ON ESTIMATING THE SPECTRAL EXPONENT OF FRACTIONAL BROWNIAN-MOTION

Three estimators of the exponent alpha in the power spectral density g(lambda) = c(g)\lambda\(-alpha) of fractional Brownian motion are evaluated. These are i) the periodogram-based estimator <(alpha)over cap>(PG); ii) the maximum likelihood estimator <(alpha)over cap>(ML); and iii) the Allan-variance-based estimator <(alpha)over cap>(AV). Large-sample properties of the mean-square error (MSE) and the associated sampling distribution are examined, with emphasis on the case alpha is an element of (1, 2). The MSE performance of <(alpha)over cap>(PG) is judged to be inferior to that of both <(alpha)over cap>(ML) and <(alpha)over cap>(AV). The rate of decrease of MSE is the same for <(alpha)over cap>(ML) and <(alpha)over cap>(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.