## 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)

KernSmooth
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.