  We propose MC+, a fast, continuous, nearly unbiased and accurate method of
penalized variable selection in high-dimensional linear regression. The LASSO
is fast and continuous, but biased. The bias of the LASSO may prevent
consistent variable selection. Subset selection is unbiased but computationally
costly. The MC+ has two elements: a minimax concave penalty (MCP) and a
penalized linear unbiased selection (PLUS) algorithm. The MCP provides the
convexity of the penalized loss in sparse regions to the greatest extent given
certain thresholds for variable selection and unbiasedness. The PLUS computes
multiple exact local minimizers of a possibly nonconvex penalized loss function
in a certain main branch of the graph of critical points of the penalized loss.
Its output is a continuous piecewise linear path encompassing from the origin
for infinite penalty to a least squares solution for zero penalty. We prove
that at a universal penalty level, the MC+ has high probability of matching the
signs of the unknowns, and thus correct selection, without assuming the strong
irrepresentable condition required by the LASSO. This selection consistency
applies to the case of $p\gg n$, and is proved to hold for exactly the MC+
solution among possibly many local minimizers. We prove that the MC+ attains
certain minimax convergence rates in probability for the estimation of
regression coefficients in $\ell_r$ balls. We use the SURE method to derive
degrees of freedom and $C_p$-type risk estimates for general penalized LSE,
including the LASSO and MC+ estimators, and prove their unbiasedness. Based on
the estimated degrees of freedom, we propose an estimator of the noise level
for proper choice of the penalty level.
