Adaptive confidence bands for Markov chains and diffusions

Estimating the invariant measure and the drift

Journal article (2016)

Authors

J. Söhl University of Cambridge

Mathias Trabs Hamburg University of Technology

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Diffusion Drift estimation Functional central limit theorem Adaptive confidence bands Ergodic Markov chain Lepski's method Stationary density

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Published Date

2016

Language

English

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Abstract

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.