Improving density estimators of discretely observed processes by interpolation

We consider density estimation for a smooth stationary process X t , t ∈ R , based on a discrete sample Y i = X Δ i , i = 0 , … , n = T / Δ . By a suitable interpolation scheme of order p, we augment data to form an approximation X p , t , t ∈ [ 0 , T ] , of the continuous-time process and base our density estimate on the augmented sample path. Our results show that this can improve the rate of convergence (measured in terms of n) of the density estimate. Among other things, this implies that recording n observations using a small Δ can be more efficient than recording n independent observations.