Sharp minimax estimation of the variance of Brownian motion corrupted with Gaussian noise

Abstract

Let Wt be a Brownian motion with ²in i.∼i.d. N(0, 1), i = 1, . . . , n, independent of Wt. σ, τ > 0 are real, unknown parameters. Suppose we observe Yi,n = σWi/n + τ ²in. In this paper we establish sharp estimators for σ2 and τ 2 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 Yi,n and can be computed explicitly taking O(n log n) 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.