Local asymptotic normality of truncated empirical processes

Abstract

Given $n$ iid copies $X_1,\dots, X_n$ of a random element $X$ in some arbitrary measurable space $S$, we are only interested in those observations that fall into some subset $D$ having but a small probability of occurrence. It is assumed that the distribution $P_X$ of $X$ belongs on $D$ to a parametric family $P_X(\cdot\capD) = P_{\vartheta}, \vartheta \epsilon \Theta \subset \mathbb{R}^d$. Nonlinear regression analysis and the peaks-over-threshold (POT) approach in extreme value analysis are prominent examples. For the POT approach on $S = \mathbb{R}$ and $P_{\vartheta}$ being a generalized Pareto distribution, it is known that the complete information about the underlying parameter $\vartheta_0$ is asymptotically contained in the number $\tau(n)$ of observations in $D$ among $X_1,\dots, X_n$, but not in their actual values. This result is formulated in terms of local asymptotic normality of the log-likelihood ratio of the point process of exceedances with $\tau(n)$ being the central sequence. In this paper we establish a necessary and sufficient condition such that $\tau(n)$ has this property for a general truncated empirical process in an arbitrary sample space and for an arbitrary parametric family. The known results are then consequences of this result. We can, moreover, characterize the influence of the actual observations in $D$ on the central sequence, if this condition is violated. Immediate applications are asymptotically optimal tests for testing $\vartheta_0$ and, if $\Theta \subset \mathbb{R}$, asymptotic efficiency of the ML-estimator $\hat{\vartheta}_n$ satisfying $P_{\hat{\vartheta}_n}(D) = \tau(n)/n$, where these statistics are based on $\tau(n)$ only.