Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T16:47:37.368Z Has data issue: false hasContentIssue false
  • English
  • Français

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: bernard.delyon@univ-rennes1.fr
*** Postal address: LTCI, Télécom ParisTech, 46 rue Barrault, 75634 Paris, Cedex 13, France.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Athreya, K. B. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245, 493–501.Google Scholar
[2]Azaïs, R., Delyon, B. and Portier, F. (2018). Integral estimation based on Markovian design. Supplementary material. Available at http://doi.org/10.1017/apr.2018.38.Google Scholar
[3]Bertail, P. and Clémençon, S. (2011). A renewal approach to Markovian U-statistics. Math. Meth. Statist. 20, 79–105.Google Scholar
[4]Chacón, J. E. and Duong, T. (2010). Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices. TEST 19, 375–398.Google Scholar
[5]Delyon, B. and Portier, F. (2016). Integral approximation by kernel smoothing. Bernoulli 22, 2177–2208.Google Scholar
[6]Duong, T. (2007). Ks: kernel density estimation and kernel discriminant analysis for multivariate data in R. J. Statist. Software 21, 16pp.Google Scholar
[7]Einmahl, U. and Mason, D. M. (2005). Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33, 1380–1403.Google Scholar
[8]Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford University Press.Google Scholar
[9]Gasser, T., Muller, H.-G. and Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. J. R. Statist. Soc. B 47, 238–252.Google Scholar
[10]Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726–748.Google Scholar
[11]Härdle, W. and Stoker, T. M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84, 986–995.Google Scholar
[12]Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Prob. 12, 224–247.Google Scholar
[13]Kosaka, Y. and Xie, S.-P. (2013). Recent global-warming hiatus tied to equatorial Pacific surface cooling. Nature 501, 403–407.Google Scholar
[14]Li, Q. and Racine, J. S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
[15]Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.Google Scholar
[16]Nolan, D. and Pollard, D. (1987). U-processes: rates of convergence. Ann. Statist. 15, 780–799.Google Scholar
[17]Novak, E. (2016). Some results on the complexity of numerical integration. In Monte Carlo and Quasi-Monte Carlo Methods, Springer, Cham, pp. 161–183.Google Scholar
[18]Nummelin, E. (1978). A splitting technique for Harris recurrent Markov chains. Z. Wahrscheinlichkeitsth. 43, 309–318.Google Scholar
[19]Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press.Google Scholar
[20]R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. Available at http://www.R-project.org/.Google Scholar
[21]Rahmstorf, S. et al. (2015). Exceptional twentieth-century slowdown in Atlantic Ocean overturning circulation. Nature Climate Change 5, 475–480.Google Scholar
[22]Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd edn. Springer, New York.Google Scholar
[23]Roget-Vial, C. (2003). Deux contributions à l'étude semi-paramétrique d'un modèle de régression. Doctoral thesis. University of Rennes 1.Google Scholar
[24]Roussas, G. G. (1969). Nonparametric estimation of the transition distribution function of a Markov process. Ann. Math. Statist. 40, 1386–1400.Google Scholar
[25]Roxy, M. K., Ritika, K., Terray, P. and Masson, S. (2014). The curious case of Indian Ocean warming. J. Climate 27, 8501–8509.Google Scholar
[26]Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall, London.Google Scholar
[27]Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York.Google Scholar
[28]Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.Google Scholar
[29]Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.Google Scholar
Supplementary material: PDF

Azaïs et al. supplementary material

Azaïs et al. supplementary material

Download Azaïs et al. supplementary material(PDF)
PDF 442.3 KB