Estimation with selected binomial information or do you really believe that dave winfield is batting .471?

George Casella, Roger L. Berger

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


Often sports announcers, particularly in baseball, provide the listener with exaggerated information concerning a player’s performance. For example, we may be told that Dave Winfield, a popular baseball player, has hit safely in 8 of his last 17 chances (a batting average of .471). This is biased, or selected information, as the “17” was chosen to maximize the reported percentage. We model this as observing a maximum success rate of a Bernoulli process and show how to construct the likelihood function for a player’s true batting ability. The likelihood function is a high-degree polynomial, but it can be computed exactly. Alternatively, the problem yields to solutions based on either the EM algorithm or Gibbs sampling. Using these techniques, we compute maximum likelihood estimators, Bayes estimators, and associated measures of error. We also show how to approximate the likelihood using a Brownian motion calculation. We find that although constructing good estimators from selected information is difficult, we seem to be able to estimate better than expected, particularly when using prior information. The estimators are illustrated with data from the 1992 Major League Baseball season.

Original languageEnglish (US)
Pages (from-to)1080-1090
Number of pages11
JournalJournal of the American Statistical Association
Issue number427
StatePublished - Sep 1994


  • Brownian motion
  • EM algorithm
  • Gibbs sampling
  • Selection bias

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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