Asymptotic Theory of Fractal Index and Scale Parameter of Irregularly Observed Gaussian Field

• Activity Number: 352
• Type: Contributed
• Date/Time: Tuesday, August 2, 2011 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Myoungji Lee*+ and Michael Stein
• Companies: The University of Chicago and The University of Chicago
• Keywords: fractal dimension ; increments ; variogram ; Gaussian random process ; least squares estimation ; fixed domain asymptotics

Abstract

The fractal dimension is a scale invariant measure of quantifying how rough or smooth a curve or surface is. Assuming that the process is stationary isotropic Gaussian and observations are even, the variogram based estimation of the fractal dimension is consistent and follows normal or Rosenblatt distribution with slower speed than root n depending on the smoothness of the field. Uniform root n convergence to normal distribution is achieved by using the high order increments in a lattice. This paper extends the work by allowing uneven observations, which is far more natural than evenly spaced data. We first expand the concept of the increment to accommodate irregularity of observations. Then using the squared increments, the least squares estimators of the fractal index and the scale parameter of the covariance function will be proposed and their consistency and asymptotic normality will be shown. Under fixed domain asymptotics, the same rates of convergence are achieved for unequally spaced data as for equally spaced data under some regularity assumptions on the amount of irregularity of locations of observations.