Sharp Minimax Estimation of the Variance of Brownian Motion Corrupted with Gaussian Noise
Tony Cai, Axel Munk and Johannes Schmidt-Hieber
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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.