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Density Estimation under Qualitative Assumptions in Higher Dimensions

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  • Polonik, W.

Abstract

We study a method for estimating a density f in Rd under assumptions which are of qualitative nature. The resulting density estimator can be considered as a generalization of the Grenander estimator for monotone densities. The assumptions on f are given in terms of shape restrictions of the density contour clusters [Gamma]([lambda]) = (x : f(x) >= [lambda]). We assume that for all [lambda] >= 0 the sets [Gamma]([lambda]) lie in a given class of measurable subsets of Rd. By choosing appropriately it is possible to model for example monotonicity, symmetry, or multimodality. The main mathematical tool for proving consistency and rates of convergence of the density estimator is empirical process theory. It turns out that the rates depend on the richness of measured by metric entropy.

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  • Polonik, W., 1995. "Density Estimation under Qualitative Assumptions in Higher Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 55(1), pages 61-81, October.
    • Handle: RePEc:eee:jmvana:v:55:y:1995:i:1:p:61-81
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    Cited by:

    1. Obereder, Andreas & Scherzer, Otmar & Kovac, Arne, 2007. "Bivariate density estimation using BV regularisation," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5622-5634, August.
    2. De-Jun Feng & Ding Feng, 2004. "On a Statistical Framework for Estimation from Random Set Observations," Journal of Theoretical Probability, Springer, vol. 17(1), pages 85-110, January.
    3. Nguyen, Hung T. & Wu, Berlin, 2006. "Random and fuzzy sets in coarse data analysis," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 70-85, November.
    4. Neumeyer, Natalie, 2007. "A note on uniform consistency of monotone function estimators," Statistics & Probability Letters, Elsevier, vol. 77(7), pages 693-703, April.
    5. 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.
    6. Polonik, Wolfgang, 1997. "Minimum volume sets and generalized quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 1-24, July.
    7. 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.
    8. Neumeyer, Natalie, 2005. "A note on uniform consistency of monotone function estimators," Technical Reports 2005,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    9. Polonik, Wolfgang & Yao, Qiwei, 2002. "Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 234-255, February.
    10. 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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