Estimation of the Scale Parameter for a Misspecified Gaussian Process Model
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Parameters of the covariance kernel of a Gaussian process model often need to be estimated from the data generated by an unknown Gaussian process. We consider fixed-domain asymptotics of the maximum likelihood estimator of the scale parameter under smoothness misspecification. If the covariance kernel of the data-generating process has smoothness ν_0 but that of the model has smoothness ν≥ν_0, we prove that the expectation of the maximum likelihood estimator is of the order N^2(ν-ν_0)/d if the N observation points are quasi-uniform in [0, 1]^d. This indicates that maximum likelihood estimation of the scale parameter alone is sufficient to guarantee the correct rate of decay of the conditional variance. We also discuss a connection the expected maximum likelihood estimator has to Driscoll's theorem on sample path properties of Gaussian processes. The proofs are based on reproducing kernel Hilbert space techniques and worst-case case rates for approximation in Sobolev spaces.
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