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Assessing the significance of the correlation between the components of a bivariate random field is of great interest in the analysis of spatial data. This problem has been addressed in the literature using suitable hypothesis testing procedures or using coefficients of spatial association between two sequences. In this paper, testing the association between autocorrelated variables is addressed for the components of a bivariate Gaussian random field using the asymptotic distribution of the maximum likelihood estimator of a specific parametric class of bivariate covariance models. Explicit expressions for the Fisher information matrix are given for a separable and a nonseparable version of the parametric class, leading to an asymptotic test. The empirical evidence supports our proposal, and as a result, in most of the cases, the new test performs better than the modified t test even when the bivariate covariance model is misspecified or the distribution of the bivariate random field is not Gaussian. Finally, to illustrate how the proposed test works in practice, we study a real dataset concerning the relationship between arsenic and lead from a contaminated area in Utah, USA.

We study composite likelihood estimation of the covariance parameters with data from a one-dimensional Gaussian process with exponential covariance function under fixed domain asymptotics. We show that the weighted pairwise maximum likelihood estimator of the microergodic parameter can be consistent or inconsistent, depending on the range of admissible parameter values in the likelihood optimization. On the contrary, the weighted pairwise conditional maximum likelihood estimator is always consistent. Both estimators are also asymptotically Gaussian when they are consistent, with asymptotic variance larger or strictly larger than that of the maximum likelihood estimator. A simulation study is presented in order to compare the finite sample behavior of the pairwise likelihood estimators with their asymptotic distributions.