Estimating the intensity of a random measure by histogram type estimators
By Y. Baraud, Lucien Birgé. 2006
47 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/9c2d8528eb3d8d65332cc58e4bf7091f33805a11
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In Applied Mathematics
- Vincent Rivoirard
- Lecu'e Guillaume
- Patricia Reynaud-Bouret
- Lerasle Matthieu
- I. Castillo
- Y. Baraud
- Francis R. Bach
- Lucien Birgé
In Artificial Intelligence
- Lecu'e Guillaume
- Patricia Reynaud-Bouret
- Vincent Rivoirard
- Lerasle Matthieu
- I. Castillo
- Richard Nickl
- Dominique Bontemps
In Statistics and Probability
- Vincent Rivoirard
- Lecu'e Guillaume
- Patricia Reynaud-Bouret
- Lerasle Matthieu
- Richard Nickl
- I. Castillo
- Guillaume Lecué
- Matthieu Lerasle
Citations in the corpus
- Learning from MOM’s principles: Le Cam’s approach (2017, methodology): “Le Cam’s approach has been used by Birg´e to define T -estimators (cf. Baraud and Birg´e (2009); Birg´e (2006, 2013)) and by Baraud, Birg´e and Sart to define ρ -estimators (cf. Baraud and Birg´e (2016); Baraud et al. (2017)).”
- Near optimal thresholding estimation of a Poisson intensity on the real line (2008, methodology): “Besides, in [10], [6] and [22], minimax performance on classical functional spaces is derived only for compactly supported signals.”