We state a pointwise central limit theorem for the linear wavelet density estimator in a more general setting than the result of Wu [12]. Furthermore, we also give a pointwise law of the iterated logarithm for this density estimator. Our proof of the law of the iterated logarithm uses the results of Mason [9] on the asymptotic behavior of the tail empirical process.
Nous énonçons une généralisation du théorème central limite ponctuel de Wu [12] pour l'estimateur linéaire par méthode d'ondelettes. Nous présentons également une loi du logarithme itéré ponctuelle pour ce même estimateur. Pour établir cette loi du logarithme itéré, nous utilisons les résultats de Mason [9] sur le comportement asymptotique du processus empirique de queue.
Revised:
Published online:
Anne Massiani 1
@article{CRMATH_2002__335_6_553_0, author = {Anne Massiani}, title = {\'Etude asymptotique locale de l'estimateur par ondelettes}, journal = {Comptes Rendus. Math\'ematique}, pages = {553--556}, publisher = {Elsevier}, volume = {335}, number = {6}, year = {2002}, doi = {10.1016/S1631-073X(02)02519-0}, language = {fr}, }
Anne Massiani. Étude asymptotique locale de l'estimateur par ondelettes. Comptes Rendus. Mathématique, Volume 335 (2002) no. 6, pp. 553-556. doi : 10.1016/S1631-073X(02)02519-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02519-0/
[1] Wavelet methods for curve estimation, J. Amer. Statist. Assoc., Volume 89 (1994), pp. 1340-1353
[2] Probability and Measure, Wiley, 1979
[3] Ten Lectures on Wavelets, SIAM, Philadelphia, 1992
[4] Functional laws of the iterated logarithm for the increments of empirical and quantile processes, Ann. Probab., Volume 20 (1992), pp. 1248-1287
[5] Fonctional laws of the iterated logarithm for local empirical processes indexed by sets, Ann. Probab., Volume 22 (1994), pp. 1619-1661
[6] Laws of the iterated logarithm for nonparametric density estimator, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, Volume 56 (1981), pp. 47-61
[7] Wavelets, Approximation, and Statistical Applications, Springer-Verlag, New York, 1998
[8] An approximation of partial sums of independent rv's and the sample df. , Z. Wahrscheinlichkeitstheorie Verw. Gebiete, Volume 32 (1975), pp. 111-131
[9] A strong invariance theorem for the tail empirical process, Ann. Inst. H. Poincaré, Volume 24 (1988), pp. 491-506
[10] A. Massiani, Pointwise asymptotic behavior of linear wavelet density estimator, Prépublication LSTA, 2002
[11] Ondelettes : Ondelettes et Opérateurs I, Hermann, Paris, 1990
[12] Asymptotic normality of the multiscale wavelet density estimator, Comm. Statist. Theory Methods, Volume 25 (1996), pp. 1957-1970
[13] On the asymptotic normality for L2-error of wavelet density estimator with application, Comm. Statist. Theory Methods, Volume 28 (1999), pp. 1093-1104
Cited by Sources:
Comments - Policy