On Conditional Expectations of Location Statistics

Abstract

Abstract Let T 1 and T 2 be two odd location statistics, such as the sample mean and the sample median. Let S 1 and S 2 be two even location-free statistics, such as the sample variance and the sample range. When sampling from a symmetric distribution, it is proved that the conditional expectation of either odd location statistic, given the even location-free statistics, is equal to θ, the midpoint of the distribution. This implies that if the random weights W 1 and W 2 are functions of even location-free statistics so that W 1 + W 2 = 1, then W 1 T 1 + W 2 T 2 is an unbiased estimator of θ.