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.