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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 $\sigma \left(t\right)=\sigma \left(0\right)-{A|t|}^{\alpha }+{o\left(|t|}^{\alpha }\right)$ as $|t|\to 0$, where $0<\alpha <2$. Conventional estimates based on an equally spaced sample of size $n$ on the interval $t\in \left[0,1\right]$ have the property that $\text{var}\left(\stackrel{^}{\alpha }\right)$ is of order ${n}^{-1}$ for $0<\alpha <3/2$, but of lower order ${n}^{2\alpha -4}$ for $\frac{3}{2}<\alpha <2$.

The motivation for writing this paper is twofold: to produce estimators of $\alpha$ which have variance of order ${n}^{-1}$ for all $\alpha \in \left(0,2\right)$ 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