Bayesian Statistics: An Introduction, 4th Edition

8.3 The baseball example

Efron and Morris’s example on baseball statistics was outlined in Section 8.1. As their primary data, they take the number of times hits Si or equivalently the batting averages Yi=Si/n of r=18 major league players as they were recorded after n=45 times at bat in the 1970 season. These were, in fact, all the players who happened to have batted exactly 45 times the day the data were tabulated. If Xi and are as in Section 8.1, so that approximately

then we have a case of the hierarchical normal model. With the actual data, we have

and so with

the empirical Bayes estimator for the takes the form

so giving estimates

We can test how well an estimator performs by comparing it with the observed batting averages. We suppose that the ith player had Ti hits and was at bat ...