On some problems of weak consistency of quasi-maximum likelihood estimates in generalized linear models

Abstract

In this paper, we explore some weakly consistent properties of quasi-maximum likelihood estimates (QMLE) concerning the quasi-likelihood equation $\sum\nolimits_{i = 1}^n {X_i (y_i - \mu (X_i^\prime \beta ))}$ for univariate generalized linear model E(y|X) = μ(X′β). Given uncorrelated residuals {e _i = Y _i − μ(X ^′ _i β₀), 1 ⩽ i ⩽ n} and other conditions, we prove that $\hat \beta _n - \beta _0 = O_p (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } _n^{ - 1/2} )$ holds, where $\hat \beta _n$ is a root of the above equation, β ₀ is the true value of parameter β and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } _n$ denotes the smallest eigenvalue of the matrix S _n = ∑ ⁿ _i=1 X _i X ^′ _i . We also show that the convergence rate above is sharp, provided independent non-asymptotically degenerate residual sequence and other conditions. Moreover, paralleling to the elegant result of Drygas (1976) for classical linear regression models, we point out that the necessary condition guaranteeing the weak consistency of QMLE is S ⁻¹ _n → 0, as the sample size n → ∞.