Impact of the sampling rate on the estimation of the parameters of fractional Brownian motion

Fractional Brownian motion is a mean-zero self-similar Gaussian process with stationary increments. Its covariance depends on two parameters, the self-similar parameter H and the variance C. Suppose that one wants to estimate optimally these parameters by using n equally spaced observations. How should these observations be distributed? We show that the spacing of the observations does not affect the estimation of H (this is due to the self-similarity of the process), but the spacing does affect the estimation of the variance C. For example, if the observations are equally spaced on [0, n] (unit-spacing), the rate of convergence of the maximum likelihood estimator (MLE) of the variance C is root n. However, if the observations are equally spaced on [0, 1] (1/n-spacing), or on [0, n(2)] (n-spacing), the rate is slower, root n/ln n. We also determine the optimal choice of the spacing Delta when it is constant, independent of the sample size n. While the rate of convergence of the MLE of C is root n p in this case, irrespective of the value of Delta, the value of the optimal spacing depends on H. It is 1 (unit-spacing) if H = 1/2 but is very large if H is close to 1.