## Cube root asymptotics.(English)Zbl 0703.62063

This paper is concerned with properties of estimators of a parameter $$\theta$$ defined by maximization of processes $P_ ng(\cdot,\theta)=(1/n)\sum_{i\leq n}g(X_ i,\theta)$ where $$\{X_ i\}$$ is a sequence of independent observations taken from a distribution P and [g($$\cdot,\theta)$$; $$\theta\in \Theta]$$ is a class of functions indexed by a subset $$\Theta$$ of $$R^ d$$. Let $$[\theta_ n]$$ be a sequence of estimators for which $P_ ng(\cdot,\theta_ n)\geq \sup_{\theta \in \Theta}P_ ng(\cdot,\theta)-o_ p(n^{-2/3}).$ Suppose $$\theta_ n$$ converges in probability to the unique $$\theta_ 0$$ that maximizes Pg($$\cdot,\theta)$$ and $$\theta_ 0$$ is an interior point of $$\Theta$$. Assume that Pg($$\cdot,\theta)$$ is twice differentiable with second derivative matrix -V at $$\theta_ 0$$, and $H(s,t)=\lim_{A\to \infty}A Pg(\cdot,\theta_ 0+s/A)g(\cdot,\theta_ 0+t/A)$ exists for each s,t in $$R^ d$$. Then under some additional regularity conditions, the process $$n^{2/3}P_ ng(\cdot,\theta_ 0+tn^{-1/3})$$ converges in distribution to a Gaussian process Z(t) with continuous sample paths, expected value $$-t'Vt/2$$, and covariance kernel H. If V is positive definite and Z has nondegenerate increments, then $$n^{1/3}(\theta_ n-\theta_ 0)$$ converges in distribution to the (almost surely unique) random vector that maximizes Z. Various examples are given.
Reviewer: L.Weiss

### MSC:

 62G20 Asymptotic properties of nonparametric inference 60F17 Functional limit theorems; invariance principles 60G15 Gaussian processes 62G99 Nonparametric inference 62F12 Asymptotic properties of parametric estimators 62M99 Inference from stochastic processes
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