Services on Demand
Journal
Article
Spanish (pdf)
Article in xml format
Article references
Automatic translation
Send this article by e-mailIndicators
-
Cited by SciELO -
Access statistics
Related links
-
Cited by Google -
Similars in
SciELO -
Similars in Google
Share
Permalink
Revista Colombiana de Estadística
Print version ISSN 0120-1751
Rev.Colomb.Estad. vol.32 no.2 Bogotá July/Dec. 2009
1Subdirección Salud Pública de Gipuzkoa, CIBER de Epidemiología y Salud Pública (CIBERESP), Donostia, Spain. Investigador postdosctoral. Email: pmcamblor@hotmail.com
2Universidad de Oviedo, Estadística e Investigación Operativa y Didáctica de la Matemática, Asturias, Spain. Catedrático. Email: norbert@uniovi.es
3Universidad de Oviedo, Estadística e Investigación Operativa y Didáctica de la Matemática, Asturias, Spain. Profesora titular. Email: teresa@uniovi.es
In this paper a Chernoff type theorem for the L1 distance between kernel estimators from two independent and identically distributed random samples is developed. The harmonic mean is used to correct the distance for inequal sample sizes case. Moreover, the proved result is used to compute the Bahadur slope of a test based on L1 distance and to compare it with the classical nonparametric Mann-Whitney test by using the Bahadur relative efficiency.
Palabras clave: Kernel estimator, Large deviation, Bahadur slope.
En este trabajo se desarrolla un teorema de tipo Chernoff para la distancia L1 entre estimadores núcleo procedentes de muestras aleatorias independientes e idénticamente distribuidas. Se usa la media armónica para corregir esta distancia en el caso de muestras de distintos tamaños. Además, se usa el resultado demostrado para el cálculo de la pendiente de Bahadur de un test para la comparación de densidades basado en la distancia L1 y se compara con el clásico test de Mann-Whitney a partir de la eficiencia relativa de Bahadur.
Key words: estimador núcleo, grandes muestras, pendiente de Bahadur.
Texto completo disponible en PDF
Referencias
1. Bahadur, R. R. & Zabell, S. L. (1979), 'Large Deviations of the Sample Mean in General Vector Space.', Annals of Probability 7(4), 587-621. [ Links ]
2. Beirlant, L., Devroye, L., Györfi, L. & Vajda, A. (2001), 'Large Deviations of Divergence Measures on Partitions', Journal of Statistical Planning and Inference 93, 1-16. [ Links ]
3. Berlinet, A., Devroye, L. & Györfi, L. (1995), 'Asymptotic Normality of the L_1-error in the Histogram Density Estimation', Statistics 26, 329-343. [ Links ]
4. Cao, R. & Lugosi, G. (2005), 'Goodness-of-fit Tests Based on teh Kernel Density Estimate', Scandinavian Journal of Statistics 32, 599-617. [ Links ]
5. Cao, R. & Van Keilegom, I. (2006), 'Empirical Likelihood Tests for Two-Sample Problems via Nonparametric Density Estimation', Canadian Journal of Statistics 34, 61-77. [ Links ]
6. Devroye, L. (1983), 'The Equivalence of Weak, Strong and Complete Convergence in L_1 for Kernel Density Estimates', Annals of Statistics 11, 896-904. [ Links ]
7. Devroye, L. (1987), A course in Density Estimation, Birkhauser, Boston, United States. [ Links ]
8. Devroye, L. & Gyorfi, L. (1985), Nonparametric Density Estimation. The L1-View, Wiley, New York, United States. [ Links ]
9. Devroye, L. & Wagner, T. J. (1979), 'The L_1 Convergence of Kernel Density Estimates', Annals of Statistics 7, 1136-1139. [ Links ]
10. Hórvath, L. (1991), 'On L_p-Norms of Multivariate Density Estimations', Annals of Statistics 19(4), 1933-1949. [ Links ]
11. Konakov, V. (1978), 'Complete Asymptotic Expansions for the Maximun Deviation of the Empirical Density Function', Theory Probability Applied 28, 495-509. [ Links ]
12. Louani, D. (1998), 'Large Deviations Limit Theorems for the Kernel Density Estimator', Scandinavian Journal of Statistics 25(1), 243-253. [ Links ]
13. Louani, D. (2000), 'Large Deviation for L_1-Distance in Kernel Density Estimation', Journal of Statistical Planning and Inference 90, 177-182. [ Links ]
14. Louani, D. (2005), 'Uniform L_1-Distance Large Deviations for Nonparametric Density Estimation', Test 14, 1-24. [ Links ]
15. Martínez-Camblor, P. (2008), 'Tests de hipótesis para contrastar la igualdad entre k poblaciones', Revista Colombiana de Estadística 31(1), 1-18. [ Links ]
16. Martínez-Camblor, P. & Corral, N. (2008), 'Weaker Conditions for Asymptotic Approximation to L_P-norms of the Kernel Estimators', InterSTAT Journal june, 1-18. [ Links ]
17. Osmoukhina, A. V. (2001), 'Large Deviations Probabilities for a Test of Symmetry Based on Kernel Density Estimator', Statistics and Probability Letters 54(4), 363-371. [ Links ]
18. Parzen, E. (1962), 'On Estimation of a Probability Density Function and Mode', Annals of Mathematical Statistics 33, 1065-1076. [ Links ]
19. Rosenblatt, M. (1956), 'Remarks on Some Nonparametric Estimates of a Density Functions', Annals of Mathematical Statistics 27, 832-837. [ Links ]
20. Silverman, B. W. (1978), 'Weak and strong uniform consistency of the kernel estimate of a density and its derivates', Annals of Statistics 6(1), 177-184. [ Links ]
21. Van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge University Press, London, England. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv32n2a07,
AUTHOR = {Martínez-Camblor, Pablo and Corral, Norberto and López, Teresa},
TITLE = {{Cramér-Chernoff Theorem for L1-norm in Kernel Density Estimator for Two Independent Samples}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2009},
volume = {32},
number = {2},
pages = {289-299}
} 
