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Statistica Sinica 12(2002), 863-883



Zhengyuan Zhu and Michael L. Stein

University of Chicago

Abstract: We study the use of increments to estimate the fractal dimension of a two-dimensional fractal Brownian surface observed on a regular grid. Linear filters are used to describe differencing of two-dimensional surfaces and generalized variograms are defined based on them. We examine the practical performance of ordinary and generalized least squares estimators based on different filters by numerically calculating their asymptotic variance and also by simulation using an exact simulation method for fractional Brownian surface proposed by Stein (2002). An extensive numerical and simulation study of the practical performance of estimators based on different collections of lags is presented for Gaussian and non-Gaussian random fields, and a comparison to the much more computationally intensive restricted maximum likelihood estimator is provided.

Key words and phrases: Fractal, fractional Gaussian random field, generalized least squares, self-similar, variogram, REML.

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