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. Tweedie’s formula, first reported by Robbins in 1956, offers a simple empirical Bayes approach for correcting selection bias. This article 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.