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Multivariate boundary kernels and a continuous least squares principle

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  • H. G. Müller
  • U. Stadtmüller

Abstract

Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean‐squared error or sup‐norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non‐negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel‐generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.

Suggested Citation

  • H. G. Müller & U. Stadtmüller, 1999. "Multivariate boundary kernels and a continuous least squares principle," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(2), pages 439-458, April.
    • Handle: RePEc:bla:jorssb:v:61:y:1999:i:2:p:439-458
    DOI: 10.1111/1467-9868.00186
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    Cited by:

    1. Cheng, Ming-Yen & Hall, Peter, 2003. "Reducing variance in nonparametric surface estimation," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 375-397, August.
    2. Cai, T. Tony & Levine, Michael & Wang, Lie, 2009. "Variance function estimation in multivariate nonparametric regression with fixed design," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 126-136, January.
    3. Bouezmarni, T. & Rombouts, J.V.K., 2009. "Semiparametric multivariate density estimation for positive data using copulas," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2040-2054, April.
    4. Chen, Song Xi & Gao, Jiti & Tang, Chenghong, 2005. "A test for model specification of diffusion processes," MPRA Paper 11976, University Library of Munich, Germany, revised Feb 2007.
    5. Jeffrey Racine, 2015. "Mixed data kernel copulas," Empirical Economics, Springer, vol. 48(1), pages 37-59, February.
    6. Marshall, Jonathan C. & Hazelton, Martin L., 2010. "Boundary kernels for adaptive density estimators on regions with irregular boundaries," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 949-963, April.

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