Efficient novel nonparametric tests and estimations with applications to biomedical studies
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Likelihood is arguably the most important concept for statistical inference in parameter models. The Neyman-Pearson lemma states that the likelihood ratio test is the most powerful test when the likelihood functions are completely known composing the likelihood ratio. However, it may not be realistic to know the likelihood functions completely. In that case, we use nonparametric methods of statistical inferences without having to assume known distributions. Empirical Likelihood (EL) is a nonparametric method of statistical inferences. In this dissertation, several methods based on the empirical likelihood are introduced. First, a nonparametric method can be utilized for the purpose of developing a goodness-of-fit test. We derive the EL ratio test for an inverse Gaussian distribution using the density-based empirical likelihood approach. Second, we propose various two-sample median tests based on the empirical likelihood concept regardless of violation of exchangeability (constant shift) assumption. Even though the existing parametric and nonparametric methods are available to test the location shift between two groups, we may not employ those methods to test location differences when constant shift is violated. Third, we propose non-parametric approaches to estimate characteristics of interest for longitudinal data modeled by autoregressive process subject to the limit of detection (LOD). Left-censored data caused by the LOD are replaced with estimated conditional expectation calculated using Kaplan-Meier estimator of error distribution. Due to complexity of deriving limiting distributions of the estimators, we suggest an alternative to evaluate the proposed estimators providing their jackknife variance estimators. Based on the confidence intervals computed with the jackknife variance estimators, we perform hypothesis testing that can detect a difference between groups. All the three nonparametric methods introduced above are applied to actual data appropriately for the purpose of demonstrating their applicability.