The Bactra Review: Occasional and eclectic book reviews by Cosma Shalizi   134 E. J. G. Pitman, Some Basic Theory for Statistical Inference

Some Basic Theory for Statistical Inference

by E. J. G. Pitman

London: Chapman and Hall, 1979
"Monographs on Applied Probability and Statistics" series

Intermediate Statistics from an Advanced Point of View

This little book has its origins in a feeling of dis-satisfaction with the literature on theoretical statistics. Pitman felt, it seems, that even the best of it was just too messy and jerry-rigged — too kludgy — to meet the standards of the rest of applied mathematics, of which it is a part.
This book is an attempt to present some of the basic mathematical results required for statistical inference with some elegance as well as precision, and at a level which will make it readable by most students of statistics.

It is a largely successful attempt. In somewhat less than a hundred pages of main text, Pitman covers many of the key issues of the theory of statistical inference: using parametric probability distributions to model real-world phenomena, using data to learn about the parameters of those models, how to gauge the ability of different statistics to enable that learning, estimation of parameters, hypothesis testing and its efficiency, maximum likelihood estimation, and the convergence of empirical distribution functions.

Throughout, Pitman is generally clear, usually giving short, direct proofs of theorems with straightforward hypotheses. (I intend to steal his proof of the Cramér-Rao inequality when I teach that.) He makes abundant and ingenious use of the Hellinger metric for distances between probability distributions (though he doesn't call it that). He also has some harsh words, in several places, for the Fisher information, with examples of how it doesn't always behave in the way the word "information" leads one to expect. (He thinks it should be called "sensitivity".) Two weaknesses did strike me, however. (Assuming all samples are independent is a weakness, but so common as to not be striking.) One is that he often silently use a curious, non-standard version of the dominated convergence theorem, which he states in an appendix; this repeatedly had me going "huh?" in mid-proof. The other is the chapter on the convergence of empirical distribution functions, and Kolmogorov-Smirnov-type tests. There he avoids "crossing the Brownian bridge", that is, using advanced probability (the functional central limit theorem, empirical processes, etc.), by forcing readers to trudge through opaque, multi-page combinatorial calculations.

Pitman presumes his readers can find their way around in theoretical statistics and measure-theoretic probability, sort of, and wants to show them how different parts of the territory fit together, why the highways go where they do, and why they can't just always take them. It's strongly recommended if any of this sounds at all interesting.

vii + 110 pp., a figure, line graphs, index of names and subjects, minimal bibliography
Probability and Statistics
Out of print. Hardback ISBN 047026554X [Buy from Powell's]
3 December 2006