Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift

Söhl, Jakob; Trabs, Mathias

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 432-462.

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) $L^{\infty }$-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.

Classification : 62G15, 60F05, 60J05, 60J60, 62M05
Mots clés : Adaptive confidence bands, diffusion, drift estimation, ergodic Markov chain, stationary density, Lepski’s method, functional central limit theorem

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