Convergence rate of MLE in generalized linear and nonlinear mixed-effects models: Theory and applications

Abstract

Generalized linear and nonlinear mixed-effects models are used extensively in the study of repeated measurements and longitudinal data analysis. Convergence rates of maximum likelihood estimates (MLEs) differ from parameter to parameter, which is not well explained in the literature. We consider the convergence rates of the MLEs for three different cases: (1) the number of subjects (clusters) n tends to infinity while the number of measurements per subject p remains finite; (2) both n and p tend to infinity; (3) n remains finite while p tends to infinity. In the three cases above, the MLE may have different convergence rates. In case (1), as we can expect, the MLE of all parameters are n -consistent under some regularity conditions; in case (2), some parameters could be np -consistent. These rates of convergence have a crucial impact on both experimental design and data analysis. Limited simulations were performed to examine the theoretical results. We illustrate applications of our results through one example and through presenting the exact convergence rates of some “approximate” MLEs such as PQL, CGEE2, etc. It is shown that the MLE for some parameters and some “approximate” MLEs may be N -consistent, here N = np is the total sample size. Also our results show that the MLE of some parameters maybe asymptotically independent of some other parameters.