Estimating a covariance function having an un unknown scale kakameter
Jay H. Beder · Communication in Statistics- Theory and Methods · 1988
The structure of a family ρ of measures corresponding to a zero-mean Gaussian process with covariance αR(s,t) is discussed. When R itself is known (R and T completely arbitrary), it is shown that ρ is either homogeneous or composed of singular measures, depending on whether the reproducing kernel Hilbert space H(R,T) is finite- or infinite-dimensional. For the case dim H < ∞ the MLE α is given; when dim H = ∞ an almost sure discriminator is constructed. More generally, it is shown that when R itself depends upon a parameter θ (not necessarily a scalar) and certain broad assumptions are met, one may describe the orthogonal decomposition of P and estimate both a and θ.