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

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) $L^{\infty }$-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.

to appear in ESAIM: Probability and Statistics

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Cited by in corpus (2)

• Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
• Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data