Sharp Minimax Estimation of the Variance of Brownian Motion Corrupted with Gaussian Noise
Tony Cai, Axel Munk and Johannes Schmidt-Hieber
Abstract:
Let W_{t} be a Brownian Motion and let
ε_{in} be iid N(0, 1), i = 1, ..., n
and independent of W_{t}. σ, τ > 0 are
real, unknown parameters. Suppose we observe
Y_{i,n}=σ W_{i/n}+ τ ε_{i,n}.
In this paper we will establish sharp estimators for
σ^{2} and τ^{2} in minimax sense, i.e. they
attain asymptotically the minimax constant. These estimators
are based on a spectral decomposition of the underlying
process Y_{i,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.