This dissertation includes three papers. In the first paper, a new statistical procedure is proposed to analyze verbal autopsy data. Verbal autopsy procedures are widely used for estimating cause-specific mortality in areas without medical death certifications. We show that the problem of estimating cause-specific mortality rate can be directly solved using the distribution of symptoms that is available from the population verbal autopsy survey and the cause-specific distribution of symptoms that can be obtained from hospital data. To solve this deconvolution problem, we offer an optimization procedure that is stable and easy to compute. Through empirical analyses in data from China and Tanzania, we illustrate the accuracy of this approach. In the second paper, we focus on the analysis of roll call and vote records data to legislative and judicial voting behaviors. Ideal point estimation is an important tool to analyze this type of data. We introduce a hierarchical ideal point estimation framework that directly models complex voting behaviors based on the characteristics of the political actors and the votes they cast. Bayesian MCMC algorithms are proposed to estimate the proposed hierarchical models. Through simulations and empirical examples we show that this framework holds good promise for resolving many unsettled issues, such as the multi-dimensional aspects of ideology, and the effects of political parties. In the third paper, we address the issue of variable selection in linear mixed effect models. Mixed effect models are fundamental tools for the analysis of longitudinal data, panel data and cross-sectional data. However, the complex nature of these models has made variable selection and parameter estimation a challenging problem. In this paper, we propose a simple iterative procedure that estimates and selects fixed and random effects for linear mixed models. In particular, we propose to utilize the partial consistency property of the random effect coefficients and select groups of random effects simultaneously via a data-oriented penalty function. We show that the proposed method is a consistent variable selection procedure and possesses the Oracle properties. Simulation studies and a real data analysis are also conducted to empirically examine the performance of this procedure.