Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T09:49:38.136Z Has data issue: false hasContentIssue false
  • English
  • Français

The law of the iterated logarithm for the multivariate kernel modeestimator

Published online by Cambridge University Press:  15 May 2003

Abdelkader Mokkadem  and
Mariane Pelletier
Abdelkader Mokkadem
Affiliation:
Département de Mathématiques, Université de Versailles-Saint-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; mokkadem@math.uvsq.fr.pelletier@math.uvsq.fr.
Mariane Pelletier
Affiliation:
Département de Mathématiques, Université de Versailles-Saint-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; mokkadem@math.uvsq.fr.pelletier@math.uvsq.fr.
Get access

Abstract

Let θ be the mode of a probability density and θn its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θn - θ. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θn - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], of θn - θ. Finally, we consider the case θ is degenerate and give the exact weak and strong convergence rate of θn - θ in the univariate framework.

Keywords

Densitymodekernel estimatorcentral limit theoremlaw of the iterated logarithm.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 1 - 21
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arcones, M.A., The law of the iterated logarithm for a triangular array of empirical processes. Electron. J. Probab. 2 (1997) 1-39. CrossRef
Berlinet, A., Gannoun, A. and Matzner-Loeber, E., Normalité asymptotique d'estimateurs convergents du mode conditionnel. Can. J. Statist. 26 (1998) 365-380. CrossRef
Chernoff, H., Estimation of the mode. Ann. Inst. Stat. Math. 16 (1964) 31-41. CrossRef
Collomb, G., Härdle, W. and Hassani, S., A note on prediction via estimation of the conditional mode function. J. Statist. Planning Inference 15 (1987) 227-236. CrossRef
Eddy, W.F., Optimum kernel estimates of the mode. Ann. Statist. 8 (1980) 870-882. CrossRef
Eddy, W.F., The asymptotic distributions of kernel estimators of the mode. Z. Warsch. Verw. Geb. 59 (1982) 279-290. CrossRef
Einmahl, U. and Mason, D.M., An empirical process approach to the uniform consistency of kernel-type functions estimators. J. Theoret. Probab. 13 (2000) 1-37. CrossRef
E. Giné and A. Guillou, Rates of strong uniform consistency for multivariate kernel density estimators, Preprint. Paris VI (2000).
Grenander, U., Some direct estimates of the mode. Ann. Math. Statist. 36 (1965) 131-138. CrossRef
Grund, B. and Hall, P., On the minimisation of L p error in mode estimation. Ann. Statist. 23 (1995) 2264-2284.
Hall, P., Laws of the iterated logarithm for nonparametric density estimators. Z. Warsch. Verw. Geb. 56 (1981) 47-61. CrossRef
Hall, P., Asymptotic theory of Grenander's mode estimator. Z. Warsch. Verw. Geb. 60 (1982) 315-334. CrossRef
Konakov, V.D., On asymptotic normality of the sample mode of multivariate distributions. Theory Probab. Appl. 18 (1973) 836-842.
Leclerc, J. and Pierre-Loti-Viaud, D., Vitesse de convergence presque sûre de l'estimateur à noyau du mode. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 637-640. CrossRef
Louani, D. and Ould-Said, E., Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis. J. Nonparametr. Statist. 11 (1999) 413-442. CrossRef
A. Mokkadem and M. Pelletier, A law of the iterated logarithm for the kernel mode estimator. Statist. Probab. Lett. (submitted).
Nadaraya, E.A., On non-parametric estimates of density functions and regression curves. Theory Probab. Appl. 10 (1965) 186-190. CrossRef
Ould-Said, E., A note on ergodic processes prediction via estimation of the conditional mode function. Scand. J. Stat. 24 (1997) 231-239. CrossRef
Parzen, E., On estimating probability density function and mode. Ann. Math. Statist. 33 (1962) 1065-1076. CrossRef
D. Pollard, Convergence of Stochastic Processes. Springer, New York (1984).
Quintela-Del-Rio, A. and Vieu, P., A nonparametric conditional mode estimate. J. Nonparametr. Statist. 8 (1997) 253-266. CrossRef
Romano, J., On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist. 16 (1988) 629-647. CrossRef
Rüschendorf, L., Consistency of estimators for multivariate density functions and for the mode. Sankhya Ser. A 39 (1977) 243-250.
Sager, T.W., Consistency in nonparametric estimation of the mode. Ann. Statist. 3 (1975) 698-706. CrossRef
Samanta, M., Nonparametric estimation of the mode of a multivariate density. South African Statist. J. 7 (1973) 109-117.
M. Samanta and A. Thavaneswaran, Nonparametric estimation of the conditional mode. Commun Stat., Theory Methods 19 (1990) 4515-4524.
Tsybakov, A.B., Recurrent estimation of the mode of a multidimensional distribution. Problems Inform. Transmission 26 (1990) 31-37.
Van Ryzin, J., On strong consistency of density estimates. Ann. Math. Statist. 40 (1969) 1765-1772. CrossRef
Venter, J.H., On estimation of the mode. Ann. Math. Statist. 38 (1967) 1446-1455. CrossRef
Vieu, P., A note on density mode estimation. Statist. Probab. Lett. 26 (1996) 297-307. CrossRef
Yamato, H., Sequential estimation of a continuous probability density function and the mode. Bull. Math. Statist. 14 (1971) 1-12.