Sharp adaptive estimation of the drift function for ergodic diffusions

Details Sharp adaptive estimation of the drift function for ergodic diffusions Sharp adaptive estimation of the drift function for ergodic diffusions

2006-02-28

arxiv.org/abs/math/0602659v1

Annals of Statistics 2005, Vol. 33, No. 6, 2507-2528

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053605000000615 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053605000000615

The global estimation problem of the drift function is considered for a large class of ergodic diffusion processes. The unknown drift $S(\cdot)$ is supposed to belong to a nonparametric class of smooth functions of order $k\geq1$, 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.

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