Hypothesis testing for the smoothness parameter of Matérn covariance model on a regular grid

We consider the hypothesis testing problem for the smoothness parameter ν in a stationary isotropic Gaussian random field with Matérn covariance. For the data observed on a regular grid, we construct the rejection region for one-tailed tests, and starting from there, we develop a chain-like testing procedure, which can determine an interval containing the true value of ν. Such an interval can help improve the performance of various estimation methods for ν, such as restricting the parameter space or validating the assumptions for the asymptotic properties of the estimator. The test statistic is built on recursive applications of the Laplace operator to the observations. For this statistic, the fixed-domain asymptotic normality is established and the forms of asymptotic mean and variance are derived. Therefore, the proposed tests are guaranteed to have correct asymptotic size under certain conditions. Simulation studies indicate that our proposed methods are efficient for moderate sample sizes. As an application of the chain-like testing procedure, we provide a method of choosing the number of differencing for a local Whittle-likelihood type estimator of ν proposed by Wu, Lim, and Xiao, and show that it can avoid obtaining inconsistent estimates of ν via a numerical experiment.