On second order minimax estimation of invariant density for ergodic diffusion

Summary There are many asymptotically first order efficient estimators in the problem of estimating the invariant density of an ergodic diffusion process nonparametrically. To distinguish between them, we consider the problem of asymptotically second order minimax estimation of this density based on a sample path observation up to the time T . It means that we have two problems. The first one is to find a lower bound on the second order risk of any estimator. The second one is to construct an estimator, which attains this lower bound. We carry out this program (bound + estimator) following Pinsker’s approach. If the parameter set is a subset of the Sobolev ball of smoothness k  > 1 and radius R  > 0, the second order minimax risk is shown to behave as − T −2 k /(2 k −1) Π̂( k , R ) for large values of T . The constant Π̂( k , R ) is given explicitly.