On estimating the spectral exponent of fractional Brownian motion

Three estimators of the exponent /spl alpha/ in the power spectral density g(/spl lambda/)=c/sub g/|/spl lambda/|/sup -/spl alpha// of fractional Brownian motion are evaluated. These are (i) the periodogram-based estimator /spl alpha//spl circ//sub PG/ (ii) the maximum likelihood estimator /spl alpha//spl circ//sub ML/; and (iii) the Allan (1966) variance-based estimator /spl alpha//spl circ//sub AV/. Large-sample properties of the mean-square error (MSE) and the associated sampling distribution are examined, /spl alpha//spl circ//sub PG/ emphasis on the case /spl alpha//spl isin/(1, 2). The MSE performance of /spl alpha//spl circ//sub PG/ is judged to be inferior to that of both /spl alpha//spl circ//sub ML/ and /spl alpha//spl circ//sub AV/. The rate of decrease of MSE is the same for /spl alpha//spl circ//sub ML/ and /spl alpha//spl circ//sub 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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>