Gaussian pseudo-likelihood estimation for stationary processes on a lattice

For a wide class of second-order stationary spatial processes on a lattice, the statistical properties of the maximum Gaussian pseudo-likelihood estimators are studied. The estimators are natural as they imitate the theoretical prototypes of spatial best linear prediction. Under certain conditions, their asymptotic normality is established with the elements of the asymptotic variance matrix being simple functions of the variable auto-covariances. A short simulation study and a data example favor the use of the Gaussian pseudo-likelihood when the spatial covariance dependence is to be estimated.     

The table auto-regressive moving-average model for (categorical) stationary series: statistical properties (causality; from the all random to the conditional random)

A strictly stationary time series is modelled directly, once the variables' realizations fit into a table: no knowledge of a distribution is required other than the prior discretization. A multiplicative model with combined random ‘Auto-Regressive’ and ‘Moving-Average’ parts is considered for the serial dependence. Based on a multi-sequence of unobserved series that serve as differences and differences of differences from the main building block, a causal version is obtained; a condition that secures an exponential rate of convergence for its expected random coefficients is presented. For the remainder, writing the conditional probability as a function of past conditional probabilities, is within reach: subject to the presence of the moving-average segment in the original equation, what could be a long process of elimination with mathematical arguments concludes with a new derivation that does not support a simplistic linear dependence on the lagged probability values.     

Modelling data observed irregularly over space and regularly in time

When the data has been collected regularly over time and irregularly over space, it is difficult to impose an explicit auto-regressive structure over the space as it is over time. We study a phenomenon on a number of fixed locations. On each location the process forms an auto-regressive time series. The second-order dependence over space is reflected by the covariance matrix of the noise process, which is ‘white’ in time but not over the space. We consider the asymptotic properties of our inference methods, when the number of recordings in time only tends to infinity.     

The auto-regression and the moving-average

We explore some relationships in the second-order properties of a causal auto-regression and an invertible moving-average process with the same polynomial. We reveal that the inverse variance matrix for random variables from the auto-regression is equal to a conditional variance matrix of Gaussian random variables from the moving-average and vice versa. While the inverse variance matrix for the auto-regression can be written explicitly, we manage to write down the exact Gaussian likelihood of consecutive observations from the moving-average process, by using the properties of the auto-regression.