Abstract
The sample mean estimator of the population mean is considered for autocorrelated superpopulations in one dimension. A pure superpopulation model is assumed. The natural conjecture, following work by Cochran and Hajek in the classical case is that the mean squared error is minimized for a convex autocorrelation function when the sample is equally spaced. This turns out to be exactly true on the circle and approximately true on the line. A special representation of convex autocorrelation functions (essentially due to Hajek) helps with the analysis. A general asymptotic solution is proposed for the non-convex case with specific solutions for moving average processes of order 2.
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