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Nonparametric estimation of the stationary density and the transition density of a Markov chain

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  • Lacour, Claire

Abstract

In this paper, we study first the problem of nonparametric estimation of the stationary density f of a discrete-time Markov chain (Xi). We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density g of (Xi,Xi+1) and so to provide an adaptive estimator of the transition density [pi]=g/f. We give bounds in L2 norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.

Suggested Citation

  • Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:2:p:232-260
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    References listed on IDEAS

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    Cited by:

    1. Gaëlle Chagny & Claire Lacour, 2015. "Optimal adaptive estimation of the relative density," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 605-631, September.
    2. Pierre Alquier & Olivier Wintenberg, 2010. "Model Selection for Weakly Dependent Time Series Forecasting," Working Papers 2010-39, Center for Research in Economics and Statistics.
    3. Martínez-Ovando Juan Carlos & Walker Stephen G., 2011. "Time-series Modelling, Stationarity and Bayesian Nonparametric Methods," Working Papers 2011-08, Banco de México.
    4. Fabienne Comte & Gwennaelle Mabon & Adeline Samson, 2017. "Spline regression for hazard rate estimation when data are censored and measured with error," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 71(2), pages 115-140, May.
    5. Gautier, Eric & Gaillac, Christophe, 2019. "Adaptive estimation in the linear random coefficients model when regressors have limited variation," TSE Working Papers 19-1026, Toulouse School of Economics (TSE).
    6. Salima El Kolei & Fabien Navarro, 2022. "Contrast estimation for noisy observations of diffusion processes via closed-form density expansions," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 303-336, July.
    7. Christophe Chesneau & Salima El Kolei & Fabien Navarro, 2017. "Parametric estimation of hidden Markov models by least squares type estimation and deconvolution," Working Papers 2017-66, Center for Research in Economics and Statistics.
    8. Sandra Plancade, 2011. "Model selection for hazard rate estimation in presence of censoring," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 313-347, November.
    9. Gaëlle Chagny, 2015. "Adaptive Warped Kernel Estimators," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 336-360, June.
    10. Comte, F. & Lacour, C. & Rozenholc, Y., 2010. "Adaptive estimation of the dynamics of a discrete time stochastic volatility model," Journal of Econometrics, Elsevier, vol. 154(1), pages 59-73, January.

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