Volume 20, 2016
|Page(s)||432 - 462|
|Published online||30 November 2016|
Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift∗
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.
Revised: 24 May 2016
Accepted: 1 July 2016
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.
Mathematics Subject Classification: 62G15 / 60F05 / 60J05 / 60J60 / 62M05
Key words: Adaptive confidence bands / diffusion / drift estimation / ergodic Markov chain / stationary density / Lepski’s method / functional central limit theorem
The authors acknowledge intensive and very helpful discussions with Richard Nickl. J.S. thanks the European Research Council (ERC) for support under Grant No. 647812. M.T. is grateful to the Statistical Laboratory of the University of Cambridge for its hospitality during a visit from February to March 2014, where this research was initiated, and to the Deutsche Forschungsgemeinschaft (DFG) for the research fellowship TR 1349/1-1. Part of the paper was carried out while M.T. was employed at the Humboldt-Universität zu Berlin.
© EDP Sciences, SMAI 2016
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