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

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  • Zhengyuan Zhu
  • Murad S. Taqqu

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 . However, if the observations are equally spaced on [0, 1] (1/n‐spacing), or on [0, n2] (n‐spacing), the rate is slower, . We also determine the optimal choice of the spacing Δ when it is constant, independent of the sample size n. While the rate of convergence of the MLE of C is in this case, irrespective of the value of Δ, 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.

Suggested Citation

  • Zhengyuan Zhu & Murad S. Taqqu, 2006. "Impact of the Sampling Rate on the Estimation of the Parameters of Fractional Brownian Motion," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(3), pages 367-380, May.
    • Handle: RePEc:bla:jtsera:v:27:y:2006:i:3:p:367-380
    DOI: 10.1111/j.1467-9892.2005.00470.x
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    Cited by:

    1. José León & Carenne Ludeña, 2015. "Difference based estimators and infill statistics," Statistical Inference for Stochastic Processes, Springer, vol. 18(1), pages 1-31, April.

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