Geodesic
Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift
Söhl, Jakob1 ; Trabs, Mathias2
1 Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CB3 0WB Cambridge, UK.
2 Department of Mathematics, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany.
ESAIM: Probability and Statistics, Tome 20 (2016), pp. 432-462

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As a starting point we prove a functional central limit theorem for estimators of the invariant measure of a geometrically ergodic Harris-recurrent Markov chain in a multi-scale space. This allows to construct confidence bands for the invariant density with optimal (up to undersmoothing) L ∞ -diameter by using wavelet projection estimators. In addition our setting applies to the drift estimation of diffusions observed discretely with fixed observation distance. We prove a functional central limit theorem for estimators of the drift function and finally construct adaptive confidence bands for the drift by using a completely data-driven estimator.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2016017
Classification : 62G15, 60F05, 60J05, 60J60, 62M05
Keywords: Adaptive confidence bands, diffusion, drift estimation, ergodic Markov chain, stationary density, Lepski’s method, functional central limit theorem

Söhl, Jakob 1 ; Trabs, Mathias 2

1 Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CB3 0WB Cambridge, UK.
2 Department of Mathematics, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany. 
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     title = {Adaptive confidence bands for {Markov} chains and diffusions: {Estimating} the invariant measure and the drift},
     journal = {ESAIM: Probability and Statistics},
     pages = {432--462},
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     year = {2016},
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Söhl, Jakob; Trabs, Mathias. Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 432-462. doi: 10.1051/ps/2016017

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