Tuning of a Bayesian estimator under discrete time observations and unknown transition density

We study the asymptotic behaviour of a Bayesian parameter estimation method on a compact one-dimensional parameter space. The estimation procedure is considered under discrete observations and unknown transition density. Here, we observe the data with constant time steps and the transition density of the data is approximated by using a kernel density estimation method applied to the Monte Carlo simulations of approximations of the theoretical random variables generating the observations. We estimate the error between the theoretical estimator, which assumes the knowledge of the transition density and its approximation which uses the simulation. We prove the strong consistency of the approximated estimator and find the order of the error. Most importantly, we give a parameter tuning result which relates the number of data, the weak error in the approximation process, the number of the Monte-Carlo simulations and the bandwidth size of the kernel density estimation. A guiding example for this situation is the use of Monte Carlo simulations of the Euler scheme for Bayesian estimation in a diffusion setting.