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Uniform in bandwidth consistency of kernel estimators of the density of mixed data. (English) Zbl 1333.60056

Summary: We establish a general uniform in bandwidth consistency result for kernel estimators of the unconditional and conditional joint density of a distribution, which is defined by a mixed discrete and continuous random variable.

MSC:

60F15 Strong limit theorems
62G07 Density estimation
62G08 Nonparametric regression and quantile regression
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[1] Aitchison, J. and Aitken, C. G. G. (1976). Multivariate binary discrimination by the kernel method., Biometrika 63 413-420. · Zbl 0344.62035
[2] Brown, P. J. and Rundell, P. W. K. (1985). Kernel estimates for categorical data., Technometrics 27 293-299. · Zbl 0611.62058
[3] Burman, P. (1987). Smoothing sparse contingency tables., Sankhyā, Ser. A 49 24-36. · Zbl 0639.62050
[4] Deheuvels, P. and Mason, D. M. (2004). General asymptotic confidence bands based on kernel-type function estimators., Stat. Inference Stoch. Process. 7 225-277. · Zbl 1125.62314
[5] Dony, J., Einmahl, U. and Mason, D. M. (2006). Uniform in bandwidth consistency of local polynomial regression function estimators., Aust. J. Stat. 35 105-120.
[6] Einmahl, U. and Mason, D. M. (1997). Gaussian approximation of local empirical processes indexed by functions., Probab. Theory Rel. 107 283-311. · Zbl 0878.60025
[7] Einmahl, U. and Mason, D. M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators., J. Theor. Probab. 13 1-37. · Zbl 0995.62042
[8] Einmahl, U. and Mason, D. M. (2005). Uniform in bandwidth consistency of kernel-type function estimators., Ann. Stat. 33 1380-1403. · Zbl 1079.62040
[9] Hall, P., Racine, J. and Li, Q. (2004). Cross-validation and the estimation of conditional probability densities., J. Am. Stat. Assoc. 99 1015-1026. · Zbl 1055.62035
[10] Hall, P. and Titterington, D. M. (1987). On smoothing sparse multinomial data., Aust. J. Stat. 29 19-37. · Zbl 0628.62039
[11] Li, Q. and Racine, J. (2003). Nonparametric estimation of distributions with categorical and continuous data., J. Multivariate Anal. 86 266-292. · Zbl 1019.62030
[12] Mason, D. M. (2012). Proving consistency of non-standard kernel estimators., Stat. Inference Stoch. Process. 15 151-176. · Zbl 1242.62028
[13] Mason, D. M. and Swanepoel, J. W. H. (2011). A general result on the uniform in bandwidth consistency of kernel-type function estimators., Test 20 72-94. · Zbl 1331.62235
[14] Nolan, D. and Marron, J. S. (1989). Uniform consistency of automatic and location-adaptive delta-sequence estimators., Probab. Theory Rel. 80 619-632. · Zbl 0644.62041
[15] Nolan, D. and Pollard, D. (1987). U-processes: Rates of convergence., Ann. Stat. 15 780-799. · Zbl 0624.60048
[16] Ouyang, D., Li, Q. and Racine, J. (2006). Cross-validation and the estimation of probability distributions with categorical data., J. Nonparametric Stat. 18 69-100. · Zbl 1087.62051
[17] Racine, J. (2008). Nonparametric econometrics: A primer., Foundations and Trends in Econometrics 3 1-88.
[18] Simonoff, J. S. (1996)., Smoothing Methods in Statistics . Springer-Verlag, New York. · Zbl 0859.62035
[19] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. With Applications to Statistics . Springer Series in Statistics . Springer-Verlag, New York. · Zbl 0862.60002
[20] Wang, M. C. andvan Ryzin, J. A. (1981). A class of smooth estimators for discrete distributions., Biometrika 68 301-309. · Zbl 0483.62027
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