Impact of the Sampling Rate on the Estimation of the Parameters of Fractional Brownian Motion

• Activity Number: 233
• Type: Contributed
• Date/Time: Tuesday, August 10, 2004 : 12:00 PM to 1:50 PM
• Sponsor: IMS
• Author(s): Zhengyuan Zhu*+ and Murad S. Taqqu
• Companies: University of North Carolina, Chapel Hill and Boston University
• Keywords: fractional Gaussian noise ; self-similarity ; long-range dependence ; maximum likelihood estimation

Abstract

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 \sqrt{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, \sqrt{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 \sqrt{n} 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.