  In parametric estimation of covariance function of Gaussian processes, it is
often the case that the true covariance function does not belong to the
parametric set used for estimation. This situation is called the misspecified
case. In this case, it has been shown that, for irregular spatial sampling of
observation points, Cross Validation can yield smaller prediction errors than
Maximum Likelihood. Motivated by this observation, we provide a general
asymptotic analysis of the misspecified case, for independent and uniformly
distributed observation points. We prove that the Maximum Likelihood estimator
asymptotically minimizes a Kullback-Leibler divergence, within the misspecified
parametric set, while Cross Validation asymptotically minimizes the integrated
square prediction error. In a Monte Carlo simulation, we show that the
covariance parameters estimated by Maximum Likelihood and Cross Validation, and
the corresponding Kullback-Leibler divergences and integrated square prediction
errors, can be strongly contrasting. On a more technical level, we provide new
increasing-domain asymptotic results for independent and uniformly distributed
observation points.
