© 2015, Institute of Mathematical Statistics. All right received. We study a two-component semiparametric mixture model where one component distribution belongs to a parametric class, while the other is symmetric but otherwise arbitrary. This semiparametric model has wide applications in many areas such as large-scale simultaneous testing/multiple testing, sequential clustering, and robust modeling. We develop a class of estimators that are surprisingly simple and are unique in terms of their construction. A unique feature of these methods is that they do not rely on the estimation of the nonparametric component of the model. Instead, the methods only require a working model of the unspecified distribution, which may or may not reflect the true distribution. In addition, we establish connections between the existing estimator and the new methods and further derive a semiparametric efficient estimator. We compare our estimators with the existing method and investigate the advantages and cost of the relatively simple estimation procedure.

High-dimensional feature selection arises in many areas of modern science. For example, in genomic research we want to find the genes that can be used to separate tissues of different classes (e.g. cancer and normal) from tens of thousands of genes that are active (expressed) in certain tissue cells. To this end, we wish to fit regression and classification models with a large number of features (also called variables, predictors). In the past decade, penalized likelihood methods for fitting regression models based on hyper-LASSO penalization have received increasing attention in the literature. However, fully Bayesian methods that use Markov chain Monte Carlo (MCMC) are still in lack of development in the literature. In this paper we introduce an MCMC (fully Bayesian) method for learning severely multi-modal posteriors of logistic regression models based on hyper-LASSO priors (non-convex penalties). Our MCMC algorithm uses Hamiltonian Monte Carlo in a restricted Gibbs sampling framework; we call our method Bayesian logistic regression with hyper-LASSO (BLRHL) priors. We have used simulation studies and real data analysis to demonstrate the superior performance of hyper-LASSO priors, and to investigate the issues of choosing heaviness and scale of hyper-LASSO priors.