ON THE POSITIVE DEFINITENESS OF THE INFORMATION MATRIX UNDER THE BINARY AND POISSON MIXED MODELS

RAHUL MUKERJEE^{1} AND BRAJENDRA C. SUTRADHAR^{2}

1Indian Institute of Management, Post Box No. 16757, Calcutta 700 027, India, e-mail: rmuk1@hotmail.com

2Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1C 5S7, e-mail: bsutradh@math.mun.ca

(Received January 11, 2000; revised January 23, 2001)

Abstract. Binary and Poisson generalized linear mixed models are used to analyse over/under-dispersed proportion and count data, respectively. As the positive definiteness of the information matrix is a required property for valid inference about the fixed regression vector and the variance components of the random effects, this paper derives the relevant necessary and sufficient conditions under both these models. It is found that the conditions for the positive definiteness are not identical for these two nonlinear mixed models and that a mere analogy with the usual linear mixed model does not dictate these conditions.

Key words and phrases: Estimating function, Fisher information matrix, generalised linear mixed model, joint estimates, likelihood estimation, positive definiteness, regression effects, variance component of the random effects.