Exploring Dependence with Data on Spatial Lattices

The application of Markov random field models to problems involving spatial data on lattice systems requires decisions regarding a number of important aspects of model structure. Existing exploratory techniques appropriate for spatial data do not provide direct guidance to an investigator about these decisions. We introduce an exploratory quantity that is directly tied to the structure of Markov random field models based on one parameter exponential family conditional distributions. This exploratory diagnostic is shown to be a meaningful statistic that can inform decisions involved in modeling spatial structure with statistical dependence terms. In this article, we develop the diagnostic, show that it has stable statistical behavior, illustrate its use in guiding modeling decisions with simulated examples, and demonstrate that these properties have use in applications

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

Parameters of Gaussian multivariate models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical when the sample size is very large, as, for example, in the case of massive georeferenced data sets. In this paper, we study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, designed to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first-order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations.Approximate likelihood Massive data sets Computational efficiency Statistical efficiency analysis Spatial statistics Autoregressive processes on a lattice

Asymptotic properties of three alternative estimators of spatial parameters

Parameters of Gaussian spatial models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical for large data sets. We study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, which are meant to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations