Constructing confidence intervals in high-dimensional models and dealing with pleiotropy in mendelian randomization

Constructing confidence intervals in high-dimensional models is a challenging task due to the lack of knowledge on the distribution of many regularized estimators. The debiased Lasso approach (Zhang and Zhang, 2014) has been proposed for constructing confidence intervals of low-dimensional parameters in high-dimensional linear models. This thesis generalizes the idea of “debiasing” to make inference in high-dimensional Cox models with time-dependent covariates. A quadratic optimization algorithm is proposed for computing the debiased Lasso estimator and its benefits are demonstrated. This thesis also studies the sample size conditions for inference in high-dimensional linear models with bootstrapped debiased Lasso. It is proved that bootstrap can further correct the bias of debiased Lasso and new sample size conditions involving the number of weak signals are obtained. In many economical and biological applications, estimating the causal effect of an exposure on an outcome is an important task. Mendelian Randomization, in particular, uses genetic variants as instruments to estimate causal effects in epidemiological studies. However, when there exist pleiotropic effects, conventional instrumental variable methods can be biased. Theoretical properties of Bayes estimators induced by single and mixture Gaussian priors are studied in the existence of pleiotropy. The methods under consideration are generalized to deal with summarized data and demonstrated in various simulation settings and on two real datasets.Ph.D.Includes bibliographical referencesby Sai L