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Contribution to the bandwidth choice for kernel density estimates. (English) Zbl 1194.62031

The authors propose a new technique of bandwidth selection for univariate kernel density estimates \(\hat f_{h,K}\) as solutions to the equation \[ h=(2nk)^{-1}\left (V(K)/\int(\text{bias}\;\hat f_{h,K})^2dx\right), \] where \(n\) is the sample size, \(k\) is the order of the kernel \(K\), \(V(K)=\int K^2(x)dx\), and \[ \text{bias}\;\hat f_{h,K}=\int \hat f_{h,K}(x-hy)K(y)dy-\hat f_{h,K}(x) \] is an estimate of the bias of \(\hat f_{h,K}\). The equation is solved iteratively using the Steffensen acceleration method. Results of simulations are presented.

MSC:

62G07 Density estimation
65C60 Computational problems in statistics (MSC2010)

Software:

KernSmooth
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References:

[1] Deheuvels, P., Estimation nonparametrique de la densité par histogrammes generalisés, Rev Stat Appl, 35, 5-42 (1977)
[2] Devroye, L., A course on density estimation (1987), Basel: Birkhauser, Basel
[3] Devroye, L., Universal smoothing factor selection in density estimation: theory and practice, Test, 6, 223-320 (1997) · Zbl 0949.62026
[4] Devroye, L.; Györfi, L., Nonparametric density estimation: the L_1 view (1985), New York: Wiley, New York · Zbl 0546.62015
[5] Engel, J.; Herrmann, E.; Gasser, T., An iterative bandwidth selector for kernel estimation of densities and their derivatives, J Nonparametr Stat, 4, 21-34 (1994) · Zbl 1380.62146
[6] Gasser, T.; Kneip, A.; Köhler, W., A flexible and fast method for automatic smoothing, J Am Stat Assoc, 86, 643-652 (1991) · Zbl 0733.62047
[7] Härdle, W.; Müller, M.; Sperlich, S.; Weewitz, A., Nonparametric and semiparametric modeling (1999), Humboldt: Universität zu Berlin, Humboldt
[8] Jones, MC; Marron, JS; Park, BU, A simple root-n bandwidth selector, Ann Stat, 19, 1919-1932 (1991) · Zbl 0745.62033
[9] Müller HG (1988) Nonparametric regrssion analysis of longitudined data. In: Lecture notes in statistics, vol 46. Springer, Berlin
[10] Müller, HG; Wang, JL, Locally addaptive hazard smoothing, Prob Theor Relat Fields, 85, 523-538 (1990) · Zbl 0677.62034
[11] Park, BU; Marron, JS, Comparison of data-driven bandwidth selectors, J Am Stat Assoc, 85, 66-72 (1990)
[12] Scott, DW, Multivariate density estimation (1992), practice and visualization. Wiley: Theory, practice and visualization. Wiley · Zbl 0850.62006
[13] Sheather, S.; Jones, MC, A reliable data-based bandwidth selection method for kernel density estimation, J R Stat Soc B, 53, 683-690 (1991) · Zbl 0800.62219
[14] Silverman, B., Density estimation for statistics and data analysis (1986), London: Chapman and Hall, London · Zbl 0617.62042
[15] Stoer, J.; Bulirsch, R., Introduction numerical analysis (1980), New York: Springer, New York · Zbl 0423.65002
[16] Szegö G (1991) Orthogonal polynomials. American Mathematical Society Colloquium Publications, Providence · Zbl 0023.21505
[17] Terrell, GR, The maximal smoothing principle in density estimation, J Am Stat Assoc, 85, 470-477 (1990)
[18] Terrell, GR; Scott, DE, Oversmoothed nonparametric density estimations, J Am Stat Assoc, 80, 209-214 (1985)
[19] Wand, MP; Jones, MC, Kernel smoothing (1995), London: Chapman and Hall, London · Zbl 0854.62043
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