Nonnegative estimation and variable selection under minimax concave penalty for sparse high-dimensional linear regression models

As a promising alternative to the Lasso penalty, the minimax concave penalty (MCP) produces nearly unbiased shrinkage estimate. In this paper, we propose the nonnegative MCP estimator in the specific high-dimensional settings where the regression coefficients are imposed with the nonnegativity constraint, which is particularly relevant when modelling nonnegative data. We prove that the asymptotic theory for the nonnegative MCP estimator requires much weaker assumptions than those in the literature. In particular, we do not impose the restrictive distributional conditions on the random errors and do not assume the irrepresentable condition or its variants on the design matrix. For the nonnegative MCP estimator, despite its good asymptotic properties, the corresponding optimization problem is non-convex, and consequently much hard to solve. Thus, we develop an efficient computation algorithm, which is a coupling of the difference convex algorithm with the multiplicative updates algorithm, for the implementation of the nonnegative MCP estimator. Simulation studies are carried out to examine superior performance of the nonnegative MCP estimator against alternative methods and a real data analysis for index tracking is also presented for illustration.