Printed in Great Britain Modality of the restricted likelihood for spatial Gaussian

Numerical simulations recently reported in the literature have shown that the profile likelihood associated with the estimation of covariance range parameters for a Gaussian field can be multimodal. Here we complement these recent results by considering covari-ances with unknown nugget, scale and range parameters. Estimation is performed in a restricted maximum likelihood framework. For unbounded sampling domains and known range parameter, conditions ensuring asymptotic unimodality of the restricted likelihood are derived. For bounded sampling domains, Monte Carlo simulations indicate that if the range parameter is known, multimodality is relatively rare unless the sample size is small. For unknown range parameter, frequencies of multimodality can be large and may increase with the sample size for the Gaussian and spherical covariance models