Advancement of Inferences for GeneticTreatment Effects under Semiparametric Models

This dissertation presents three methodological advancements in semiparametric modeling and inference, with applications in longitudinal data analysis and individualized treatment rules (ITRs).

First, we enhance the semiparametric profile estimator for analyzing longitudinal data, addressing the challenge of within-subject correlation. By incorporating a nonparametric operator-regularized approach for estimating the covariance function, we develop a refined estimator that significantly improves efficiency over traditional local kernel smoothing methods, which assume an independent correlation structure. We further introduce an Empirical Likelihood (EL)-based inference method and demonstrate, through simulations and an application to the Genetic Analysis Workshop 18 dataset, that our approach attains the semiparametric efficiency bound and outperforms existing methods.

Second, we propose a rank-based inference procedure for ITRs under a semiparametric single-index varying coefficient model, where the nonparametric coefficient function is assumed to be monotone increasing. By leveraging maximum rank correlation, our method circumvents direct estimation of the nonparametric function, thereby mitigating potential biases. For hypothesis testing, we derive the asymptotic distribution of the proposed estimator using de-biasing techniques. Monte Carlo simulations and an application to the ACTG175 dataset confirm the effectiveness of our approach.

Finally, we develop a jackknife empirical likelihood ratio test to enhance hypothesis testing in the semiparametric single-index varying coefficient model. Existing methods often rely on plug-in variance-covariance estimators that approximate indicator functions using a sigmoid transformation, which are computationally complex and difficult to implement. Our proposed test offers a much simpler computational approach while achieving the same effectiveness. Extensive simulations and real-data analysis using the ACTG175 dataset further demonstrate the efficiency and practicality of our method.

Together, these contributions enhance the efficiency and reliability of semiparametric estimation and inference, particularly in the contexts of longitudinal data analysis and individualized treatment decision-making.