Sharp Minimax Estimation of the Variance of Brownian Motion Corrupted with Gaussian NoiseTony Cai, Axel Munk and Johannes Schmidt-Hieber
Sharp Minimax Estimation of the Variance of Brownian Motion Corrupted with Gaussian NoiseTony Cai, Axel Munk and Johannes Schmidt-Hieber
- Abstract: Let Wt be a Brownian Motion and let εin be iid N(0, 1), i = 1, ..., n and independent of Wt. σ, τ > 0 are real, unknown parameters. Suppose we observe Yi,n=σ Wi/n+ τ εi,n. In this paper we will establish sharp estimators for σ2 and τ2 in minimax sense, i.e. they attain asymptotically the minimax constant. These estimators are based on a spectral decomposition of the underlying process Yi,n and can be computed explicitly in O(nlog n) operations. A proof for the minimax lower bound is given. Further we show that these estimators are asymptotically normal.
- Paper: pdf file.