Estimating the fractal dimension of a locally self similar Gaussian process using increments

Research Report SRR95-034

Estimating the fractal dimension of a locally self similar Gaussian process using increments

John T. Kent and Andrew T.A. Wood

Abstract: Consider the problem of estimating the parameter $\alpha$ of a stationary Gaussian process with covariance function $\sigma(t) = \sigma(0) - A |t|^\alpha + o(|t|^\alpha)$ as $|t| \rightarrow 0$, where $0 &le \alpha < 2$. Conventional estimates based on an equally-spaced sample of size $n$ on the interval $t \in [0, 1]$ have the property that var$(\hat{\alpha})$ is of order $n^{-1}$ for $0 < \alpha < {\frac {3}{2}}$, but of lower order $n^{2\alpha-4}$ for ${\frac {3}{2}} < \alpha < 2$. The motivation for writing this paper is twofold: (a) to produce estimators of $\alpha$ which have variance of order $n^{-1}$ for all $\alpha \in (0,2)$; and (b) to gain a better understanding of a simulation anomaly, whereby estimators of $\alpha$ with variance of order $n^{2\alpha-4}$ perform well in simulations when $\alpha$ is close to 2.

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