WEAK CONSISTENCY OF MODIFIED VERSIONS OF BAYESIAN
INFORMATION CRITERION IN A SPARSE LINEAR REGRESSION
Abstract: We consider the regression model in the situation when the number of available
regressors
is much bigger than the sample size
and the number of nonzero coefficients
is small (the sparse regression). To choose the regression model, we need to identify the
nonzero coefficients. However, in this situation the classical model selection criteria for the
choice of predictors like, e.g., the Bayesian Information Criterion (BIC) overestimate the
number of regressors. To address this problem, several modifications of BIC have
been recently proposed. In this paper we prove weak consistency of some of these
modifications under the assumption that both
and
as well as
go to infinity.
2000 AMS Mathematics Subject Classification: Primary: 62J05; Secondary:
92D20.
Keywords and phrases: Sparse linear regression, mBIC, mBIC2, consistency.