Asymptotics for lasso-type estimators

We consider the asymptotic behavior ofregression estimators that minimize the residual sum of squares plus a penalty proportional to |ⱼ|^. for some > 0. These estimators include the Lasso as a special case when = 1. Under appropriate conditions, we show that the limiting distributions can have positive probability mass at 0 when the true value of the parameter is 0. We also consider asymptotics for “nearly singular” designs.