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Keywords:
kernel; optimum kernel; optimum boundary kernel
Summary:
Kernel smoothers belong to the most popular nonparametric functional estimates used for describing data structure. They can be applied to the fix design regression model as well as to the random design regression model. The main idea of this paper is to present a construction of the optimum kernel and optimum boundary kernel by means of the Gegenbauer and Legendre polynomials.
References:
[1] Gasser, T. L., Müller, H.-G., Mammitzsch, V.: Kernels for nonparametric curve estimation. J. R. Stat. Soc. B47 (1985), 238-251. MR 0816088
[2] Granovsky, B. L., Müller, H.-G.: Optimizing kernel methods: A unifying variational principle. Int. Stat. Rev. 59 (1991), 373-388.
[3] Granovsky, B. L., Müller, H.-G., Pfeifer, C.: Some remarks on optimal kernel function. Stat. Decis. 13 (1995), 101-116. MR 1342732
[4] Härdle, W.: Applied Nonparametric Regression. Cambridge University Press Cambridge (1990). MR 1161622
[5] Horová, I.: Gegenbauer polynomials, optimal kernels and Stancu operators. Approximation Theory and Function Series Budapest (Hungary), 1995 P. Vértesi et al. János Bolyai Mathematical Socity Budapest (1996), 227-235. MR 1432671
[6] Horová, I.: Some remarks on kernels. J. Comput. Anal. Appl. 2 (2000), 253-263. MR 1778550
[7] Horová, I.: Optimization problems connected with kernel estimates. Signal Processing, Communications and Computer Science World Scientific and Engineering Society Press (2002), 339-334.
[8] Kolaček, J., Poměnková, J.: Comparative study of boundary effects for kernel smoothing. Austr. J. Stat. 35 (2006), 281-288.
[9] Mammitzsch, V.: The fluctuation of kernel estimators under certain moment conditions. Proc. ISI 1985 (1985), 17-18.
[10] Poměnková, J.: Gasser-Müller's estimate, LI. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis 3 (2004), Czech. MR 2159138
[11] Poměnková, J.: Some aspects of smoothing the regression function. PhD. thesis University of Ostrava Ostrava (2005), Czech.
[12] Poměnková, J.: Optimal kernels. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, LII (2004), 69-77 Czech.
[13] Poměnková, J.: Optimum choice of the bandwidth using ${\rm AMSE}$ for the Gasser-Müller estimator. Applications of Mathematics and Statistics in Economy University of Economics and Faculty of Informatics and Statistics Praha (2004), 192-198.
[14] Szegö, G.: Orthogonal Polynomials. American Mathematical Society Colloquium Publications, Vol. 23. Am. Math. Soc. New York (1939).
[15] Wand, M. P., Jones, M. C.: Kernel Smoothing. Chapman & Hall London (1995). MR 1319818 | Zbl 0854.62043
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