The Asymptotic Variance of the Continuous-Time Kernel Estimator with Applications to Bandwidth Selection

We derive simple expressions for the asymptotic variance of the kernel-density estimator of a stationary continuous-time process in one and d dimensions and relate convergence rates to sample path smoothness. Important applications include methods for selecting optimal smoothing parameters and construction of confidence bands for testing hypotheses about the density. In a simulation study the results are applied to bandwidth selection for discrete-time processes that can be modelled as continuous-time processes sampled at a high rate.