Abstract: We establish central limit theorems for function-valued estimators defined as a zero point of a function-valued random criterion function. Our approach is based on a differential identity that applies when the random criterion function is linear in terms of the empirical measure. We do not require linearity of the statistical model in the unknown parameter, even though the result is most applicable for models with convex linearity that can be boundedly extended to the linear span of the parameter space. Three examples are given to illustrate the application of these theorems: a simplified frailty model which is nonlinear in the unknown parameter; the multiplicative censoring and double censoring models which are bounded convex linear in the unknown parameter.
Key words and phrases: Estimating equations, M-estimators, nonparametric maximum likelihood, stochastic equicontinuity, self-consistency, weak convergence.