A family of minimax rates for density estimators in continuous time

Abstract

In continuous time, rates of convergence of density estimators fluctuate with the nature of observed sample paths. In this paper, we give a family of rates reached by the kernel estimator and we show that these rates are minimax. Finally, we study applications of these results for specific classes of processes including the Gaussian ones.

Nonparametric trend coefficient estimation for multidimensional diffusions

“…This proposition follows from the results by Blanke and Bosq [3], where intermediate rates for kernel density estimators are provided under slightly weaker conditions than the Castellana and Leadbetter one. These rates depend on the local behavior of the joint density f u (x, y), when u is small.…”

“…These rates depend on the local behavior of the joint density f u (x, y), when u is small. Using results in [9] and [12], it is easy to see that for all S ∈ S d , process (1) satisfies the conditions presented in [3]. From [3], it follows also that the previous rates are sharp, since a lower bound for the quadratic risk can be proved.…”

“…Using results in [9] and [12], it is easy to see that for all S ∈ S d , process (1) satisfies the conditions presented in [3]. From [3], it follows also that the previous rates are sharp, since a lower bound for the quadratic risk can be proved. Moreover, these rates turn out to be minimax in the sense that if the kernel estimator reaches an intermediate rate over a class of processes, than no better estimator exists over that class.…”

Optimal sampling for density estimation in continuous time

“…The case α > 1 typically occurs for higher dimensions d > 1. One may found more detailed examples in Blanke and Bosq (2000). Moreover for preliminary estimation of local smoothness coefficients for continuous time processes, we refer to Blanke (2002).…”

“…a mean-square differentiable Gaussian process, the rate decreases as (ln T /T ) (see Bosq 1997, Sköld andHössjer 1999). In fact according to regularity of sample paths, the kernel estimator reaches a family of rates that are minimax (Blanke and Bosq, 2000).…”