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.  Zbl1079.62014
  2. P. Baufays and J.-P. Rasson, A new geometric discriminant rule. Comput. Stat. Q.2 (1985) 15–30.  Zbl0616.62084
  3. P. Billingsley, Convergence of Probability measures. Wiley (1968).  Zbl0172.21201
  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.  Zbl0479.62028
  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).  Zbl1053.62092
  9. J. Geffroy, Sur un problème d'estimation géométrique. Publications de l'Institut de Statistique de l'Université de ParisXIII (1964) 191–200.  Zbl0129.32301
  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.  Zbl1043.62105
  11. S. Girard and P. Jacob, Projection estimates of point processes boundaries. J. Statist. Planning Inference116 (2003), 1–15.  Zbl1023.62098
  12. S. Girard and P. Jacob, Extreme values and kernel estimates of point processes boundaries. ESAIM: PS8 (2005) 150–168 .  Zbl1154.60330
  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.  Zbl1094.60039
  14. S. Girard and L. Menneteau, Smoothed extreme value estimators of non uniform boundaries with applications to star-shaped supports estimation. Submitted. Zbl1135.62038
  15. A. Hardy and J.P. Rasson, Une nouvelle approche des problèmes de classification automatique. Statist. Anal. Données7 (1982) 41–56.  Zbl0505.62040
  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.  Zbl0890.62029
  17. P. Hall, B.U. Park and S.E. Stern, On polynomial estimators of frontiers and boundaries. J. Multivariate Anal.66 (1998) 71–98.  Zbl1127.62358
  18. W. Härdle, Applied nonparametric regression. Cambridge University Press, Cambridge (1990).  Zbl0714.62030
  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.  Zbl0863.62030
  21. J.A. Hartigan, Clustering Algorithms. Wiley, Chichester (1975).  Zbl0372.62040
  22. W. Kallenberg, Intermediate efficiency theory and examples. Ann. Statist.11 (1983) 170–182.  Zbl0512.62057
  23. W. Kallenberg, On moderate deviation theory in estimation. Ann. Statist.11 (1983) 498–504.  Zbl0515.62027
  24. A.P. Korostelev, L. Simar and A.B. Tsybakov, Efficient estimation of monotone boundaries. Ann. Statist.23 (1995) 476–489.  Zbl0829.62043
  25. A.P. Korostelev and A.B. Tsybakov, Minimax theory of image reconstruction, in Lecture Notes in Statistics82, Springer-Verlag, New York (1993).  Zbl0833.62039
  26. A.P. Korostelev and A.B. Tsybakov, Asymptotic efficiency of the estimation of a convex set. Problems Inform. Transmission30 (1994) 317–327.  Zbl0926.94007
  27. E. Mammen and A.B. Tsybakov, Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist.23 (1995) 502–524.  Zbl0834.62038
  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.  Zbl1074.62031
  30. V.V. Petrov, Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability, (1995) 4.  Zbl0826.60001
  31. G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley, New York (1986).  Zbl1170.62365
  32. G.P. Tolstov, Fourier series. 2nd ed. New York: Dover Publications (1976).  Zbl0358.42001
  33. A.B. Tsybakov, On nonparametric estimation of density level sets. Ann. Statist.25 (1997) 948–969.  Zbl0881.62039

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