On stable laws for estimating functions and derived estimators (Q1088324)

Let \(\{P_{\theta}\}_{\theta \in \Theta}\) be a family of probability measures on (X,A) where \(\Theta\) \((\subset {\mathbb{R}}^ p)\) is a parameter space. Let \(\{X_ i\), \(i\geq 1\}\) be a sequence of independent random vectors such that under \(P_{\theta}\), \(X_ i\) has a probability density function \(f_ i(x,\theta)\). Let \(\eta_ i(x,\theta)\) be \({\mathbb{R}}^ p\)-valued function on \(X\times \Theta\). Define an estimating function by \(S_ n(t)=\sum^{n}_{i=1}\eta_ i(X_ i,t)\) \((t\in {\mathbb{R}}^ p)\) and consider a derived estimator \(T_ n=T(X_ 1,...,X_ n)\) of \(\theta\) such that for some given \(\alpha\) (1/2\(\leq \alpha \leq 1)\), \(n^{-\alpha} S_ n(T_ n)\to 0\) in probability as \(n\to \infty\). It is well known that the maximum likelihood estimator, Huber's M- estimators, etc., belong to this class of estimators derived from suitable estimating functions. Let \(U_ i(X_ i,\theta,d)=\eta_ i(X_ i,\theta +d)-\eta_ i(X_ i,\theta)\) and put \(-\Gamma_ i=\lim_{d\downarrow 0}\{d^{-1}:\) \(E_{\theta}[U_ i(X_ i,\theta,de_ 1),...,U_ i(X_ i,\theta,de_ p)]\}\) where each \(e_ j\) \((\in {\mathbb{R}}^ p)\) has 1 in the j-th position and 0 elsewhere. Further, let \({\bar \Gamma}_ n=n^{-1}\sum^{n}_{i=1}\Gamma_ i.\) In this paper, under suitable conditions it is shown that if \(n^{- \alpha} S_ n(\theta)\) has asymptotically a stable law G with centering parameter 0, then \(n^{1-\alpha} {\bar \Gamma}_ n(T_ n-\theta)\) has asymptotically the law G. Stable laws for M-estimators of location are also considered.