A Note on Confidence Sets for Random Variables
Lionel E. Weiss · The Annals of Mathematical Statistics · 1955
under equivalence [S, P]. A subfield equivalent to a statistic need not itself be a statistic. In an attempt to avoid this difficulty, one may define a pseudo-statistic as any subfield equivalent to a statistic. If Lemma 3 remained valid for pseudostatistics in the sense that a member of CT, is a pseudo-statistic if and only if it is equivalent to Sn., this would establish the desired result. The following example shows that this stronger version of Lemma 3 is not correct. Let S, be the class of all Lebesgue sets on the real line and So the class of all Lebesgue sets differing only by a set 0 from a set symmetric with respect to the origin. Clearly, { x} E So for all x so that So E C1r . Also So is a pseudostatistic since it is equivalent to the subfield induced by T(x) = lxl. But clearly So and S, are not equivalent.