Prof. Shuva Gupta : Asymptotic properties of the estimates of $l_{1}$ penalized regression

Posted by Min Yang , part of the Statistics and Data Science Seminar.

At

Oct. 24, 2012, 4 p.m.

In

SEO 636

Abstract

Here we investigate two problems concerning the asymptotic properties of an $l_{1}$ penalized regression. In the first problem we study the asymptotic distribution of the Lasso estimator for regression models with dependent errors. The asymptotic distribution of the Lasso estimator for regression models with independent errors has been investigated by Knight and Fu. Here we extend these results to regression models with a general weak dependence structure. We determine the asymptotic distribution of the Lasso estimator when the number of parameters M is fixed and the number of observations, n, converges to $\infty$. We show that, for an appropriate choice of the tuning parameter of the method, this asymptotic distribution reduces to a multivariate normal distribution. We also provide some illustrative examples. In the second problem, we deal with the asymptotic distribution of residual empirical process of residuals in an adaptive lasso setting. We study the asymptotic properties of the empirical residual process and then use it to investigate the problem of goodness of fit when p<n but increases with n. We explore different applications of residual empirical process eg test of goodness of fit. This work was largely motivated by that of Chen and Lockhart (2001).