## The Annals of Statistics

### Local asymptotic normality of truncated empirical processes

Michael Falk

#### 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.

#### Article information

Source
Ann. Statist., Volume 26, Number 2 (1998), 692-718.

Dates
First available in Project Euclid: 31 July 2002

https://projecteuclid.org/euclid.aos/1028144855

Digital Object Identifier
doi:10.1214/aos/1028144855

Mathematical Reviews number (MathSciNet)
MR1626087

Zentralblatt MATH identifier
0930.62017

Subjects
Primary: 62F05: Asymptotic properties of tests 62F12: Asymptotic properties of estimators
Secondary: 60G55: Point processes

#### Citation

Falk, Michael. Local asymptotic normality of truncated empirical processes. Ann. Statist. 26 (1998), no. 2, 692--718. doi:10.1214/aos/1028144855. https://projecteuclid.org/euclid.aos/1028144855

#### References

• ANDERSON, P. K., BORGAN, Ø., GILL, R. D. and KEIDING, N. 1992. Statistical Models Based on Counting Processes. Springer, New York. Z.
• BALKEMA, A. A. and DE HAAN, L. 1974. Residual lifetime at great age. Ann. Probab. 2 792 804. Z.
• FALK, M. 1995a. On testing the extreme value index via the POT-Method. Ann. Statist. 23 2013 2035. Z.
• FALK, M. 1995b. LAN of extreme order statistics. Ann. Inst. Statist. Math. 47 693 717. Z.
• FALK, M. and MAROHN, F. 1993. Asy mptotically optimal tests for conditional distributions. Ann. Statist. 21 45 60. Z.
• FALK, M., HUSLER, J. and REISS, R.-D. 1994. Laws of Small Numbers: Extreme and Rare ¨ Events. Birkhauser, Basel. ¨ Z.
• HAJEK, J. 1962. Asy mptotically most powerful rank order tests. Ann. Math. Statist. 33 ´ 1224 1147. Z.
• HAJEK, J. 1970. A characterisation of limiting distributions of regular estimates. Z. Wahrsch. ´ Verw. Gebiete 12 21 55. Z.
• HILL, B. M. 1975. A simple approach to inference about the tail of a distribution. Ann. Statist. 3 1163 1174. Z.
• HOPFNER, R. 1994. On tail parameter estimation in certain point process models. J. Statist. ¨ Plann. Inference. To appear. Z.
• HOPFNER, R. 1997. Two comments on parameter estimation in stable processes. Math. Methods ¨ Statist. 6 125 134. Z.
• HOPFNER, R. and JACOD J. 1994. Some remarks on the joint estimation of the index and the ¨ scale parameter for stable processes. In Asy mptotic Statistics. Proceedings of the Fifth Z. Prague Sy mposium 1993 P. Mandl and M. Huskova, eds. 273 284. physica, Heidelberg. Z.
• IBRAGIMOV, L. A. and HAS'MINSKII, R. Z. 1981. Statistical Estimation. Asy mptotic Theory. Springer, New York. Z.
• JANSSEN, A. and MAROHN, F. 1994. On statistical information of extreme order statistics, local, extreme value alternatives, and Poisson point processes. J. Multivariate Anal. 48 1 30. Z.
• KARR, A. F. 1991. Point Processes and Their Statistical Inference, 2nd ed. Dekker, New York. Z.
• LECAM, L. 1956. On the asy mptotic theory of estimation and testing hy pothesis. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 129 156. Univ. California Press, Berkeley. Z.
• LECAM, L. 1986. Asy mptotic Methods in Statistical Decision Theory. Springer, New York. Z.
• LECAM, L. and YANG, G. L. 1990. Asy mptotics in Statistics. Some Basic Concepts. Springer, New York. Z.
• MAROHN, F. 1995. Contributions to a local approach in extreme value statistics. Habilitation thesis, Katholische Univ. Eichstatt. ¨ Z.
• NISHIy AMA, Y. 1995. Local asy mptotic normality of a sequential model for marked point processes and its application. Ann. Inst. Statist. Math. 47 195 209. Z.
• PFANZAGL, J. 1994. Parametric Statistical Theory. de Gruy ter, Berlin. Z.
• PICKANDS III, J. 1975. Statistical inference using extreme order statistics. Ann. Statist. 3 119 131.
• REISS, R.-D. 1993. A Course on Point Processes. Springer, New York. Z.
• Ry CHLIK, T. 1992. Weak limit theorems for stochastically largest order statistics. In Order Z Statistics and Nonparametrics: Theory and Applications P. K. Sen and I. A. Salama,. eds. 141 154. North-Holland, Amsterdam. Z.
• SAKSENA, S. K. and JOHNSON, A. M. 1984. Best unbiased estimators for the parameters of a two-parameter Pareto distribution. Metrika 31 77 83. Z.
• STRASSER, H. 1985. Mathematical Theory of Statistics. de Gruy ter, Berlin. Z.
• WEI, X. 1995. Asy mptotically efficient estimators of the index of regular variation. Ann. Statist. 23 2036 2058.