THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Statistics for Spatial Data.

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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Multiple comparisons with the best, which is applicable to single-stage experiments, is introduced as a method for choosing the best of a finite number of system designs. Examples are given.

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Optimization over a finite number of system designs with one-stage sampling and multiple comparisons with the best

Selecting the optimal policy using simulation is subject to input model risk when input models that mimic real-world randomness in the simulation have estimation error due to finite sample sizes. I...

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Input–Output Uncertainty Comparisons for Discrete Optimization via Simulation

Some new quantile estimators that employ a control variate are introduced. The properties of these estimators do not depend on the usual assumption of joint normality between the random variable of interest and the control. Empirical results are presented.

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Control variates for quantile estimation

The focus on mean (long-run average) performance as the primary output measure produced by simulation experiments diminishes the usefulness of simulation for characterizing risk. Confidence intervals on means are often misinterpreted as measures of future risk, when in fact they are measures of error. We introduce the measure of risk & error (MORE) plot as a way to display and make intuitive the concepts of risk and error and thus support sound experiment design and correct decision making.

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The more plot: displaying measures of risk & error from simulation output

A comprehensive guide to applying three well-known variance reduction techniques is given, including point and interval estimators, software requirements, and guidelines for experiment design.

Barry L. Nelson

Papers

Optimization over a finite number of system designs with one-stage sampling and multiple comparisons with the best

Input–Output Uncertainty Comparisons for Discrete Optimization via Simulation

Control variates for quantile estimation

The more plot: displaying measures of risk & error from simulation output

Variance reduction for simulation practitioners