Estimating the amount of sparsity in two-point mixture models

We consider the problem of estimating the fraction of nonzero means in a sparse normal mixture model in the region where variable selection is possible. The focus is on the situation in which the proportion of nonzero means is very small. The proposed estimator is shown to be nearly rate optimal in the asymptotically minimax sense. Using this estimator, one can also consistently estimate the sparsity parameter in sparse normal mixtures, whose knowledge, in particular, is required to carry out the so-called almost full variable selection procedure. The advantage of using the new estimator is illustrated analytically and numerically. The obtained results can be extended to some nonnormal mixtures.