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

Measuring risks in asset portfolios has been one of the central topics in the financial industry. Since the introduction of coherent risk measures, studies on risk measurement have flourished and measures beyond value-at-risk, such as expected shortfall, have been adopted by academics and practitioners. However, the complexity of financial products makes it very difficult and time consuming to perform the numerical tasks necessary to compute these risk measures. In this paper, we introduce a stochastic kriging metamodel-based method for efficient estimation of risks and their sensitivities. In particular, this method uses gradient estimators of assets in a portfolio and gives the best linear unbiased predictor of the risk sensitivities with minimum mean squared error. Numerical comparisons of the proposed method with two other stochastic kriging based approaches demonstrate the promising role that the proposed method can play in the estimation of financial risk.

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Stochastic kriging for conditional value-at-risk and its sensitivities

Many indifference-zone Ranking-and-Selection (R&S) procedures have been invented for choosing the best simulated system. To obtain the desired Probability of Correct Selection (PCS), existing procedures exploit knowledge about the particular combination of system performance measure (e.g., mean, probability, variance, quantile) and assumed output distribution (e.g., normal, exponential, Poisson). In this article, we take a step toward general-purpose R&S procedures that work for many types of performance measures and output distributions, including situations where different simulated alternatives have entirely different output distribution families. There are only two versions of our procedure: with and without the use of common random numbers. To obtain the required PCS we exploit intense computation via bootstrapping, and to mitigate the computational effort we create an adaptive sample-allocation scheme that guides the procedure to quickly reach the necessary sample size. We establish the asym...

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General-purpose ranking and selection for computer simulation

Suppose that one wishes to evaluate the distribution of profit and loss (P&L) resulting from a dynamic trading strategy. A straightforward method is to simulate thousands of paths (i.e., time series) of relevant financial variables and to track the resulting P&L at every time at which the trading strategy rebalances its portfolio. In many cases, this requires numerical computation of portfolio weights at every rebalancing time on every path, for example, by a nested simulation performed conditional on market conditions at that time on that path. Such a two-level simulation could involve many millions of simulations to compute portfolio weights, and thus be too computationally expensive to attain high accuracy. We show that response surface methodology enables a more efficient simulation procedure: in particular, it is possible to do far fewer simulations by using kriging to model portfolio weights as a function of underlying financial variables.

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Response surface methodology for simulating hedging and trading strategies

Nelson and Staum derived ranking-and-selection (R&S) procedures that employ control-variate (CV) estimators instead of sample means to obtain greater statistical efficiency However, control-variate estimators require more computational effort than sample means, and effective controls must be identified In this paper, we present a new CV screening procedure to avoid much of the computation cost along with a better paired CV model than that of Nelson and Staum We also present a two-stage CV combined procedure that captures the ability to eliminate inferior systems in the first stage and the statistical efficiency of control variates for selection in the second stage Some guidelines about control-variate selection and an empirical evaluation are provided

http://users.iems.northwestern.edu/~staum/TsaiNelsonStaum_ScreeningWithControlVariates.pdf

Combined screening and selection of the best with control variates

We introduce the topic of this book, explain what we mean by stochastic computer simulation and provide examples of application areas. We motivate the remaining chapters in the book through two in-depth examples. These examples also help clarify several concepts and techniques that are pervasive in simulation theory and practice.

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Barry L. Nelson

Papers

Stochastic kriging for conditional value-at-risk and its sensitivities

General-purpose ranking and selection for computer simulation

Response surface methodology for simulating hedging and trading strategies

Combined screening and selection of the best with control variates

Stochastic Computer Simulation