Statisticians have long worked on ways of estimating population parameters, and methods such as maximum likelihood and least squares are well established. These methods provide “good” estimators, often in the sense that the estimators are unbiased and have a small standard error. They work well for sets of independent random variables, but with a lack of independence the work gets much harder. Both Bayesian and classical statisticians have been engaged in this work.
It came as a surprise when Charles Stein and his graduate assistant Willard James showed in the 1960’s that it was possible to have estimators with smaller standard errors than had been thought for a long time. They showed that in more than three dimensions there exist estimators with smaller standard errors than the usual maximum likelihood estimators.
Much work has been done on Stein estimators since that time. This book contains four lectures given by different people on Stein estimators in Singapore in 2003. The first lecture was by Louis H. Y. Chen and Qi-Man Shao entitled “Stein’s method for normal approximations,” the second was by Torkel Erhardsson entitled “Stein’s method for Poisson and compound Poisson approximation,” the third by Aihua Xia entitled “Stein’s method and Poisson process approximation, and the final paper by Gesine Reinert entitled “Three general approaches to Stein’s method.”
The four papers give a good sense of the work currently being done on Stein estimators. The papers will be rewarding for statisticians with a mature background in mathematics currently working on Stein estimators. The papers may not be as rewarding for those who only want to get an understanding of what Stein estimators are all about, in spite of the title of the book. The many proofs in the papers distract from the getting an introduction to the Stein method.
Gudmund R. Iversen (firstname.lastname@example.org) is professor emeritus of statistics at Swarthmore College. Among his interests are statistics education and uses of Bayesian statistics.