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

Strong consistency for maximum quasi-likelihood estimators of regression pa rameters in generalized Linear regression models is studied. Results parall el to the elegant work of Lai, Robbins and Wei and Lai and Wei on least squ ares estimation under both fixed and adaptive designs are obtained. Let y(1 ),..., y(n) and x(1),...,x(n) be the observed responses and their correspon ding design points (p x 1 vectors), respectively. For fixed designs, it is shown that if the minimum eigenvalue of Sigma x(i)x'(i) goes to infinity, t hen the maximum quasi-likelihood estimator for the regression parameter vec tor is strongly consistent. For adaptive designs, it is shown that a suffic ient condition for strong consistency to hold is that the ratio of the mini mum eigenvalue of Sigma x(i)x'(i) to the logarithm of the maximum eigenvalu es goes to infinity. Use of the results for the adaptive design case in qua ntal response experiments is also discussed.