## Asymptotic properties of nonlinear mixed-effects models.(English)Zbl 0897.62067

Gregoire, Timothy G. (ed.) et al., Modelling longitudinal and spatially correlated data: methods, applications, and future directions. Proceedings of the conference, Nantucket, USA, October 1996. New York: Springer. Lect. Notes Stat., Springer-Verlag. 122, 49-62 (1997).
Summary: The asymptotic properties of four estimators for nonlinear mixed-effects models are investigated: maximum likelihood estimator (MLE), an estimator based on the first-order approximation to the expectation function, the two-stage, and the Lindstrom-Bates estimators [M. J. Lindstrom and D. M. Bates, Biometrics 46, 673-687 (1990)]. Two asymptotic situations are considered: (i) when $$N\to \infty$$ and $$n_i$$ is finite and (ii) $$N\to\infty$$ and $$\min n_i\to \infty$$, where $$N$$ is the number of individuals and $$n_i$$ is the number of observations on the $$i$$th individual. For a simple one-parameter balanced exponential model only the MLE is consistent when the numbers of observations per individual, $$\{n_i\}$$, are finite. The estimator based on the first-order approximation is always inconsistent, i.e., has a systematic bias. The asymptotic bias for the other three estimators is evaluated when $$N\to\infty$$ and $$n_i=\text{const}$$. When $$N\to\infty$$ and $$\min n_i\to\infty$$, the MLE, Lindstrom-Bates and two-stage estimators are equivalent. The bias for the two-stage estimator based on a second-order approximation is evaluated for the general nonlinear mixed-effects model, and a bias-corrected version of this estimator is proposed.
For the entire collection see [Zbl 0868.00060].

### MSC:

 62J02 General nonlinear regression 62H12 Estimation in multivariate analysis