Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T09:25:05.989Z Has data issue: false hasContentIssue false
  • English
  • Français

Multidimensional limit theorems for smoothed extreme value estimates of pointprocesses boundaries

Published online by Cambridge University Press:  08 May 2008

Ludovic Menneteau
Ludovic Menneteau*
Affiliation:
Place Eugène Bataillon, 34095 Montpellier Cedex 5, France; mennet@math.univ-montp2.fr
Get access

Abstract

In this paper, we give sufficient conditions to establish central limittheorems and moderate deviation principle for a class of support estimates ofempirical and Poisson point processes. The considered estimates are obtained bysmoothing some bias corrected extreme values of the point process. We show howthe smoothing permits to obtain Gaussian asymptotic limits and thereforepointwise confidence intervals. Some unidimensional and multidimensionalexamples are provided.

Keywords

Functional estimatecentral limit theoremmoderate deviation principlesextreme valuesshape estimation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 273 - 307
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adell, J.A. and Jodrá, P., The median of the Poisson distribution. Metrika 61 3 (2005) 337–346. CrossRef
Baufays, P. and Rasson, J.-P., A new geometric discriminant rule. Comput. Stat. Q. 2 (1985) 15–30.
P. Billingsley, Convergence of Probability measures. Wiley (1968).
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 Eds., North Holland, Amsterdam (1984).
J.D. Deuschel and D.W. Stroock, Large Deviations. Pure and Applied Mathematics, 137. Boston, MA Academic Press (1989).
Devroye, L.P. and Wise, G.L., Detection of abnormal behavior via non parametric estimation of the support. SIAM J. Appl. Math. 38 (1980) 448–480. CrossRef
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Bartlett, Boston and London (1993).
L. Gardes, Estimating the support of a Poisson process via the Faber-Schauder basis and extrems values. Publications de l'Institut de Statistique de l'Université de Paris XLVI 43–72 (2002).
J. Geffroy, Sur un problème d'estimation géométrique. Publications de l'Institut de Statistique de l'Université de Paris XIII (1964) 191–200.
Gijbels, I., Mammen, E., Park, B.U. and Simar, L., On estimation of monotone and concave frontier functions. J. Amer. Statist. Assoc. 94 (1999) 220–228. CrossRef
S. Girard and P. Jacob, Projection estimates of point processes boundaries. J. Statist. Planning Inference 116 (2003), 1–15.
Girard, S. and Jacob, P., Extreme values and kernel estimates of point processes boundaries. ESAIM: PS 8 (2005) 150–168 . CrossRef
Girard, S. and Menneteau, L., Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries. J. Statist. Planning Inference 135 (2005) 433–460. CrossRef
S. Girard and L. Menneteau, Smoothed extreme value estimators of non uniform boundaries with applications to star-shaped supports estimation. Submitted.
Hardy, A. and Rasson, J.P., Une nouvelle approche des problèmes de classification automatique. Statist. Anal. Données 7 (1982) 41–56.
Hall, P., Nussbaum, M. and Stern, S.E., On the estimation of a support curve of indeterminate sharpness. J. Multivariate Anal. 62 (1997) 204–232. CrossRef
Hall, P., Park, B.U. and Stern, S.E., On polynomial estimators of frontiers and boundaries. J. Multivariate Anal. 66 (1998) 71–98. CrossRef
W. Härdle, Applied nonparametric regression. Cambridge University Press, Cambridge (1990).
Härdle, W., Hall, P. and Simar, L., Iterated bootstrap with application to frontier models. J. Productivity Anal. 6 (1995) 63–76.
Härdle, W., Park, B.U. and Tsybakov, A.B., Estimation of a non sharp support boundaries. J. Multivariate Anal. 43 (1995) 205–218. CrossRef
J.A. Hartigan, Clustering Algorithms. Wiley, Chichester (1975).
Kallenberg, W., Intermediate efficiency theory and examples. Ann. Statist. 11 (1983) 170–182. CrossRef
Kallenberg, W., On moderate deviation theory in estimation. Ann. Statist. 11 (1983) 498–504. CrossRef
Korostelev, A.P., 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, in Lecture Notes in Statistics 82, Springer-Verlag, New York (1993).
Korostelev, A.P. and Tsybakov, A.B., Asymptotic efficiency of the estimation of a convex set. Problems Inform. Transmission 30 (1994) 317–327.
Mammen, E. and Tsybakov, A.B., Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 (1995) 502–524. CrossRef
L. Menneteau, Limit theorems for piecewise constant kernel smoothed estimates of point process boundaries. Technical Report (2007).
Mokkadem, A. and Pelletier, M., Moderate deviations for the kernel mode estimator and some applications. J. Statist. Planning Inference 135 (2005) 276–299. CrossRef
V.V. Petrov, Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability, (1995) 4.
G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley, New York (1986).
G.P. Tolstov, Fourier series. 2nd ed. New York: Dover Publications (1976).
Tsybakov, A.B., On nonparametric estimation of density level sets. Ann. Statist. 25 (1997) 948–969. CrossRef