Geodesic
Empirical estimator of the regularity index of a probability measure
Kybernetika, Tome 48 (2012) no. 4, pp. 589-599 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The index of regularity of a measure was introduced by Beirlant, Berlinet and Biau [1] to solve practical problems in nearest neighbour density estimation such as removing bias or selecting the number of neighbours. These authors proved the weak consistency of an estimator based on the nearest neighbour density estimator. In this paper, we study an empirical version of the regularity index and give sufficient conditions for its weak and strong convergence without assuming absolute continuity or other global properties of the underlying measure.
The index of regularity of a measure was introduced by Beirlant, Berlinet and Biau [1] to solve practical problems in nearest neighbour density estimation such as removing bias or selecting the number of neighbours. These authors proved the weak consistency of an estimator based on the nearest neighbour density estimator. In this paper, we study an empirical version of the regularity index and give sufficient conditions for its weak and strong convergence without assuming absolute continuity or other global properties of the underlying measure.
Classification : 62G05
Keywords: regularity index; Lebesgue point; small ball probability 
@article{KYB_2012_48_4_a1,
     author = {Berlinet, Alain and Servien, R\'emi},
     title = {Empirical estimator of the regularity index of a probability measure},
     journal = {Kybernetika},
     pages = {589--599},
     year = {2012},
     volume = {48},
     number = {4},
     mrnumber = {3013392},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2012_48_4_a1/}
}
TY  - JOUR
AU  - Berlinet, Alain
AU  - Servien, Rémi
TI  - Empirical estimator of the regularity index of a probability measure
JO  - Kybernetika
PY  - 2012
SP  - 589
EP  - 599
VL  - 48
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/KYB_2012_48_4_a1/
LA  - en
ID  - KYB_2012_48_4_a1
ER  - 
%0 Journal Article
%A Berlinet, Alain
%A Servien, Rémi
%T Empirical estimator of the regularity index of a probability measure
%J Kybernetika
%D 2012
%P 589-599
%V 48
%N 4
%U http://geodesic.mathdoc.fr/item/KYB_2012_48_4_a1/
%G en
%F KYB_2012_48_4_a1
Berlinet, Alain; Servien, Rémi. Empirical estimator of the regularity index of a probability measure. Kybernetika, Tome 48 (2012) no. 4, pp. 589-599. http://geodesic.mathdoc.fr/item/KYB_2012_48_4_a1/

[1] J. Beirlant, A. Berlinet, G. Biau: Higher order estimation at Lebesgue points. Ann. Inst. Statist. Math. 60 (2008), 651-677. | DOI | MR | Zbl

[2] A. Berlinet, S. Levallois: Higher order analysis at Lebesgue points. In: G. G. Roussas Festschrift - Asymptotics in Statistics and Probability (M. L. Puri, ed.), 2000, pp. 17-32.

[3] A. Berlinet, R. Servien: Necessary and sufficient condition for the existence of a limit distribution of the nearest neighbour density estimator. J. Nonparametr. Statist. 23 (2011), 633-643. | DOI | MR

[4] L. Devroye, G. Lugosi: Combinatorial Methods in Density Estimation. Springer, New York 2001. | MR | Zbl

[5] R. M. Dudley: Real Analysis and Probability. Chapman and Hall, New York 1989. | MR | Zbl

[6] W. Rudin: Real and Complex Analysis. McGraw-Hill, New York 1987. | MR | Zbl