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On the risk of estimates for block decreasing densities

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  • Biau, Gérard
  • Devroye, Luc

Abstract

A density f=f(x1,...,xd) on [0,[infinity])d is block decreasing if for each j[set membership, variant]{1,...,d}, it is a decreasing function of xj, when all other components are held fixed. Let us consider the class of all block decreasing densities on [0,1]d bounded by B. We shall study the minimax risk over this class using n i.i.d. observations, the loss being measured by the L1 distance between the estimate and the true density. We prove that if S=log(1+B), lower bounds for the risk are of the form C(Sd/n)1/(d+2), where C is a function of d only. We also prove that a suitable histogram with unequal bin widths as well as a variable kernel estimate achieve the optimal multivariate rate. We present a procedure for choosing all parameters in the kernel estimate automatically without loosing the minimax optimality, even if B and the support of f are unknown.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:86:y:2003:i:1:p:143-165
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    References listed on IDEAS

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    1. Polonik, W., 1995. "Density Estimation under Qualitative Assumptions in Higher Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 55(1), pages 61-81, October.
    2. 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.
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    Cited by:

    1. Berlinet, Alain & Biau, Gérard & Rouvière, Laurent, 2005. "Optimal L1 bandwidth selection for variable kernel density estimates," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 116-128, September.
    2. Pavlides, Marios G. & Wellner, Jon A., 2012. "Nonparametric estimation of multivariate scale mixtures of uniform densities," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 71-89.
    3. Gao, Fuchang & Wellner, Jon A., 2007. "Entropy estimate for high-dimensional monotonic functions," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1751-1764, October.

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