Symmetrically distributed and unbiased estimators in linear models

Let Y be distributed symmetrically about Xβ. Natural generalizations of odd location statistics, say T‘Y’, and even location-free statistics, say W‘Y’, that were used by Hogg ‘1960, 1967)’ are introduced. We show that T‘Y’ is distributed symmetrically about β and thus E[T‘Y’] = β and that each element of T‘Y’ is uncorrelated with each element of W‘Y’. Applications of this result are made to R-estiraators and the result is extended to a multivariate linear model situation.