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.


62M09 Non-Markovian processes: estimation
62E20 Asymptotic distribution theory in statistics