Maximum Quasi-likelihood Estimate of Generalized Linear Models with Measurement Errors in Its Explanatory Variables

Abstract

The asymptotic properties of the maximum quasi-likelihood estimate (MQLE) in generalized linear models with measurement errors in covariates are studied. Both the fixed and the adaptive designs cases are considered. The strong consistency and convergence rates of MQLE for both designs are reported, which are parallel to the past work of Lai, Robbins and Wei, Lai and Wei in the least square linear regression and, to Chen, Hu and Ying in the generalized linear model. In addition, we propose a sequential procedure for constructing confidence set of the regression parameters based on the strong consistent MQLE. The proposed sequential procedures can be shown to be asymptotic consistent and efficient in the sense of Chow and Robbins (1965) for both the fixed and the adaptive design cases. The results rely on the (conditional) moment conditions of measurement error only, and no knowledge on the distribution of measurement error is required.