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New Multivariate Product Density Estimators

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  • Devroye, Luc
  • Krzyzak, Adam

Abstract

Let X be an d-valued random variable with unknown density f. Let X1, ..., Xn be i.i.d. random variables drawn from f. The objective is to estimate f(x), where x=(x1, ..., xd). We study the pointwise convergence of two new density estimates, the Hilbert product kernel estimate where Xi=(Xi1, ..., Xid), and the Hilbert k-nearest neighbor estimate where X(k)=(X(k) 1, ..., X(k) d), and X(k) is the kth nearest neighbor of x when points are ordered by increasing values of the product [product operator]dj=1 xj-X(k) j, and k=o(log n), k-->[infinity]. The auxiliary results needed permit us to formulate universal consistency results (pointwise and in L1) for product kernel estimates with different window widths for each coordinate, and for rectangular partitioning and tree estimates. In particular, we show that locally adapted smoothing factors for product kernel estimates may make the kernel estimate inconsistent even under standard conditions on the bandwidths.

Suggested Citation

  • Devroye, Luc & Krzyzak, Adam, 2002. "New Multivariate Product Density Estimators," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 88-110, July.
    • Handle: RePEc:eee:jmvana:v:82:y:2002:i:1:p:88-110
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    References listed on IDEAS

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    1. Devroye, Luc & Krzyzak, Adam, 1999. "On the Hilbert kernel density estimate," Statistics & Probability Letters, Elsevier, vol. 44(3), pages 299-308, September.
    2. Devroye, Luc & Györfi, Laszlo & Krzyzak, Adam, 1998. "The Hilbert Kernel Regression Estimate," Journal of Multivariate Analysis, Elsevier, vol. 65(2), pages 209-227, May.
    3. Sanz de Santamaría Alejandro & Rojas Santiago & Guzmán Enrique, 1981. "Las migraciones laborales en regiones de economía campesina : una propuesta metodológica para su estudio empírico," Revista Desarrollo y Sociedad, Universidad de los Andes,Facultad de Economía, CEDE, January.
    4. Mack, Y. P. & Rosenblatt, M., 1979. "Multivariate k-nearest neighbor density estimates," Journal of Multivariate Analysis, Elsevier, vol. 9(1), pages 1-15, March.
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    2. Lei Yang & Xianyi Wu, 2013. "Estimation of Dirichlet process priors with monotone missing data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(4), pages 787-807, December.

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