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

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 is an element of [1/5, 1/2] [12, 14]. A major innovation in our proof is the use of the injective function.