Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs

Abstract

Strong consistency for maximum quasi-likelihood estimators of regression parameters in generalized linear regression models is studied. Results parallel to the elegant work of Lai, Robbins and Wei and Lai and Wei on least squares estimation under both fixed and adaptive designs are obtained. Let $y_1,\dots, y_n$ and $x_1,\dots, x_n$ be the observed responses and their corresponding design points ($p \times 1$ vectors), respectively. For fixed designs, it is shown that if the minimum eigenvalue of $\Sigma x_i x^\prime_i$ goes to infinity, then the maximum quasi-likelihood estimator for the regression parameter vector is strongly consistent. For adaptive designs, it is shown that a sufficient condition for strong consistency to hold is that the ratio of the minimum eigenvalue of $\Sigma x_i \x^\prime_i$ to the logarithm of the maximum eigenvalues goes to infinity. Use of the results for the adaptive design case in quantal response experiments is also discussed.