Traditional methods of analysis in brain imaging based on Gaussian random field theory may leave small, but significant changes in the signal level undetected, because the assumption of Gaussianity is not fulfilled. In group comparisons, the number of subjects in each group is usually small so the alternative strategy of using a non-parametric test may not be appropriate either because of low power. We propose to use a flexible, yet tractable model for a random field, based on kernel smoothing of a so-called Lévy basis. The resulting field may be Gaussian but there are many other possibilities, e.g. random fields based on Gamma, inverse Gaussian and normal inverse Gaussian (NIG) Lévy bases. We show that it is easy to estimate the parameters of the model and accordingly to assess by simulation the quantiles of a test statistic. A finding of independent interest is the explicit form of the kernel function that induces a covariance function belonging to the Matérn family.

Keywords: Covariance, cumulant, Gaussian random field, Matérn covariance function, non-Gaussian random field, normal inverse Gaussian Lévy basis