Dans cette Note, nous présentons le polygone des fréquences comme estimateur de la densité pour des champs aléatoires indexés sur un treillis. Nous déterminons la largeur de cellule optimale qui minimise asymptotiquement l'erreur moyenne quadratique intégrée (IMSE). On montre également que, sous des conditions très générales, le polygone des fréquences atteint la même vitesse de convergence pour l'IMSE que les estimateurs à noyaux. Il peut aussi atteindre la vitesse optimale de la convergence uniforme sous des conditions générales. En conséquence, du point de vue de la convergence uniforme ou de l'IMSE, le polygone des fréquences est un très bon estimateur de la densité.
The purpose of this Note is to investigate the frequency polygon as a density estimator for stationary random fields indexed by multidimensional lattice points space. Optimal bin widths which asymptotically minimize integrated errors (IMSE) are derived. Under mild regularity assumptions, frequency polygons achieve the same rate of convergence to zero of the IMSE as kernel estimators. They can also attain the rate of uniform convergence under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence.
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Michel Carbon 1
@article{CRMATH_2006__342_9_693_0, author = {Michel Carbon}, title = {Polygone des fr\'equences pour des champs al\'eatoires}, journal = {Comptes Rendus. Math\'ematique}, pages = {693--696}, publisher = {Elsevier}, volume = {342}, number = {9}, year = {2006}, doi = {10.1016/j.crma.2006.02.019}, language = {fr}, }
Michel Carbon. Polygone des fréquences pour des champs aléatoires. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 693-696. doi : 10.1016/j.crma.2006.02.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.019/
[1] Spatial kernel density estimation, Math. Methods Statist., Volume 12 (2003), pp. 371-390
[2] On the central limit theorem for stationary random fields, Ann. Probab., Volume 10 (1982), pp. 1047-1050
[3] M. Carbon, Frequency Polygons for Random Fields. Lucarne bleue, CREST, 2005-04, 2005
[4] Frequency polygon for weakly dependent processes, Statist. Probab. Lett., Volume 33 (1997), pp. 1-13
[5] Kernel density estimation for random fields: the
[6] Kernel density estimation for random fields (Density estimation for random fields), Statist. Probab. Lett., Volume 36 (1997), pp. 115-125
[7] X. Guyon, Estimation d'un champ par pseudo-vraisemblance conditionnelle : Etude asymptotique et application au cas Markovien, in : Proc. 6th Franco-Belgian Meeting of Statisticians, 1987
[8] Vitesse de convergence du théorème de la limite centrale pour des champs faiblement dépendants, Z. Wahrsch. Verw. Gebiete (1984), pp. 297-314
[9] Density estimation for spatial linear processes, Bernouilli, Volume 7 (2001), pp. 657-688
[10] Kernel density estimation for spatial linear processes: the
[11] Local linear spatial regression, Ann. Statist., Volume 32 (2004) no. 6, pp. 2469-2500
[12] Convergence of block spins defined on random fields, J. Statist. Phys., Volume 22 (1980), pp. 673-684
[13] Stationary Sequences and Random Fields, Birkhäuser, Boston, 1985
[14] Frequency polygons, theory and applications, J. Amer. Statist. Assoc., Volume 80 (1985), pp. 348-354
[15] Optimal uniform rate of convergence for non parametric estimators of a density function and its derivative (M.H. Revzi; J.S. Rustagi; D. Siegmund, eds.), Recent Advances in Statistics: Papers in Honor of H. Chernoff, 1983, pp. 393-406
[16] On the rates in the central limit theorem for weakly dependent random fields, Z. Wahrsch. Verw. Gebiete, Volume 62 (1983), pp. 477-480
[17] Kernel density estimation on random fields, J. Multivariate Anal., Volume 34 (1990), pp. 37-53
[18] Nearest neighbor estimators for random fields, J. Multivariate Anal., Volume 44 (1993), pp. 23-46
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- Kernel spatial density estimation in infinite dimension space, Metrika, Volume 76 (2013) no. 1, pp. 19-52 | DOI:10.1007/s00184-011-0374-4 | Zbl:1256.62016
- On the asymptotic normality of frequency polygons for strongly mixing spatial processes, Statistical Inference for Stochastic Processes, Volume 16 (2013) no. 3, pp. 193-206 | DOI:10.1007/s11203-013-9086-x | Zbl:1292.62054
- Asymptotic normality of frequency polygons for random fields, Journal of Statistical Planning and Inference, Volume 140 (2010) no. 2, pp. 502-514 | DOI:10.1016/j.jspi.2009.07.028 | Zbl:1177.62111
- Frequency polygons for continuous random fields, Statistical Inference for Stochastic Processes, Volume 13 (2010) no. 1, p. 55 | DOI:10.1007/s11203-009-9038-7
- Kernel regression estimation for continuous spatial processes, Mathematical Methods of Statistics, Volume 16 (2007) no. 4, pp. 298-317 | DOI:10.3103/s1066530707040023 | Zbl:1140.62071
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