Estimation of the Amount of Sparsity in Normal Mixture Models

Abstract

In this thesis, we are interested in estimating the amount of sparsity in sparse normal mixture models. The estimation problem in hand emerges naturally in the context of variable selection in high-dimensional settings. Handling this problem, we have modified the wellknown procedure of Cai et al. (2007) on estimating the proportion of nonzero means in normal mixture models in such a way that the new procedure is more efficient in theory, less complicated in construction, and less asymptotic in applications. The analytical findings obtained in the thesis are supported via simulations. The simulation study testifies that the new procedure not only displays a better convergence rate but also requires smaller sample size in order to work as designed. Theorem 7 of Chapter 3 that establishes the upper bound on the risk of the newly proposed estimator is the main result of the thesis. This result shows the superiority of our estimator over the estimator of Cai et al. (2007), and it is new.