Published online by Cambridge University Press: 09 April 2009
In [4] Halmos considers the following situation. Let be a class of distribution functions over a given (Borel) subset E of the real line, and F a function over . He investigates which functions F admit estimates that are unbiased over and what are all possible such estimates for any given F. In particular he shows that on the basis of a sample (of size n) one can always obtain an estimate of the first moment which is unbiased in and that the central moments Fm of order m ≧ 2 have estimates which are unbiased in if and only if n ≧ m, provided satisfies the following properties: Fm exists and is finite for all distributions in and includes all distributions which assign probability one to a finite number of points of E. Halmos also finds that symmetric estimates which are unbiased on are unique1 and have smaller variances on than unsymmetric unbiased estimates.