Equivalence of Measures for Two-Dimensional Ornstein Uhlenbeck Processes and Matern Random Fields

• Activity Number: 663
• Type: Contributed
• Date/Time: Thursday, August 4, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistics and the Environment
• Author(s): Stefano Castruccio*+ and Debashis Mondal
• Companies: The University of Chicago and The University of Chicago
• Keywords: Matern Class ; Equivalent Measures ; Infill Asymptotics ; Markov Random Fields

Abstract

We will present some results about equivalence of measures for generalized Ornstein Uhlenbeck on the plane (Matern random fields with smoothness parameter zero). In particular, we will show that the range parameter of this field is not estimable, but the variance is. The usual theorems for pointwise defined random fields are not useful in this scenario, so we will use a Stochastic PDE approach to exploit and reinterpret some results previously known from probability theory. Furthermore, we will consider an infill asymptotic scheme via a Gaussian Markov Random Field approximation, and discuss convergence of the maximum likelihood estimator of the variance parameter for any choice of the range. Finally, with a similar approach, we will discuss related results for Matern random fields with rational spectrum in all dimensions, with particular emphasis on the open problem in four dimensions.