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

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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

Framework Convention on Climate Change

Response Surface Methodology (RSM) is a metamodel- based optimization method. Its strategy is to explore small subregions of the parameter space in succession instead of attempting to explore the entire parameter space directly. This method has been widely used in simulation optimization. However, RSM has two significant shortcomings: Firstly, it is not automated. Human involvements are usually required in the search process. Secondly, RSM is heuristic without convergence guarantee. This paper proposes Stochastic Trust Region Gradient-Free Method (STRONG) for simulation optimization with continuous decision variables to solve these two problems. STRONG combines the traditional RSM framework with the trust region method for deterministic optimization to achieve convergence property and eliminate the requirement of human involvement. Combined with appropriate experimental designs and specifically efficient screening experiments, STRONG has the potential of solving high-dimensional problems efficiently.

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Stochastic trust region gradient-free method (strong): a new response-surface-based algorithm in simulation optimization

We consider a problem of ranking and selection via simulation in the context of personalized decision making, where the best alternative is not universal but varies as a function of some observable covariates. The goal of ranking and selection with covariates (R&S-C) is to use simulation samples to obtain a selection policy that specifies the best alternative with certain statistical guarantee for subsequent individuals upon observing their covariates. A linear model is proposed to capture the relationship between the mean performance of an alternative and the covariates. Under the indifference-zone formulation, we develop two-stage procedures for both homoscedastic and heteroscedastic simulation errors, respectively, and prove their statistical validity in terms of average probability of correct selection. We also generalize the well-known slippage configuration, and prove that the generalized slippage configuration is the least favorable configuration for our procedures. Extensive numerical experiments are conducted to investigate the performance of the proposed procedures, the experimental design issue, and the robustness to the linearity assumption. Finally, we demonstrate the usefulness of R&S-C via a case study of selecting the best treatment regimen in the prevention of esophageal cancer. We find that by leveraging disease-related personal information, R&S-C can substantially improve patients' expected quality-adjusted life years by providing patient-specific treatment regimen.

Ranking and Selection with Covariates for Personalized Decision Making

Specifying a proper input distribution is often a challenging task in simulation modeling. In practice, there may be multiple plausible distributions that can fit the input data reasonably well, es...

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Distributionally Robust Selection of the Best

The indifference-zone (IZ) formulation of ranking and selection (R&S) is the foundation of many procedures that have been useful for choosing the best among a finite number of simulated alternatives. Of course, simulation models are imperfect representations of reality, which means that a simulation-based decision, such as choosing the best alternative, is subject to model risk. In this paper we explore the impact of model risk due to input uncertainty on IZ R&S. "Input uncertainty" is the result of having estimated ("fit") the simulation input models to observed real-world data. We find that input uncertainty may force the user to revise, or even abandon, their objectives when employing a R&S procedure, or it may have very little effect on selecting the best system even when the marginal input uncertainty is substantial.

Input uncertainty and indifference-zone ranking & selection

Selecting the solution with the largest or smallest mean of a primary performance measure from a finite set of solutions while requiring secondary performance measures to satisfy certain constraints is called constrained selection of the best CSB in the simulation ranking and selection literature. In this paper, we consider CSB problems with secondary performance measures that must satisfy probabilistic constraints, and we call such problems chance constrained selection of the best CCSB. We design procedures that first check the feasibility of all solutions and then select the best among all the sample feasible solutions. We prove the statistical validity of these procedures for variations of the CCSB problem under the indifference-zone formulation. Numerical results show that the proposed procedures can efficiently handle CCSB problems with up to 100 solutions, each with five chance constraints.

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L. Jeff Hong

Papers

Stochastic trust region gradient-free method (strong): a new response-surface-based algorithm in simulation optimization

Ranking and Selection with Covariates for Personalized Decision Making

Distributionally Robust Selection of the Best

Input uncertainty and indifference-zone ranking & selection

Chance Constrained Selection of the Best