Statistics is a powerful tool in different scientific fields by providing statistical supports in experimental designs, data processing and statistical inference. In this thesis, we conduct theoretical and methodological statistical research with applications in biological and genomic areas.

In Chapter 2, we study the statistical testing problems with order-restricted null hypothesis, where the null parameter space is a union of two disjoint convex cones. We derive the likelihood ratio test and the intersection-union test, and show that the likelihood ratio test is uniformly more powerful than the intersection-union test. We also demonstrate the situation in which the uniformly more powerful tests are constructed, and discuss the applicability of the uniformly more powerful tests to real data analyses.

In Chapter 3, we propose four testing procedures for detecting the monotonic changes in multivariate gene expression distributions. We consider cases in which the treatment factor is ordinal and can be naturally ordered. The proposed procedures focus the detection powers to genes with monotonic departures from mean equality. Also, the proposed methods are able to deal with small sample sizes and high-dimensional distributions.

In Chapter 4, we propose a new methodology, based on a Hidden Markov Model with a mixture emission distribution, to detect copy number variations between different genomics using next generation sequencing read counts. This method demonstrates an improvement comparing to existing methods. We use this method to identify copy number variations between two maize genotypes, and the result is concordant to previous genomic studies using microarray data.

This thesis concludes in Chapter 5, which provides a discussion of future research directions.