The Annals of Statistics

Adaptive invariant density estimation for ergodic diffusions over anisotropic classes

Claudia Strauch

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Consider some multivariate diffusion process $\mathbf{X}=(X_{t})_{t\geq0}$ with unique invariant probability measure and associated invariant density $\rho$, and assume that a continuous record of observations $X^{T}=(X_{t})_{0\leq t\leq T}$ of $\mathbf{X}$ is available. Recent results on functional inequalities for symmetric Markov semigroups are used in the statistical analysis of kernel estimators $\widehat{\rho}_{T}=\widehat{\rho}_{T}(X^{T})$ of $\rho$. For the basic problem of estimation with respect to $\mathrm{sup}$-norm risk under anisotropic Hölder smoothness constraints, the proposed approach yields an adaptive estimator which converges at a substantially faster rate than in standard multivariate density estimation from i.i.d. observations.

Article information

Ann. Statist., Volume 46, Number 6B (2018), 3451-3480.

Received: March 2017
Revised: October 2017
First available in Project Euclid: 11 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M05: Markov processes: estimation 62G07: Density estimation 62G20: Asymptotic properties

Ergodic diffusion anisotropic density estimation adaptation


Strauch, Claudia. Adaptive invariant density estimation for ergodic diffusions over anisotropic classes. Ann. Statist. 46 (2018), no. 6B, 3451--3480. doi:10.1214/17-AOS1664.

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Supplemental materials

  • Supplement to “Adaptive invariant density estimation for ergodic diffusions over anisotropic classes”. The supplementary file contains the proofs of the $\mathrm{sup}$-norm oracle-type inequality stated in Theorem 3.3, the upper bound in Theorem 3.4, the lower bound from Theorem 3.6 and of the results for estimators based on discrete observations stated in Theorem 4.1 and Theorem 4.3, respectively.