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Estimating the intensity of a random measure by histogram type estimators
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  • Published: 04 January 2008

Estimating the intensity of a random measure by histogram type estimators

  • Yannick Baraud1 &
  • Lucien Birgé2 

Probability Theory and Related Fields volume 143, pages 239–284 (2009)Cite this article

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Abstract

The purpose of this paper is to estimate the intensity of some random measure N on a set \({\mathcal{X}}\) by a piecewise constant function on a finite partition of \({\mathcal{X}}\) . Given a (possibly large) family \({\mathcal{M}}\) of candidate partitions, we build a piecewise constant estimator (histogram) on each of them and then use the data to select one estimator in the family. Choosing the square of a Hellinger-type distance as our loss function, we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound. Moreover, the selection procedure leads to a final estimator satisfying some oracle-type inequality, with, as usual, a possible loss corresponding to the complexity of the family \({\mathcal{M}}\) . When this complexity is not too high, the selected estimator has a risk bounded, up to a universal constant, by the smallest risk bound obtained for the estimators in the family. For suitable choices of the family of partitions, we deduce uniform risk bounds over various classes of intensities. Our approach applies to the estimation of the intensity of an inhomogenous Poisson process, among other counting processes, or the estimation of the mean of a random vector with nonnegative components.

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Author information

Authors and Affiliations

  1. Laboratoire J-A Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Yannick Baraud

  2. Laboratoire de Probabilités et Modèles Aléatoires, boîte 188, Université Paris VI, 4 place Jussieu, 75252, Paris Cedex 05, France

    Lucien Birgé

Authors
  1. Yannick Baraud
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  2. Lucien Birgé
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Correspondence to Lucien Birgé.

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Baraud, Y., Birgé, L. Estimating the intensity of a random measure by histogram type estimators. Probab. Theory Relat. Fields 143, 239–284 (2009). https://doi.org/10.1007/s00440-007-0126-6

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  • Received: 06 June 2006

  • Revised: 30 October 2007

  • Published: 04 January 2008

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0126-6

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Keywords

  • Model selection
  • Histogram
  • Discrete data
  • Poisson process
  • Intensity estimation
  • Adaptive estimation

Mathematics Subject Classification (2000)

  • 62G05
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