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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 ..."
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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
dashes show the 100 largest z[i] values Frequency
, 1000
"... • Observe zi ∼ N(µi, 1) for i = 1, 2,..., N • Select the m biggest ones: z(1)> z(2)> z(3)> · · ·> z(m) • Question: µ values? What can we say about their corresponding • Selection Bias selected z’s. The µ’s will usually be smaller than the ..."
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• Observe zi ∼ N(µi, 1) for i = 1, 2,..., N • Select the m biggest ones: z(1)> z(2)> z(3)> · · ·> z(m) • Question: µ values? What can we say about their corresponding • Selection Bias selected z’s. The µ’s will usually be smaller than the
• Marginal Density
, 1000
"... • Observe zi ∼ N(µi, 1) for i = 1, 2,..., N • Select the m biggest ones: z(1)> z(2)> z(3)> · · ·> z(m) • Question: µ values? What can we say about their corresponding • Selection Bias selected z’s. The µ’s will usually be smaller than the ..."
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• Observe zi ∼ N(µi, 1) for i = 1, 2,..., N • Select the m biggest ones: z(1)> z(2)> z(3)> · · ·> z(m) • Question: µ values? What can we say about their corresponding • Selection Bias selected z’s. The µ’s will usually be smaller than the