The Annals of Statistics

Sharp adaptive estimation of the drift function for ergodic diffusions

Arnak Dalalyan

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The global estimation problem of the drift function is considered for a large class of ergodic diffusion processes. The unknown drift S(⋅) is supposed to belong to a nonparametric class of smooth functions of order k≥1, but the value of k is not known to the statistician. A fully data-driven procedure of estimating the drift function is proposed, using the estimated risk minimization method. The sharp adaptivity of this procedure is proven up to an optimal constant, when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density.

Article information

Ann. Statist., Volume 33, Number 6 (2005), 2507-2528.

First available in Project Euclid: 17 February 2006

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Zentralblatt MATH identifier

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

Ergodic diffusion invariant density minimax drift estimation Pinsker’s constant sharp adaptivity


Dalalyan, Arnak. Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Statist. 33 (2005), no. 6, 2507--2528. doi:10.1214/009053605000000615.

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