Cube root asymptotics
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.
scientific article; zbMATH DE number 4153723