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Variable Kernel estimates: On the impossibility of tuning the parameters

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For the standard kernel density estimate, it is known that one can tune the bandwidth such that the expected L1 error is within a constant factor of the optimal L1 error (obtained when one is allowed to choose the bandwidth with knowledge of the density). In this paper, we pose the same problem for variable bandwidth kernel estimates where the bandwidths are allowed to depend upon the location. We show in particular that for positive kernels on the real line, for any data-based bandwidth, there exists a density for which the ratio of expected L1 error over optimal L1 error tends to infinity. Thus, the problem of tuning the variable bandwidth in an optimal manner is ``too hard''. Moreover, from the class of counterexamples exhibited in the paper, it appears that placing conditions on the densities (monotonicity, convexity, smoothness) does not help.

Suggested Citation

  • Luc Devroye & Gábor Lugosi, 1998. "Variable Kernel estimates: On the impossibility of tuning the parameters," Economics Working Papers 325, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:325
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    References listed on IDEAS

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    1. Sheather, Simon, 1983. "A data-based algorithm for choosing the window width when estimating the density at a point," Computational Statistics & Data Analysis, Elsevier, vol. 1(1), pages 229-238, March.
    2. Sheather, Simon J., 1986. "An improved data-based algorithm for choosing the window width when estimating the density at a point," Computational Statistics & Data Analysis, Elsevier, vol. 4(1), pages 61-65, June.
    3. Duc Devroye & J. Beirlant & R. Cao & R. Fraiman & P. Hall & M. Jones & Gábor Lugosi & E. Mammen & J. Marron & C. Sánchez-Sellero & J. Uña & F. Udina & L. Devroye, 1997. "Universal smoothing factor selection in density estimation: theory and practice," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 6(2), pages 223-320, December.
    4. Luc Devroye & Gábor Lugosi & Frederic Udina, 1998. "Inequalities for a new data-based method for selecting nonparametric density estimates," Economics Working Papers 281, Department of Economics and Business, Universitat Pompeu Fabra.
    5. Hall, Peter & Schucany, William R., 1989. "A local cross-validation algorithm," Statistics & Probability Letters, Elsevier, vol. 8(2), pages 109-117, June.
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    Cited by:

    1. Cristina Butucea, 2001. "Numerical results concerning a sharp adaptive density estimator," Computational Statistics, Springer, vol. 16(2), pages 271-298, July.
    2. Amatulli, Giuseppe & Peréz-Cabello, Fernando & de la Riva, Juan, 2007. "Mapping lightning/human-caused wildfires occurrence under ignition point location uncertainty," Ecological Modelling, Elsevier, vol. 200(3), pages 321-333.
    3. Biau, Gérard & Devroye, Luc, 2003. "On the risk of estimates for block decreasing densities," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 143-165, July.

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    Keywords

    Density estimation; variable Kernel estimate; convergence; smoothing factor; minimax lower bounds; asymptotic optimality;
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