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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 α of a stationary Gaussian process with covariance function σ(t)=σ(0)-A|t| α +o(|t| α ) as |t|0, where 0<α<2. Conventional estimates based on an equally spaced sample of size n on the interval t[0,1] have the property that var(α ^) is of order n -1 for 0<α<3/2, but of lower order n 2α-4 for 3 2<α<2.

The motivation for writing this paper is twofold: to produce estimators of α which have variance of order n -1 for all α(0,2) and to gain a better understanding of a simulation anomaly, whereby estimators of α with variance of order n 2α-4 perform well in simulations when α is close to 2.

62M09Non-Markovian processes: estimation
62E20Asymptotic distribution theory in statistics