LEAST SQUARES ESTIMATION AND VARIABLE SELECTION UNDER MINIMAX 
CONCAVE PENALTY 
Cun-Hui Zhang, Rutgers University 
Statistical inference about a sparse high-dimensional signal vector is well understood if 
the signal is directly observed with white noise. If most components of the signal are 
zero, threshold estimators at level univ= p   2 logpfind the exact set of nonzeros with 
high probability when the minimum absolute value of the nonzero components is slightly 
greater than univ. If the signal vector belongs to a small‘rball of radiusR, threshold 
estimators at a certain level mmapproximately attain the minimax risk. We show that 
in linear regression models withndata points andp > nunknowns, these results can be 
extended with concave penalized LSE, provided that the number of significant components 
orRr= r   mmof the signal vector is no greater than a certaind and the order of thisd could 
be as high asn=log(p=n). Implications of such results on sparse recovery will be discussed 
if time permits.