Model selection for Poisson processes

Birgé, Lucien

doi:10.1214/074921707000000265

Cited by 20 publications

(25 citation statements)

References 31 publications

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“…copies of the first point of a Poisson point process on [0, ∞) with intensity ϕ(t), what is a good estimate of ϕ(t) −1 ? Poisson process intensity estimation has a large literature, see for example [3,22,31] and references therein, but the (natural) data assumed in this area is one realization of the point process, or the point process observed up to some fixed time, or i.i.d. copies of such data, which does not fit our framework.…”

Section: Related Theoretical Workmentioning

confidence: 99%

Can one hear the shape of a population history?

Kim

Mossel

Rácz

et al. 2015

Theoretical Population Biology

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Reconstructing past population size from present day genetic data is a major goal of population genetics. Recent empirical studies infer population size history using coalescent-based models applied to a small number of individuals. Here we provide tight bounds on the amount of exact coalescence time data needed to recover the population size history of a single, panmictic population at a certain level of accuracy. In practice, coalescence times are estimated from sequence data and so our lower bounds should be taken as rather conservative.

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Section: Related Theoretical Workmentioning

confidence: 99%

Can one hear the shape of a population history?

Kim

Mossel

Rácz

et al. 2015

Theoretical Population Biology

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“…The importance of the distance H in model selection for Poisson processes is explicit in [10]. In order to define our estimators we assume that…”

Section: A General Statistical Frameworkmentioning

confidence: 99%

Estimating the intensity of a random measure by histogram type estimators

Baraud¹,

Birgé

2008

Probab. Theory Relat. Fields

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The purpose of this paper is to estimate the intensity of some random measure N on a set X by a piecewise constant function on a finite partition of X . Given a (possibly large) family 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 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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“…The proof of Proposition 3 in [6] can be readily adapted to our framework, whatever α > 0. In the proof of that proposition, the assumption α ∈ (0, 1] is only used to bound k 1 (α) and C(α).…”

Section: Adaptivity Of the D-estimatormentioning

confidence: 99%

“…The collection of linear spaces we consider has been chosen for its potential qualities of approximation, as suggested by approximation results for real-valued functions due to DeVore and Yu [14] and DeVore (cf. [6]). Adapting the proofs to our framework, we prove that our collection of spaces has indeed good approximation qualities with respect to R r -valued functions defined on {1, .…”

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confidence: 99%

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Estimating a discrete distributionviahistogram selection

Akakpo

2011

ESAIM: PS

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Abstract. Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.Mathematics Subject Classification. 62G05, 62C20, 41A17.

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Model selection for Poisson processes

Cited by 20 publications

References 31 publications

Can one hear the shape of a population history?

Can one hear the shape of a population history?

Estimating the intensity of a random measure by histogram type estimators

Estimating a discrete distributionviahistogram selection

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