A User's Guide to Measure-Theoretic Probability

Use of Tables, and Applications), with 236 pages of text, 110 pages of tables; and a CD-ROM with software (written in Java) that implements the methods presented in the book. According to the Preface, the authors were motivated to write the book in part because of a consulting project. They “noticed that the recommended methods [for estimating proportions] are dif cult to handle for practitioners and, moreover, they are inaccurate and yield often useless results. ” I have a problem with these words all appearing in the same sentence. The methods that are at times useless (e.g., an interval based on a normal approximation) are not dif cult to handle, and methods that can be dif cult to handle (e.g., an exact con dence interval without the right software) do not yield useless results. At any rate, the authors decided to basically start from scratch and develop their own theory of estimation. This theory is based on a geometric approach and seems closely related to  ducial intervals as developed in several articles by Neyman. One important feature of the authors’ method is that it allows one to specify bounds on the possible parameter values. This of course can also be done within the existing paradigms; a frequentist can use a constrained maximum likelihood estimator, and a Bayesian can specify a prior with support on only part of the parameter space. I think the authors are more than a little bit presumptuous when they say (p. xiv) that “the handbook represents a  rst step toward developing statistics as a (natural) science, and further steps must follow. ” Later (on p. xvii), we learn that “it may also be looked upon as a  rst step in reconciling classical statistics with Bayes statistics. ” Apparently this reconciliation will require the Bayesians to make all the concessions, given the authors’ reverence for Jerzy Neyman. They even display Neyman’s portrait in the user interface for their software. Besides con dence intervals (which the authors sometimes call “measurement intervals” for reasons that are unclear to me), the authors also discuss “prediction regions. ” Interestingly, a prediction region does not require any data; rather, it is like an acceptance region for a null hypothesis on the Bernoulli parameter p. Again, we can specify bounds on the parameter values. I do not care much for the terminology, because it suggests that we are predicting a future result rather than testing a hypothesis. The book is a tough read. The writing is hard to follow and somewhat overblown and confrontational, and the notation is thick. There are parts that I think are incorrect; for example, in discussing the exact con dence interval for a Bernoulli parameter in Section 3.4. 1, the claim is made that an inferior interval estimator is obtained if one uses only the sample sum when the sample of individual outcomes is known. In other words, a suf cient statistic is claimed to be insuf cient. This error is of no consequence, however, because the authors’ own tables and software require only the sample sum. In the end, I believe that an engineer or scientist will not be making a serious mistake if he or she uses the accompanying software to obtain a con dence interval. I base this belief on the fact that the few examples I tried yielded results quite close to Bayes credibility intervals based on a uniform prior. It is indeed interesting to have software that can output a critical region for testing a Bernoulli parameter with an interval constraint on its value. However, as Tietjen (1986, p. 37) says, “ ducial intervals are understood by only a few and used by fewer still. ” My guess is that that also will be the fate of this work.