Asymptotic properties of maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with misspecified variance function

Abstract

The quasi-likelihood function proposed by Wedderburn [Quasi-likelihood functions, generalized linear models, and the Gauss–Newton method. Biometrika. 1974;61:439–447] broadened the application scope of generalized linear models (GLM) by specifying the mean and variance function instead of the entire distribution. However, in many situations, complete specification of variance function in the quasi-likelihood approach may not be realistic. Following Fahrmeir's [Maximum likelihood estimation in misspecified generalized linear models. Statistics. 1990;21:487–502] treating with misspecified GLM, we define a quasi-likelihood nonlinear models (QLNM) with misspecified variance function by replacing the unknown variance function with a known function. In this paper, we propose some mild regularity conditions, under which the existence and the asymptotic normality of the maximum quasi-likelihood estimator (MQLE) are obtained in QLNM with misspecified variance function. We suggest computing MQLE of unknown parameter in QLNM with misspecified variance function by the Gauss–Newton iteration procedure and show it to work well in a simulation study.

Keywords:

• asymptotic normality
• existence
• maximum quasi-likelihood estimator
• quasi-likelihood nonlinear models with misspecified variance function
• Gauss–Newton iteration method

2010 Mathematics Subject Classification:

• 62F12
• 62J02
• 62J12

Acknowledgements

We gratefully acknowledge the constructive remarks of the Associate Editor and the referees which have greatly improved the manuscript. This work was partially supported by Natural Science Foundation of China (11361013, 11101432, 11171293), Science and Technology Foundation of Guizhou Province ((2008)2249).