Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries

Ludovic Menneteau

ESAIM: Probability and Statistics (2008)

  • Volume: 12, page 273-307
  • ISSN: 1292-8100

Abstract

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In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided.

How to cite

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Menneteau, Ludovic. "Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries." ESAIM: Probability and Statistics 12 (2008): 273-307. <http://eudml.org/doc/250410>.

@article{Menneteau2008,
abstract = { In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided. },
author = {Menneteau, Ludovic},
journal = {ESAIM: Probability and Statistics},
keywords = {Functional estimate; central limit theorem; moderate deviation principles; extreme values; shape estimation; functional estimate},
language = {eng},
month = {5},
pages = {273-307},
publisher = {EDP Sciences},
title = {Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries},
url = {http://eudml.org/doc/250410},
volume = {12},
year = {2008},
}

TY - JOUR
AU - Menneteau, Ludovic
TI - Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries
JO - ESAIM: Probability and Statistics
DA - 2008/5//
PB - EDP Sciences
VL - 12
SP - 273
EP - 307
AB - In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided.
LA - eng
KW - Functional estimate; central limit theorem; moderate deviation principles; extreme values; shape estimation; functional estimate
UR - http://eudml.org/doc/250410
ER -

References

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  1. J.A. Adell and P. Jodrá, The median of the Poisson distribution. Metrika 613 (2005) 337–346.  
  2. P. Baufays and J.-P. Rasson, A new geometric discriminant rule. Comput. Stat. Q.2 (1985) 15–30.  
  3. P. Billingsley, Convergence of Probability measures. Wiley (1968).  
  4. 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).  
  5. J.D. Deuschel and D.W. Stroock, Large Deviations. Pure and Applied Mathematics, 137. Boston, MA Academic Press (1989).  
  6. L.P. Devroye and G.L. Wise, Detection of abnormal behavior via non parametric estimation of the support. SIAM J. Appl. Math.38 (1980) 448–480.  
  7. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Bartlett, Boston and London (1993).  
  8. 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 ParisXLVI 43–72 (2002).  
  9. J. Geffroy, Sur un problème d'estimation géométrique. Publications de l'Institut de Statistique de l'Université de ParisXIII (1964) 191–200.  
  10. I. Gijbels, E. Mammen, B.U. Park and L. Simar, On estimation of monotone and concave frontier functions. J. Amer. Statist. Assoc.94 (1999) 220–228.  
  11. S. Girard and P. Jacob, Projection estimates of point processes boundaries. J. Statist. Planning Inference116 (2003), 1–15.  
  12. S. Girard and P. Jacob, Extreme values and kernel estimates of point processes boundaries. ESAIM: PS8 (2005) 150–168 .  
  13. S. Girard and L. Menneteau, Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries. J. Statist. Planning Inference135 (2005) 433–460.  
  14. S. Girard and L. Menneteau, Smoothed extreme value estimators of non uniform boundaries with applications to star-shaped supports estimation. Submitted. 
  15. A. Hardy and J.P. Rasson, Une nouvelle approche des problèmes de classification automatique. Statist. Anal. Données7 (1982) 41–56.  
  16. P. Hall, M. Nussbaum and S.E. Stern, On the estimation of a support curve of indeterminate sharpness. J. Multivariate Anal.62 (1997) 204–232.  
  17. P. Hall, B.U. Park and S.E. Stern, On polynomial estimators of frontiers and boundaries. J. Multivariate Anal.66 (1998) 71–98.  
  18. W. Härdle, Applied nonparametric regression. Cambridge University Press, Cambridge (1990).  
  19. W. Härdle, P. Hall and L. Simar, Iterated bootstrap with application to frontier models. J. Productivity Anal.6 (1995) 63–76.  
  20. W. Härdle, B.U. Park and A.B. Tsybakov, Estimation of a non sharp support boundaries. J. Multivariate Anal.43 (1995) 205–218.  
  21. J.A. Hartigan, Clustering Algorithms. Wiley, Chichester (1975).  
  22. W. Kallenberg, Intermediate efficiency theory and examples. Ann. Statist.11 (1983) 170–182.  
  23. W. Kallenberg, On moderate deviation theory in estimation. Ann. Statist.11 (1983) 498–504.  
  24. A.P. Korostelev, L. Simar and A.B. Tsybakov, Efficient estimation of monotone boundaries. Ann. Statist.23 (1995) 476–489.  
  25. A.P. Korostelev and A.B. Tsybakov, Minimax theory of image reconstruction, in Lecture Notes in Statistics82, Springer-Verlag, New York (1993).  
  26. A.P. Korostelev and A.B. Tsybakov, Asymptotic efficiency of the estimation of a convex set. Problems Inform. Transmission30 (1994) 317–327.  
  27. E. Mammen and A.B. Tsybakov, Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist.23 (1995) 502–524.  
  28. L. Menneteau, Limit theorems for piecewise constant kernel smoothed estimates of point process boundaries. Technical Report (2007).  
  29. A. Mokkadem and M. Pelletier, Moderate deviations for the kernel mode estimator and some applications. J. Statist. Planning Inference135 (2005) 276–299.  
  30. V.V. Petrov, Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability, (1995) 4.  
  31. G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley, New York (1986).  
  32. G.P. Tolstov, Fourier series. 2nd ed. New York: Dover Publications (1976).  
  33. A.B. Tsybakov, On nonparametric estimation of density level sets. Ann. Statist.25 (1997) 948–969.  

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