On parameters of increasing dimensions

In statistical analyses the complexity of a chosen model is often related t o the size of available data. One important question is whether the asympto tic distribution of the parameter estimates normally derived by taking, the sample size to infinity for a fixed number of parameters would remain vali d if the number of parameters in the model actually increases with the samp le size. A number of authors have addressed this question fur the linear mo dels. The component-wise asymptotic normality of the parameter estimate rem ains valid if the dimension of the parameter space grows more slowly than s ome root of the sample size. In this paper, we consider M-estimators of gen eral parametric models. Our results apply to not only linear regression but also other estimation problems such as multivariate location and generaliz ed linear models. Examples are given to illustrate the applications in diff erent settings. (C) 2000 Academic Press.