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Extreme values and kernel estimates of point processes boundaries

Published online by Cambridge University Press:  15 September 2004

Stéphane Girard  and
Pierre Jacob
Stéphane Girard
Affiliation:
SMS/LMC, Université Grenoble 1, BP 53, 38041 Grenoble Cedex 9, France; [email protected].
Pierre Jacob
Affiliation:
EPS/I3M, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier Cedex 5, France; [email protected].
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Abstract

We present a method for estimating the edge of a two-dimensionalbounded set, given a finite random set of points drawn from the interior.The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditionsfor various kinds of convergence and asymptotic normality.We propose a method of reducing the negative bias and edge effects,illustrated by some simulations.

Keywords

Kernel estimatesextreme valuesPoisson processshape estimation.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 150 - 168
Copyright
© EDP Sciences, SMAI, 2004

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References

H. Abbar and Ch. Suquet, Estimation L 2 du contour d'un processus de Poisson homogène sur le plan. Pub. IRMA Lille 31 II (1993).
D. Bosq, Contribution à la théorie de l'estimation fonctionnelle. Publications de l'Institut de Statistique de l'Université de Paris XIX (1977) 1–96.
D. Bosq and J.P. Lecoutre, Théorie de l'estimation fonctionnelle. Economica, Paris (1987).
Cowling, A. and Hall, P., On pseudodata methods for removing boundary effects in kernel density estimation. J. Roy. Statist. Soc. B 58 (1996) 551–563.
D. Deprins, L. Simar and H. Tulkens, Measuring Labor Efficiency in Post Offices, in The Performance of Public Enterprises: Concepts and Measurements, M. Marchand, P. Pestieau and H. Tulkens Ed., North Holland, Amsterdam (1984).
L. Gardes, Estimating the support of a Poisson process via the Faber-Shauder basis and extreme values. Publications de l'Institut de Statistique de l'Université de Paris XXXXVI (2002) 43–72.
J. Geffroy, Sur un problème d'estimation géométrique. Publications de l'Institut de Statistique de l'Université de Paris XIII (1964) 191–200.
Girard, S., On the asymptotic normality of the L 1 error for Haar series estimates of Poisson point processes boundaries. Statist. Probab. Lett. 66 (2004) 81–90. CrossRef
Girard, S. and Jacob, P., Extreme values and Haar series estimates of point processes boundaries. Scand. J. Statist. 30 (2003) 369–384. CrossRef
Girard, S. and Jacob, P., Projection estimates of point processes boundaries. J. Statist. Plann. Inference 116 (2003) 1–15. CrossRef
Hall, P., Park, B.U. and Stern, S.E., On polynomial estimators of frontiers and boundaries. J. Multiv. Analysis 66 (1998) 71–98. CrossRef
Knight, K., Limiting distributions of linear programming estimators. Extremes 4 (2001) 87–103. CrossRef
Härdle, W., Park, B.U. and Tsybakov, A.B., Estimation of a non sharp support boundaries. J. Multiv. Analysis 43 (1995) 205–218. CrossRef
Härdle, W., Hall, P. and Simar, L., Iterated bootstrap with application to frontier models. J. Productivity Anal. 6 (1995) 63–76.
Hardy, A. and Rasson, J.P., Une nouvelle approche des problèmes de classification automatique. Statistique et analyse des données 7 (1982) 41–56.
P. Jacob, Estimation du contour discontinu d'un processus ponctuel sur le plan. Publications de l'Institut de Statistique de l'Université de Paris XXIX (1984) 1–26.
Jacob, P. and Suquet, P., Estimating the edge of a Poisson process by orthogonal series. J. Statist. Plann. Inference 46 (1995) 215–234. CrossRef
Korostelev, A., Simar, L. and Tsybakov, A.B., Efficient estimation of monotone boundaries. Ann. Statist. 23 (1995) 476–489. CrossRef
A.P. Korostelev and A.B. Tsybakov, Minimax theory of image reconstruction. Lect. Notes Statist. 82 (1993).
Mammen, E. and Tsybakov, A.B., Asymptotical minimax recovery of set with smooth boundaries. Ann. Statist. 23 (1995) 502–524. CrossRef
R.D. Reiss, A course on point processes. Springer series in statistics (1993).
Renyi, A. and Sulanke, R., Uber die konvexe Hülle von n zufälligen gewählten Punkten. Z. Wahrscheinlichkeitstheorie verw. Geb. 2 (1963) 75–84. CrossRef