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Sets of Estimates of Location

Edward E. Leamer

Abstract

If independent observations x are drawn from the distribution located at @m, f (x; @m)=c"3 exp[-g(x -@m)], and if g is symmetric and strictly convex, then the maximum likelihood estimate of @m lies between the smallest and largest folded sample observations. If the distribution has fatter tails than a normal distribution, then the maximum likelihood estimate lies between the smallest and largest means of trimmed subsamples. If the distribution is assumed to be symmetric and unimodal, the centers of tight clusters of observations can be maximum likelihood estimates. If observations are not independent, then there is no bound: given any example any number is a maximum likelihood estimate for some sampling distribution. Stationary is not sufficient to bound the estimate between the minimum and maximum observations.