Terry Speed wrote a column in the latest IMS Bulletin (the one I received a week ago) about the choice of the denominator in the variance estimator. That is, should s² involve n (number of observations), n-1 (degrees of freedom), n+1 or anything else in its denominator? I find the question more interesting than the answer (sorry, Terry!) as it demonstrates quite forcibly that there is not a single possible choice for this estimator of the variance but that instead the “optimal” estimator is determined by the choice of the optimality criterion: this makes for a wonderful (if rather formal) playground for a class on decision theoretic statistics. And I often use it on my students. Non-Bayesian mathematical statistics courses often give the impression that there is a natural (single) estimator, when this estimator is based on an implicit choice of an optimality criterion. (This issue is illustrated in the books of Chang and of Vasishth and Broe I discussed earlier. As well as by the Stein effect, of course.) I thus deem it worthwhile to impress upon all users of statistics that there is no such single optimal choice, that unbiasedness is not a compulsory property—just as well since most parameters cannot be estimated in an unbiased manner!—, and that there is room for a subjective choice of a “best” estimator, as paradoxical as it may sound to non-statisticians.
Archive for mathematical statistics
In a consequent package I got from CRC Press last month for CHANCE, there was this short book by Brian S. Everitt, A whistle-stop tour of statistics… Nice cover! The book is like an introductory undergraduate statistics course, except that it is much much terser and shorter, using 200 pages in A5 format with plenty of pictures. (It could also have been called a primer or a guidebook.) The table of contents is as follows
Some basics and describing data
Analysis of variance models
Linear regression models
Logistic regression and the generalized linear model
Longitudinal data and their analysis
Multivariate data and their analysis
There is nothing fundamentally wrong with the book, except that… I cannot fathom its purpose! Nor its readership. Again, it is way too short and terse to be used in an undergraduate course or for self-study. And it does not bring a new light on those standard topics when compared with most of introductory statistics books, being mostly traditional (even though it briefly mentions Bayesian inference on pp. 88-89). While the book is itself a summary of statistical methodology, it still finds room for a summary of the current notions at the end of each chapter. So I thus remain completely puzzled by the point in publishing A whistle-stop tour of statistics..!