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On visual distances in density estimation: the Hausdorff choice

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  • Cuevas, Antonio
  • Fraiman, Ricardo

Abstract

We consider a "visual" metric between multivariate densities that is defined in terms of the Hausdorff distance between their hypographs. This distance has been first proposed and analyzed by Beer (1982) in the non-probabilistic context of approximation theory. We suggest the use of this distance in density estimation as a weaker, more flexible alternative to the supremum metric: it also has a direct visual interpretation but does not require very restrictive continuity assumptions. A further Hausdorff-based distance is also proposed and analyzed. We obtain consistency results, and a convergence rate, for the usual kernel density estimators with respect to these metrics provided that the underlying density is not too discontinuous. These results can be seen as a partial extension to the "qualitative smoothing" setup (see Marron and Tsybakov, 1995) of the classical analogous theorems with respect to the supremum metric.

Suggested Citation

  • Cuevas, Antonio & Fraiman, Ricardo, 1998. "On visual distances in density estimation: the Hausdorff choice," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 333-341, November.
    • Handle: RePEc:eee:stapro:v:40:y:1998:i:4:p:333-341
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    References listed on IDEAS

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    1. van Es, Bert, 1997. "A note on the integrated squared error of a kernel density estimator in non-smooth cases," Statistics & Probability Letters, Elsevier, vol. 35(3), pages 241-250, October.
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    Cited by:

    1. Baíllo, Amparo & Cuevas, Antonio, 2004. "Image estimators based on marked bins," DES - Working Papers. Statistics and Econometrics. WS ws045114, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Manté, Claude, 2015. "Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree," Computational Statistics & Data Analysis, Elsevier, vol. 84(C), pages 68-84.

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