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Efficient density estimation for ergodic diffusion processes. (English) Zbl 0953.62085

The problem of asymptotically efficient estimation of the density of the invariant measure of a diffusion process is considered. An efficient estimator is defined by the minimax lower bound on the risk of all estimators. The author shows that the local-time and kernel-type estimators are asymptotically efficient for the loss functions with polynomial majorants. The asymptotic behavior of a wide class of unbiased estimators with the same limit variances is also discussed.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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