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Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. (English) Zbl 0889.62072
Summary: 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 $$|t|\to 0$$, where $$0<\alpha<2$$. Conventional estimates based on an equally spaced sample of size $$n$$ on the interval $$t\in[0,1]$$ have the property that $$\text{var} (\widehat \alpha)$$ is of order $$n^{-1}$$ for $$0<\alpha <3/2$$, but of lower order $$n^{2\alpha -4}$$ for $${3\over 2} <\alpha <2$$.
The motivation for writing this paper is twofold: to produce estimators of $$\alpha$$ which have variance of order $$n^{-1}$$ for all $$\alpha\in (0,2)$$ and 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.

##### MSC:
 62M09 Non-Markovian processes: estimation 62E20 Asymptotic distribution theory in statistics
##### Keywords:
filtering; increments; intrinsic process; misspecification bias