2004/05/27 by Grace Chan, Andrew T. A. Wood · 1 citation
Economics, Econometrics and Finance · Environmental Science · Mathematics · Physics and Astronomy · #Financial Risk and Volatility Modeling #Hydrology and Drought Analysis #Theoretical and Computational Physics #math.ST #msc:62E20 #msc:62M99 #stat.TH
paper · pdf · doi:10.1214/009053604000000346
published as Annals of Statistics 2004, Vol. 32, No. 3, 1222-1260
openalex publication_date 2004/05/27 · arxiv created 2004/06/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We present the asymptotic distribution theory for a class of increment-based estimators of the fractal dimension of a random field of the form gX(t), where g:R→R is an unknown smooth function and X(t) is a real-valued stationary Gaussian field on Rd, d=1 or 2, whose covariance function obeys a power law at the origin. The relevant theoretical framework here is “fixed domain” (or “infill”) asymptotics. Surprisingly, the limit theory in this non-Gaussian case is somewhat richer than in the Gaussian case (the latter is recovered when g is affine), in part because estimators of the type considered may have an asymptotic variance which is random in the limit. Broadly, when g is smooth and nonaffine, three types of limit distributions can arise, types (i), (ii) and (iii), say. Each type can be represented as a random integral. More specifically, type (i) can be represented as the integral of a certain random function with respect to Lebesgue measure; type (ii) can be represented as the integral of a second random function with respect to an independent Gaussian random measure; and type (iii) can be represented as a Wiener–Itô integral of order 2. Which type occurs depends on a combination of the following factors: the roughness of X(t), whether d=1 or d=2 and the order of the increment which is used. Another notable feature of our results is that, even though the estimators we consider are based on a variogram, no moment conditions are required on the observed field gX(t) for the limit theory to hold. The results of a numerical study are also presented.