Advanced and novel parametric and nonparametric likelihood statistical techniques with applications in epidemiology

The likelihood approach provides a basis for many important procedures and methods in statistical inference. When data distributions are completely known, the parametric likelihood approach is unarguably a powerful statistical tool that can provide optimal statistical inference. In such cases, by virtue of the Neyman-Pearson lemma, the likelihood ratio tests are the most powerful decision rules. The parametric likelihood methods cannot be applied properly if assumptions on the forms of distributions of data do not hold. Often, in the context of likelihood applications, the use of the misspecified parametric forms of data distributions may result in inaccurate statistical conclusions. The empirical likelihood (EL) methodology has been well addressed in the literature as a nonparametric counterpart of its powerful parametric likelihood approach. The objective of this dissertation is to develop several powerful parametric likelihood methods and nonparametric approaches using the EL concept. Measurement error (ME) problems can cause bias or inconsistency of statistical inferences. When investigators are unable to obtain correct measurements of biological assays, special techniques to quantify MEs need to be applied. In this dissertation, we present both parametric likelihood and EL methods for dealing with data subject to MEs based on repeated measures sampling strategies and hybrid sampling designs (a mixture of pooled and unpooled data). Utilizing the density-based EL methodology, we also propose different efficient nonparametric tests that approximate most powerful Neyman-Pearson test statistics. We first introduce the EL ratio based goodness-of-fit test for the inverse Gaussian model. Then we extend and adapt the density-based EL approach to compare two samples based on paired data. We present exact nonparametric tests for composite hypotheses to detect various differences related to treatment effects in study groups based on paired measurements. Next, we review and extend parametric retrospective and sequential Shiryaev-Roberts based policies, carrying out different contexts of the non-asymptotic optimal properties of the procedures. We propose techniques to construct novel and efficient retrospective tests for multiple change-points detection problems. Finally, future works will be discussed.