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Impact of the Sampling Rate on the Estimation of the Parameters of Fractional Brownian Motion
Author(s) -
Zhu Zhengyuan,
Taqqu Murad S.
Publication year - 2006
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2005.00470.x
Subject(s) - mathematics , fractional brownian motion , covariance , estimator , statistics , rate of convergence , brownian motion , gaussian , variance (accounting) , zero (linguistics) , gaussian process , mathematical analysis , channel (broadcasting) , physics , linguistics , accounting , philosophy , quantum mechanics , electrical engineering , business , engineering
. 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, n 2 ] ( 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.