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

Stochastic kriging is a new metamodeling technique for effectively representing the mean response surface implied by a stochastic simulation; it takes into account both stochastic simulation noise and uncertainty about the underlying response surface of interest. We show theoretically, through some simplified models, that incorporating gradient estimators into stochastic kriging tends to significantly improve surface prediction. To address the issue of which type of gradient estimator to use, when there is a choice, we briefly review stochastic gradient estimation techniques; we then focus on the properties of infinitesimal perturbation analysis and likelihood ratio/score function gradient estimators and make recommendations. To conclude, we use simulation experiments with no simplifying assumptions to demonstrate that the use of stochastic kriging with gradient estimators provides more reliable prediction results than stochastic kriging alone.

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Enhancing Stochastic Kriging Metamodels with Gradient Estimators

Providing accurate and automated input-modeling support is one of the challenging problems in the application of computer simulation of stochastic systems. The models incorporated in current input-modeling software packages often fall short because they assume independent and identically distributed processes, even though dependent time-series input processes occur naturally in the simulation of many real-life systems. Therefore, this paper introduces a statistical methodology for fitting stochastic models to dependent time-series input processes. Specifically, an automated and statistically valid algorithm is presented to fit autoregressive-to-anything (ARTA) processes with marginal distributions from the Johnson translation system to stationary univariate time-series data. ARTA processes are particularly well suited to driving stochastic simulations. The use of this algorithm is illustrated with examples.

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Fitting Time-Series Input Processes for Simulation

Foundations and Methods of Stochastic Simulation

This chapter describes tools and techniques that are useful for optimization via simulation—maximizing or minimizing the expected value of a performance measure of a stochastic simulation—when the decision variables are discrete. Ranking and selection, globally and locally convergent random search and ordinal optimization are covered, along with a collection of “enhancements” that may be applied to many different discrete optimization via simulation algorithms. We also provide strategies for using commercial solvers.

Discrete Optimization via Simulation

This paper considers the determination of the relative merits of two or more system designs via stochastic simulation experiments by constructing simultaneous interval estimates of certain differences in expected performance. Tukey's all-pairwise-comparisons procedure, Hsu's multiple-comparisons-with-the-best procedure, and Dunnett's multiple-comparisons-with-a-control procedure are standard methods for making such comparisons. We propose refinements for all three procedures through the use of two variance reduction techniques: common random numbers and control variates. We show that the proposed procedures are better than the standard multiple-comparison procedures in the sense that they have a larger probability of containing the true difference and, at the same time, not containing zero when a difference exists.

Barry L. Nelson

Papers

Enhancing Stochastic Kriging Metamodels with Gradient Estimators

Fitting Time-Series Input Processes for Simulation

Foundations and Methods of Stochastic Simulation

Discrete Optimization via Simulation

Using Common Random Numbers and Control Variates in Multiple-Comparison Procedures