Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 273-307.

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.

DOI : https://doi.org/10.1051/ps:2007039
Classification : Primary 60G70,  Secondary 62M30,  62G05
Mots clés : functional estimate, central limit theorem, moderate deviation principles, extreme values, shape estimation
@article{PS_2008__12__273_0,
     author = {Menneteau, Ludovic},
     title = {Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries},
     journal = {ESAIM: Probability and Statistics},
     pages = {273--307},
     publisher = {EDP-Sciences},
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     year = {2008},
     doi = {10.1051/ps:2007039},
     mrnumber = {2404032},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007039/}
}
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Menneteau, Ludovic. Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 273-307. doi : 10.1051/ps:2007039. http://www.numdam.org/articles/10.1051/ps:2007039/

[1] J.A. Adell and P. Jodrá, The median of the Poisson distribution. Metrika 61 3 (2005) 337-346. | MR 2230380 | Zbl 1079.62014

[2] P. Baufays and J.-P. Rasson, A new geometric discriminant rule. Comput. Stat. Q. 2 (1985) 15-30. | Zbl 0616.62084

[3] P. Billingsley, Convergence of Probability measures. Wiley (1968). | MR 233396 | Zbl 0172.21201

[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). | MR 997938 | Zbl 0705.60029

[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. | MR 579432 | Zbl 0479.62028

[7] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Bartlett, Boston and London (1993). | MR 1202429 | Zbl 0793.60030

[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 Paris XLVI 43-72 (2002). | MR 1925608 | Zbl 1053.62092

[9] J. Geffroy, Sur un problème d'estimation géométrique. Publications de l'Institut de Statistique de l'Université de Paris XIII (1964) 191-200. | MR 202237 | Zbl 0129.32301

[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. | MR 1689226 | Zbl 1043.62105

[11] S. Girard and P. Jacob, Projection estimates of point processes boundaries. J. Statist. Planning Inference 116 (2003), 1-15. | MR 1997128 | Zbl 1023.62098

[12] S. Girard and P. Jacob, Extreme values and kernel estimates of point processes boundaries. ESAIM: PS 8 (2005) 150-168 . | Numdam | MR 2085612 | Zbl 1154.60330

[13] S. Girard and L. Menneteau, Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries. J. Statist. Planning Inference 135 (2005) 433-460. | MR 2200479 | Zbl 1094.60039

[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. Données 7 (1982) 41-56. | Numdam | MR 685752 | Zbl 0505.62040

[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. | MR 1473874 | Zbl 0890.62029

[17] P. Hall, B.U. Park and S.E. Stern, On polynomial estimators of frontiers and boundaries. J. Multivariate Anal. 66 (1998) 71-98. | MR 1648521 | Zbl 1127.62358

[18] W. Härdle, Applied nonparametric regression. Cambridge University Press, Cambridge (1990). | MR 1161622 | Zbl 0714.62030

[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. | Zbl 0863.62030

[21] J.A. Hartigan, Clustering Algorithms. Wiley, Chichester (1975). | MR 405726 | Zbl 0372.62040

[22] W. Kallenberg, Intermediate efficiency theory and examples. Ann. Statist. 11 (1983) 170-182. | MR 684874 | Zbl 0512.62057

[23] W. Kallenberg, On moderate deviation theory in estimation. Ann. Statist. 11 (1983) 498-504. | MR 696062 | Zbl 0515.62027

[24] A.P. Korostelev, L. Simar and A.B. Tsybakov, Efficient estimation of monotone boundaries. Ann. Statist. 23 (1995) 476-489. | MR 1332577 | Zbl 0829.62043

[25] A.P. Korostelev and A.B. Tsybakov, Minimax theory of image reconstruction, in Lecture Notes in Statistics 82, Springer-Verlag, New York (1993). | MR 1226450 | Zbl 0833.62039

[26] A.P. Korostelev and A.B. Tsybakov, Asymptotic efficiency of the estimation of a convex set. Problems Inform. Transmission 30 (1994) 317-327. | MR 1310060 | Zbl 0926.94007

[27] E. Mammen and A.B. Tsybakov, Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 (1995) 502-524. | MR 1332579 | Zbl 0834.62038

[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 Inference 135 (2005) 276-299. | MR 2200470 | Zbl 1074.62031

[30] V.V. Petrov, Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability, (1995) 4. | MR 1353441 | Zbl 0826.60001

[31] G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley, New York (1986). | MR 838963

[32] G.P. Tolstov, Fourier series. 2nd ed. New York: Dover Publications (1976). | MR 425474 | Zbl 0358.42001

[33] A.B. Tsybakov, On nonparametric estimation of density level sets. Ann. Statist. 25 (1997) 948-969. | MR 1447735 | Zbl 0881.62039

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