On spectral estimation for a homogeneous random process on the circle

A homogeneous random process on the circle {X(P): P[set membership, variant]S} is a process whose mean is zero and whose covariance function depends only on the angular distance [theta] between the points, i.e. E{X(P)}[reverse not equivalent]0 and E{X(P)X(Q)}=R([theta]). We assume that the homogeneous process X(P) is observed at a finite number of points, equally spaced on the circle. Given independent realizations of the process, we first propose unbiased estimates for the parameters of the aliased spectrum and for the covariance function. We assume further that the process is Gaussian. The exact distribution of the spectral estimates and the asymptotic distribution of the estimates of the covariance function are derived. Finally, it is shown that the estimates proposed are in fact the maximum likelihood estimates and that they have minimum variance in the class of unbiased estimates.