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 vari- able selection in high-dimensional settings. Handling this problem, we have modified the well-known procedure of Cai et al. 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. The main result, Theorem 7, shows the superiority of our estimator over the estimator of Cai et al., and it is new.