Asymptotic Properties of Quasi-Maximum Likelihood Estimates in Generalized Linear Models

In this article, we consider the quasi-likelihood equation Sigma(n)(i=1) X-i(y(i) - mu(X-i'beta)) = 0 for generalized linear models (GLMs). Under some mild conditions, including the convergent system {e(i) = y(i) - mu(X-i'beta(0)), i >= 1} which is defined by Lai et al. (1979), we obtain the asymptotic existence of the solution (beta) over cap (n) to the above equation and show that (beta) over cap (n) - beta(0) = O((lambda) over bar (1/2)(n) (log (lambda) over bar (n))(delta/2)/(lambda) under bar (n)) a.s., where beta(0) is the true value of parameter beta and (lambda) under bar (n)((lambda) over bar (n)) denotes the smallest (largest) eigenvalue of Sigma(n)(i=1) XiXt' satisfying ((lambda) over bar (1/2)(n) (log (lambda) over bar (n))(delta/2))/(lambda) under bar (n) -> 0 as n -> infinity for given delta > 1. We also present the asymptotic normality of (beta) over cap (n) for univariate GLMs, based on which "studentized" large sample confidence intervals for beta(0) are constructed. Simulation results and related remarks are given.