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Tweedie’s Formula and Selection Bias
"... We suppose that the statistician observes some large number of estimates zi, each with its own unobserved expectation parameter µi. The largest few of the zi’s are likely to substantially overestimate their corresponding µi’s, this being an example of selection bias, or regression to the mean. Tweed ..."
Abstract

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We suppose that the statistician observes some large number of estimates zi, each with its own unobserved expectation parameter µi. The largest few of the zi’s are likely to substantially overestimate their corresponding µi’s, this being an example of selection bias, or regression to the mean. Tweedie’s formula, first reported by Robbins in 1956, offers a simple empirical Bayes approach for correcting selection bias. This paper investigates its merits and limitations. In addition to the methodology, Tweedie’s formula raises more general questions concerning empirical Bayes theory, discussed here as “relevance ” and “empirical Bayes information. ” There is a close connection between applications of the formula and James–Stein estimation. Keywords: Bayesian relevance, empirical Bayes information, James–Stein, false discovery rates, regret, winner’s curse
Sufficiency, Exponential Families, and Algebraically Independent Numbers
 Math. Meth. Statist
, 1999
"... We construct a continuous, strictly increasing, and bounded function T which maps Lebesgue almost the entire real line onto an algebraically independent set of real numbers. It follows that P n i=1 T (X i ) is a uniformly continuous onedimensional sufficient statistic for every IIDmodel P n = ..."
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We construct a continuous, strictly increasing, and bounded function T which maps Lebesgue almost the entire real line onto an algebraically independent set of real numbers. It follows that P n i=1 T (X i ) is a uniformly continuous onedimensional sufficient statistic for every IIDmodel P n = fP\Omega n : P 2 Pg such that each P 2 P has a Lebesgue density. It also follows that the oneparameter exponential family P with Lebesgue densities proportional to, say, exp(\Gammax 2 + #T (x)) is such that for the corresponding P n the order statistic is complete. Hence, in spite of exponentiality, P n does not admit a sufficient reduction beyond the trivial one. Under suitable stronger regularity conditions, such as continuous differentiability of T , none of the above is possible. 1 Introduction The present paper contributes to the understanding of the statistical notions "sufficiency " and "exponential families" by showing that continuity is not a suitable regularity condit...