Abstract: Let be a Brownian motion with , independent of . are real, unknown parameters. Suppose we observe In this paper we establish sharp estimators for and in minimax sense, i.e. they attain the minimax constant asymptotically. A short and direct proof for the minimax lower bound is given. These estimators are based on a spectral decomposition of the underlying process and can be computed explicitly taking operations. We outline how these estimators can be generalized from Brownian motion to processes with independent increments. Further we show that the spectral estimators presented are asymptotically normal.
Key words and phrases: Asymptotic normality, Brownian motion, deconvolution, minimax, spectral estimators, statistical inverse problems, variance estimation, oracle estimator.