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Central limit theorem for asymmetric kernel functionals. (English) Zbl 1095.62041
Summary: Asymmetric kernels are quite useful for the estimation of density functions with bounded support. Gamma kernels are designed to handle density functions whose supports are bounded from one end only, whereas beta kernels are particularly convenient for the estimation of density functions with compact support. These asymmetric kernels are nonnegative and free of boundary bias. Moreover, their shape varies according to the location of the data point, thus also changing the amount of smoothing. This paper applies the central limit theorem for degenerate U-statistics to compute the limiting distribution of a class of asymmetric kernel functionals.

MSC:
62G07 Density estimation
60F05 Central limit and other weak theorems
62G20 Asymptotic properties of nonparametric inference
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[1] Aït-Sahalia, Y., Bickel, P. J. and Stoker, T. M. (2001). Goodness-of-fit tests for kernel regression with an application to option-implied volatilities,Journal of Econometrics,105, 363–412. · Zbl 1004.62042 · doi:10.1016/S0304-4076(01)00091-4
[2] Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates,Annals of Statistics,1, 1071–1095. · Zbl 0275.62033 · doi:10.1214/aos/1176342558
[3] Brown, B. M. and Chen, S. X. (1999). Beta-Bernstein smoothing for regression curves with compact support,Scandinavian Journal of Statistics,26, 47–59. · Zbl 0921.62048 · doi:10.1111/1467-9469.00136
[4] Chen, S. X. (1999). Beta kernel estimators for density functions,Computational Statistics and Data Analysis,31, 131–145. · Zbl 0935.62042 · doi:10.1016/S0167-9473(99)00010-9
[5] Chen, S. X. (2000). Probability density function estimation using gamma kernels,Annals of the Institute of Statistical Mathematics,52, 471–480. · Zbl 0960.62038 · doi:10.1023/A:1004165218295
[6] Chen, S. X., Härdle, W. and Li, M. (2003). An empirical likelihood goodness-of-fit test for time series,Journal of the Royal Statistical Society B,65, 663–678. · Zbl 1063.62064 · doi:10.1111/1467-9868.00408
[7] Fernandes, M. and Grammig, J. (2005). Nonparametric specification tests for conditional duration models,Journal of Econometrics,127, 35–68. · Zbl 1337.62216 · doi:10.1016/j.jeconom.2004.06.003
[8] Hall, P. (1984). Central limit theorem for integrated squared error multivariate nonparametric density estimators,Journal of Multivariate Analysis,14, 1–16. · Zbl 0528.62028 · doi:10.1016/0047-259X(84)90044-7
[9] Härdle, W. and Mammen, E. (1993). Comparing nonparametric vs. parametric regression fits,Annals of Statistics,21, 1926–1947. · Zbl 0795.62036 · doi:10.1214/aos/1176349403
[10] Koroljuk, V. S. and Borovskich, Y. V. (1994).Theory of U-statistics, Kluwer Academic Publishers, Dordrecht.
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