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Integral estimation based on Markovian design

Part of: Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  16 November 2018

Romain Azaïs ,
Bernard Delyon  and
François Portier
Romain Azaïs*
Affiliation:
Inria Nancy and Institut Élie Cartan de Lorraine
Bernard Delyon*
Affiliation:
Univ Rennes, CNRS, IRMAR
François Portier*
Affiliation:
Télécom ParisTech and University of Paris-Saclay
*
* Postal address: Team BIGS, Inria Nancy ‒ Grand-Est Research Centre, 615 rue du Jardin Botanique, 54600 Villers-lès-Nancy, France.
** Postal address: CNRS, IRMAR - UMR 6625, University of Rennes 1, F-35000 Rennes, France. Email address: [email protected]
*** Postal address: LTCI, Télécom ParisTech, 46 rue Barrault, 75634 Paris, Cedex 13, France.
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Abstract

Suppose that a mobile sensor describes a Markovian trajectory in the ambient space and at each time the sensor measures an attribute of interest, e.g. the temperature. Using only the location history of the sensor and the associated measurements, we estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g. Monte Carlo, the proposed approach does not require any knowledge of the distribution of the sensor trajectory. We establish probabilistic bounds on the convergence rates of the estimator. These rates are better than the traditional `root n'-rate, where n is the sample size, attached to other probabilistic integration methods. For finite sample sizes, we demonstrate the favorable behavior of the procedure through simulations and consider an application to the evaluation of the average temperature of oceans.

Keywords

Integral approximation; Markov chainNummelin splitting techniqueinvariant density estimationkernel smoothing

MSC classification

Primary: 62M05: Markov processes
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 833 - 857
Copyright
Copyright © Applied Probability Trust 2018 

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