## UNLV Theses, Dissertations, Professional Papers, and Capstones

5-1-2015

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

Hokwon Cho

Malwane Ananda

Jessica Jaynes

Paulo Pinheiro

82

#### Abstract

The binomial distribution is one of the most commonly and widely occurring probabilistic phenomena in our lives. Since observations from independent Bernoulli trials yield a dichotomous type, the distribution of sequences provides the basis and clue for statistical formulations of a wide variety of problems.

Occasionally, the core of biomedical studies is related to the comparison and evaluation of the risks of events or outcomes of interest in comparing populations under study. For instance, one wishes to compare two groups of subjects drawn from two independent populations. Then, two sample proportions play central roles in those comparisons. One of the most useful ways to make comparisons for the relative risk is to take a ratio, also referred to as the risk ratio. In addition, a measure of reduction of the two proportions is considered.

In this thesis, we consider sequential methods of inferences for the ratio of two independent binomial probabilities, the risk ratio, in two populations for comparison. We obtain approximate confidence intervals and optimal sample sizes for the risk ratio and measure of reduction, respectively. Since there does not exist an unbiased estimator of the risk ratio, the procedure is developed based on a slightly modified maximum likelihood estimator. Then, we explore properties of the proposed estimator using the standard criteria, such as unbiasedness, asymptotic variance, and the normality. For further investigation, we study the first-order asymptotic expansions and large sample properties using the asymptotic results. Then, the finite sample behavior will be examined through numerical studies. Monte Carlo experiment is performed for the various scenarios of parameters of two populations.

Through illustrations, we compare the performance of the proposed methods, which is Wald-based confidence intervals, with the likelihood-ratio confidence intervals in light of length, sample sizes, and invariance. Then, we extend the proposed sequential procedure to two-stage sampling design, which has a pilot sampling stage and a stage of gathering all remaining observations if needed. The two-stage procedure is naturally a little more versatile and practical than pure sequential in terms of sample size and stopping time in many situations. Again, through numerical studies, we study the advantages and usefulness of the two- stage method as well.

Consequently, by providing more comprehensive study of dynamic sampling plans for studying the risk ratio, we hope to contribute various inferential methods to the risk ratio and related problems.

#### Disciplines

Statistics and Probability

pdf

English