Estimation of the Scale Parameter for a Misspecified Gaussian Process\n Model

Parameters of the covariance kernel of a Gaussian process model often need to\nbe estimated from the data generated by an unknown Gaussian process. We\nconsider fixed-domain asymptotics of the maximum likelihood estimator of the\nscale parameter under smoothness misspecification. If the covariance kernel of\nthe data-generating process has smoothness $\\nu_0$ but that of the model has\nsmoothness $\\nu \\geq \\nu_0$, we prove that the expectation of the maximum\nlikelihood estimator is of the order $N^{2(\\nu-\\nu_0)/d}$ if the $N$\nobservation points are quasi-uniform in $[0, 1]^d$. This indicates that maximum\nlikelihood estimation of the scale parameter alone is sufficient to guarantee\nthe correct rate of decay of the conditional variance. We also discuss a\nconnection the expected maximum likelihood estimator has to Driscoll's theorem\non sample path properties of Gaussian processes. The proofs are based on\nreproducing kernel Hilbert space techniques and worst-case case rates for\napproximation in Sobolev spaces.\n