Maximum likelihood estimation in logistic regression models with a diverging number of covariates

Abstract

Binary data with high-dimensional covariates have become more and more common in many disciplines. In this paper we consider the maximum likelihood estimation for logistic regression models with a diverging number of covariates. Under mild conditions we establish the asymptotic normality of the maximum likelihood estimate when the number of covariates $p$ goes to infinity with the sample size $n$ in the order of $p=o(n)$. This remarkably improves the existing results that can only allow $p$ growing in an order of $o(n^{\alpha})$ with $\alpha\in[1/5,1/2]$ [12, 14]. A major innovation in our proof is the use of the injective function.