On Distribution-Free Statistics
Z. W. Birnbaum, Herman Rubin · The Annals of Mathematical Statistics · 1954
Abstract : The concepts of distribution-free statistics and statistics of structure (d) were defined relative to families Omega and Omega' of cumulative probability functions. A counter example was presented to disprove the conjecture that every distribution-free statistic, symmetric in X1, X2...., Xn (a sample of a 1-dimensional random variable X with a continuous cumulative probability function F) with Omega = Omega' = Omega 2, has structure (d). Strongly distribution-free statistics in a family Omega* with respect to Omega' were defined, and a proof was presented of the equivalence of statistics of structure (d) and strongly distribution-free statistics under the condition that Omega = Omega' = Omega*.