapter, we discuss the motivation behind the work presented in this thesis. Also, we introduce the definitions used throughout the paper. In the second chapter, we derive test statistics for the likelihood ratio test and the score test for the geometric distribution. For the score test, we consider the score test using both the observed information matrix and the expected information matrix, and obtain the score test statistic *zO *and *zI *.

Chapter 3 discusses the likelihood ratio test and the score test for the inflated binomial distribution. The main parameter of interest is *w*, so *p *is a nuisance parameter in this case. We derive the likelihood ratio test statistics and the score test statistics to test *w*. In both tests, the nuisance parameter *p *is estimated using maximum likelihood estimator *p*ˆ. We also consider the score test using both the observed and the expected information matrices.

Chapter 4 focuses on the score test in the inflated binomial distribution. We generate data to follow the zero inflated binomial distribution by using the package R. We plot the graph of the ratio of the two score test statistics for the sample data, *zI /zO *, in terms of different values of *n*0, the number of zero values in the sample.

In chapter 5, we discuss and compare the use of the score test using two types of information matrices. We perform a simulation study to estimate the two types of errors when applying the test to the geometric distribution and the inflated binomial distribution. We plot the percentage of the two errors by fixing different parameters, such as the probability *p *and the number of trials *m*.

Finally, we conclude by briefly summarizing the results in chapter 6.

#### Disciplines

Applied Statistics | Other Applied Mathematics | Probability

#### Recommended Citation

Dai, Xiaogang, "Score Test and Likelihood Ratio Test for Zero-Inflated Binomial Distribution and Geometric Distribution" (2018). *Masters Theses & Specialist Projects.* Paper 2447.

https://digitalcommons.wku.edu/theses/2447