Exact asymptotics for estimating the marginal density of discretely observed diffusion processes

Abstract

We derive sharp asymptotic minimax bounds (that is, bounds which concern the exact asymptotic constant of the risk) for nonparametric density estimation based on discretely observed diffusion processes. We study two particular problems for which there already exist such results in the case of independent and identically distributed observations, namely, minimax density estimation in Sobolev classes with L₂-loss and in Hölder classes with L_∞-loss. We derive independently lower and upper bounds for the asymptotic minimax risks and show that they coincide with the classic efficiency bounds. We prove that these bounds can be attained by usual kernel density estimators. The lower bounds are obtained by analysing the problem of estimating the marginal density in certain families of processes, $\left\{\right\{X_i^f\},f\in {calF}_n\}$ , which are shrinking neighbourhoods of some central process, $\left\{X_i^{f_0}\right\}$ , in the sense that the set of densities {\cal F}_n forms a shrinking neighbourhood centred around f₀.