Strong consistency of maximum quasi-likelihood estimate in generalized linear models via a last time

In this paper, by using a last-time random variable, we show the strong consistency for the maximum quasi-likelihood estimate in generalized linear models with adaptive design variables and general link functions. Our approach is based on the Leray-Schauder Theorem and a last-time theorem. The last time that we defined here is based on a sum of martingale differences instead of independent random variables. Under some slightly stronger assumptions on the adaptive design variables, we obtain the almost sure convergence as well as the convergence rate of the estimate.