Empirical Likelihood

true for calculating moments and generating functions. Graphing also is easy using Mathematica’s powerful graphical capabilities. A few pages are devoted to copulas; again, these are easy to implement. Chapter 7 (26 pp.) covers unbiased estimation of population moments, moments of moments, and augmented symmetrics and power sums. Moment conversion formulas reappear and are extended to the k statistics (unbiased estimators of cumulants) and the h statistics (unbiased estimators of central moments). The power of this book is amply conveyed when the former are easily extended to the generalized k statistics known as “polykays” (plural of poly-k); and in an apparent novelty, the latter are extended to “poly-aches” (plural of poly-h). The accuracy of mathStatica/Mathematica is illustrated by revealing errors in long-used formulas in published references (e.g., Eq. 12.70 in Stuart and Ord 1986; the product-cumulant formula Š4325). This formula contains some 12 terms, and the correct version (with a demonstration of correctness) is given in this chapter. Examples of published errors in other references occur throughout the book. While testing mathStatica, the authors compiled a lengthy list of such errors that is available at the book’s website, www.mathstatica. com. Chapter 8, “Asymptotic Distributions” (24 pp.), covers convergence in distribution, asymptotic distribution, the central-limit theorem, convergence in probability, van Beek’s bound, and the inequalities of Chebyshev and Markov. A distinct pedagogical advantage of using the computer to teach these concepts is that the student can  ddle with n and other parameters, and observe the convergence rates. With only a couple of exceptions, all cases considered are i.i. d. , so the treatment is at an elementary level. Chapter 9, “Statistical Decision Theory” (24 pp.), covers loss and risk, mean squared error as risk, and order statistics. With the power of a computer, it becomes easy to compute and graph risk and other such quantities. The mathStatica function OrderStat automates the calculation of the pdf or joint pdf of an order statistic, and it is used extensively in this chapter. Chapter 10, on unbiased parameter estimation (24 pp.) covers Fisher information, best unbiased estimators, suf cient statistics, and minimum-variance unbiased estimation, including the Rao–Blackwell theorem. As always, a bene t of mathStatica/Mathematica is the elimination of technical drudgery. Example 2 of this chapter, computing the information lost due to censoring of a Poisson variable with parameter ‹, is almost trivial. Thus, exploring various aspects of the problem (e.g., graphing relative information against ‹) becomes easy. Chapter 11, “Principles of Maximum Likelihood Estimation” (30 pp.), covers the likelihood function, maximum likelihood estimation, and properties of maximum likelihood estimators (MLEs). This chapter constitutes preparation for Chapter 12 (42 pp.), which covers use of the FindMaximum command and various optimization algorithms, including BFGS and Newton–Raphson. It pays admirable attention to numerical detail. The three basic types of MLE standard errors (Fisher, Hessian, and OPG) are introduced and compared. An extended autoregressive conditional heteroscedasticity (ARCH) example is used illustratively. I can af rm that doing nonlinear estimation is much easier with mathStatica than with Mathematica alone. Various appendices, notes, and references (64 pp.) conclude the text. The entire book, chapter by chapter, is duplicated as Mathematica notebooks. There is no need to read a sequence of commands in the text and then type them into the computer; the reader can open up the relevant chapter notebook and click on the commands. The reader can interactively modify and experiment with the examples in the text. For readers who do not already have it, the book comes with a 30-day trial version of Mathematica. Sixty days would have been better, and unfortunately this trial version also does not allow the user to save worksheets. Each chapter concludes with several exercises, 101 altogether. A detailed solutions manual is available to instructors of courses using the text. The small number of exercises should not cause concern; any instructor, having seen the exercises and the detailed solutions, would have little trouble conjuring up more exercises. Moreover, any mathematical statistics text will be  lled with problems that can be assigned for solution with mathStatica. The preface makes the salient point that “when one is armed with mathStatica, the whole notion of what is dif cult changes,” and this theme recurs throughout the book. Already a couple of examples have been given. Here is one more: Moments of moments are conceptually easy, and for higher orders are notoriously dif cult without mathStatica. With it, they are no harder than invoking a single command. This book will be of immediate use to four classes of persons: (1) any person who would do more of the Hogg–Craig–style basics, if only it were much easier, (2) people who work with real data whose needs are ill met by traditional statistical packages (e.g., actuaries); (3) persons whose computational needs consistently bump into the bounds of traditional software (e.g., econometricians working with esoteric distributions), and (4) teachers who want to innovate the traditional mathematical statistics curriculum. This book has the potential to revolutionize the teaching of mathematical statistics. Typically, a professor using the text by Hogg and Craig (1995) might choose a few problems from each section and assign them, knowing that tedious mathematical drudgery will occupy much of the student’s time. However, if Mathematical Statistics with Mathematica is used as a supplement, then the professor can simply ask students to solve all or most of the problems in the chapter. Again, if the student does not know the Hogg and Craig material, then mathStatica will be of little use. I expect that in a couple of years, some journal will publish a “report from the  eld” that describes the collective experience of professors who have chosen to use this book in conjunction with a standard  rst-year text. Such a report will make for most interesting—and perhaps revolutionary—reading. While writing an earlier article (McCullough 2000), Internet searches convinced me (no great surprise) that very few persons were using Mathematica for statistical purposes. I shall be equally surprised if the present volume does not change this. These internet searches put me in touch with one of the authors of the book, and I wound up beta-testing mathStatica. Other than a review copy of the book, I received nothing other than a wonderful time playing with the included software.