On estimating the spectral exponent of fractional Brownian motion

Three estimators of the exponent α in the power spectral density g(λ)=c_g|λ|^-α of fractional Brownian motion are evaluated. These are (i) the periodogram-based estimator αˆ_PG (ii) the maximum likelihood estimator αˆ_ML; and (iii) the Allan (1966) variance-based estimator αˆ_AV. Large-sample properties of the mean-square error (MSE) and the associated sampling distribution are examined, αˆ_PG emphasis on the case α∈(1, 2). The MSE performance of αˆ_PG is judged to be inferior to that of both αˆ_ML and αˆ_AV. The rate of decrease of MSE is the same for αˆ_ML and αˆ_AV; the former estimator has smaller MSE, while the latter is less sensitive to departures from the power-law model and is considerably easier to compute