Extreme values and kernel estimates of point processes boundaries
ESAIM: Probability and Statistics, Tome 8 (2004), pp. 150-168.

We present a method for estimating the edge of a two-dimensional bounded 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 conditions for various kinds of convergence and asymptotic normality. We propose a method of reducing the negative bias and edge effects, illustrated by some simulations.

DOI : 10.1051/ps:2004008
Classification : 60G70, 62G05, 62G20, 62M30
Mots clés : kernel estimates, extreme values, Poisson process, shape estimation 
@article{PS_2004__8__150_0,
     author = {Girard, St\'ephane and Jacob, Pierre},
     title = {Extreme values and kernel estimates of point processes boundaries},
     journal = {ESAIM: Probability and Statistics},
     pages = {150--168},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2004},
     doi = {10.1051/ps:2004008},
     mrnumber = {2085612},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2004008/}
}
TY  - JOUR
AU  - Girard, Stéphane
AU  - Jacob, Pierre
TI  - Extreme values and kernel estimates of point processes boundaries
JO  - ESAIM: Probability and Statistics
PY  - 2004
SP  - 150
EP  - 168
VL  - 8
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2004008/
DO  - 10.1051/ps:2004008
LA  - en
ID  - PS_2004__8__150_0
ER  - 
%0 Journal Article
%A Girard, Stéphane
%A Jacob, Pierre
%T Extreme values and kernel estimates of point processes boundaries
%J ESAIM: Probability and Statistics
%D 2004
%P 150-168
%V 8
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2004008/
%R 10.1051/ps:2004008
%G en
%F PS_2004__8__150_0
Girard, Stéphane; Jacob, Pierre. Extreme values and kernel estimates of point processes boundaries. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 150-168. doi : 10.1051/ps:2004008. http://www.numdam.org/articles/10.1051/ps:2004008/

[1] 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).

[2] 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. | Zbl

[3] D. Bosq and J.P. Lecoutre, Théorie de l'estimation fonctionnelle. Economica, Paris (1987).

[4] A. Cowling and P. Hall, On pseudodata methods for removing boundary effects in kernel density estimation. J. Roy. Statist. Soc. B 58 (1996) 551-563. | Zbl

[5] 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).

[6] 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. | Zbl

[7] J. Geffroy, Sur un problème d'estimation géométrique. Publications de l'Institut de Statistique de l'Université de Paris XIII (1964) 191-200. | Zbl

[8] S. Girard, 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. | Zbl

[9] S. Girard and P. Jacob, Extreme values and Haar series estimates of point processes boundaries. Scand. J. Statist. 30 (2003) 369-384. | Zbl

[10] S. Girard and P. Jacob, Projection estimates of point processes boundaries. J. Statist. Plann. Inference 116 (2003) 1-15. | Zbl

[11] P. Hall, B.U. Park and S.E. Stern, On polynomial estimators of frontiers and boundaries. J. Multiv. Analysis 66 (1998) 71-98. | Zbl

[12] K. Knight, Limiting distributions of linear programming estimators. Extremes 4 (2001) 87-103. | Zbl

[13] W. Härdle, B.U. Park and A.B. Tsybakov, Estimation of a non sharp support boundaries. J. Multiv. Analysis 43 (1995) 205-218. | Zbl

[14] W. Härdle, P. Hall and L. Simar, Iterated bootstrap with application to frontier models. J. Productivity Anal. 6 (1995) 63-76.

[15] A. Hardy and J.P. Rasson, Une nouvelle approche des problèmes de classification automatique. Statistique et analyse des données 7 (1982) 41-56. | Numdam | Zbl

[16] 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. | Zbl

[17] P. Jacob and P. Suquet, Estimating the edge of a Poisson process by orthogonal series. J. Statist. Plann. Inference 46 (1995) 215-234. | Zbl

[18] A. Korostelev, L. Simar and A.B. Tsybakov, Efficient estimation of monotone boundaries. Ann. Statist. 23 (1995) 476-489. | Zbl

[19] A.P. Korostelev and A.B. Tsybakov, Minimax theory of image reconstruction. Lect. Notes Statist. 82 (1993). | MR | Zbl

[20] E. Mammen and A.B. Tsybakov, Asymptotical minimax recovery of set with smooth boundaries. Ann. Statist. 23 (1995) 502-524. | Zbl

[21] R.D. Reiss, A course on point processes. Springer series in statistics (1993). | MR | Zbl

[22] A. Renyi and R. Sulanke, Uber die konvexe Hülle von n zufälligen gewählten Punkten. Z. Wahrscheinlichkeitstheorie verw. Geb. 2 (1963) 75-84. | Zbl

Cité par Sources :