Efficient Estimation of a Regression Parameter for Certain Second Order Processes
0. Summary. The problem of estimation of a single regression parameter for a process with fixed known regression function and unknown covariance is attacked using a Hilbert space representation of the process. Some general results are obtained which characterize efficiency classes of covariances-that is, classes for each of which there exists a single estimate that is efficient for all members. These results are applied to both the discrete parameter and the continuous parameter stationary process with rational spectral density. Some special results are also obtained concerning the efficiency of the least square estimate. 1. Introduction. Let x(t) be a second order complex-valued process with mean value function zero and covariance (1.1) E[x(t)x(s)] = R(t, s), and suppose that the process (1.2) y(t) = kp (t) + x(t) is observed for the parameter t in a subset Cr of the real line. The function r(t) is known, and the parameter k is to be estimated. The subsets of interest will be the intervals (- oo < t < T) and (O < t ? T) for the continuous parameter process and the integers (t = T, T -1, ..) and (t = T, T-l, -,* ,0) for the discrete parameter process. A linear unbiased estimate with finite variance will be represented as a linear functional (1.3) E T = kT[y(t), t e CTJ, which is the limit in quadratic mean of unbiased finite linear combinations of the y(t) process, that is, Mm