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How easy is a given density to estimate? (English) Zbl 0937.62584
Summary: In data analytic applications of density estimation one is usually interested in estimating the density over its support. However, common estimators such as the basic kernel estimator use a single smoothing parameter over the whole of the support. While this will be adequate for some densities there will be other densities that will be very difficult to estimate using this approach. The purpose of this article is to quantify how easy a particular density is to estimate using a global smoothing parameter. By considering the asymptotic expected \(L_1\) error we obtain a scale invariant functional that is useful for measuring degree of estimation difficulty. Implications for the transformation kernel density estimators, which attempt to overcome the inadequacy of the basic kernel estimator, are also discussed.

MSC:
62G05 Nonparametric estimation
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[1] Devroye, L.; Györfi, L.: Nonparametric density estimation: the L1 view. (1985) · Zbl 0546.62015
[2] Devroye, L.; Wand, M. P.: On the effect of density shape on the performance of its kernel estimate. Statistics (1993) · Zbl 0808.62033
[3] Hall, P.; Marron, J. S.: Estimation of integrated squared density derivatives. Statist. probab. Lett. 6, 109-115 (1987) · Zbl 0628.62029
[4] Hall, P.; Wand, M. P.: Minimizing L1 distance in nonparametric density estimation. J. mult. Analysis. 26, 59-88 (1988) · Zbl 0673.62030
[5] Izenman, A. J.: Recent developments in nonparametric density estimation. J. amer. Statist. assoc. 86, 205-224 (1991) · Zbl 0734.62040
[6] Jones, M. C.; Sheather, S. J.: Using non-stochastic terms to advantage in kernel-based estimation of integrated squared density derivatives. Statist. probab. Lett. 11, 511-514 (1991) · Zbl 0724.62040
[7] Marron, J. S.; Wand, M. P.: Exact mean integrated squared error. Ann. statist. 20, 712-736 (1992) · Zbl 0746.62040
[8] Park, B. U.; Marron, J. S.: Comparison of data-given bandwidth selectors. J. amer. Statist. assoc. 85, 66-72 (1990)
[9] Ruppert, D.; Wand, M. P.: Correcting for kurtosis in density estimation. Austral. J. Statist., 19-29 (1992) · Zbl 0751.62019
[10] Silverman, B. W.: Density estimation for statistics and data analysis. (1986) · Zbl 0617.62042
[11] Terrell, G. R.: The maximal smoothing principle in density estimation. J. amer. Statist. assoc. 85, 470-477 (1990)
[12] Wand, M. P.; Marron, J. S.; Ruppert, D.: Transformations in density estimation. J. amer. Statist. assoc. 86, 343-352 (1991) · Zbl 0742.62046
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