Statistica Sinica 2(2000), 497-515

VARIANCE ESTIMATION IN HIGH DIMENSIONAL

REGRESSION MODELS

Snigdhansu Chatterjee and Arup Bose

Indian Statistical Institute, Calcutta

Abstract: We treat the problem of variance estimation of the least squares estimate of the parameter in high dimensional linear regression models by using the Uncorrelated Weights Bootstrap ($UBS$). We find a representation of the $UBS$ dispersion matrix and show that the bootstrap estimator is consistent if $p^{2}/n \rightarrow 0$ where $p$ is the dimension of the parameter and $n$ is the sample size. For fixed dimension we show that the $UBS$ belongs to the $R$-class as defined in Liu and Singh (1992).

Key words and phrases: Bootstrap, dimension asymptotics, jackknife, many parameter regression, variance estimation.