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

Classical ranking-and-selection (R&S) procedures cannot be applied directly to select the best decision in the presence of distributional ambiguity. In this paper we propose a robust selection-of-the-best (RSB) formulation which compares decisions based on their worst-case performances over a finite set of possible distributions and selects the decision with the best worst-case performance. To solve the RSB problems, we design two-layer R&S procedures, either two-stage or fully sequential, under the indifference-zone formulation. The procedure identifies the worst-case distribution in the first stage and the best decision in the second. We prove the statistical validity of these procedures and test their performances numerically.

/pdf/robust-selection-of-the-best-2sgbw2pmoy.pdf

Robust selection of the best

Value-at-risk and conditional value at risk are two widely used risk measures, employed in the financial industry for risk management purposes. This tutorial discusses Monte Carlo methods for estimating value-at-risk, conditional value-at-risk and their sensitivities. By relating the mathematical representation of value-at-risk to that of conditional value-at-risk, it provides a unified view of simulation methodologies for both risk measures and their sensitivities.

/pdf/monte-carlo-estimation-of-value-at-risk-conditional-value-at-463ajyo0fl.pdf

Monte Carlo estimation of value-at-risk, conditional value-at-risk and their sensitivities

Consider the problem of estimating the expectation of a non linear function of a conditional expectation. This function is allowed to be non-differentiable and discontinuous at a finite set of points to capture practical settings. We develop a nested simulation strategy to estimate this via simulation and identify bias and optimized mean square error allocation. We show that this mean square error converges to zero at the rate Γ−2/3, as Γ → ∞, where Γ denotes the available computational budget. We also consider combining nested simulation technique with kernel based estimation methods. We note that while the kernel based method have a better convergence rate when the underlying random process has dimensionality less than or equal to three, pure nested simulation may be preferred when this dimension is above four.

/pdf/estimating-the-mean-of-a-non-linear-function-of-conditional-26fdbgimpb.pdf

Estimating the mean of a non-linear function of conditional expectation

Robust optimization (RO) is a common approach to tractably obtain safeguarding solutions for optimization problems with uncertain constraints. In this paper, we study a statistical framework to integrate data into RO, based on learning a prediction set using (combinations of) geometric shapes that are compatible with established RO tools, and a simple data-splitting validation step that achieves finite-sample nonparametric statistical guarantees on feasibility. We demonstrate how our required sample size to achieve feasibility at a given confidence level is independent of the dimensions of both the decision space and the probability space governing the stochasticity, and discuss some approaches to improve the objective performances while maintaining these dimension-free statistical feasibility guarantees.

Learning-based Robust Optimization: Procedures and Statistical Guarantees

Ranking-and-selection (R&S) procedures are often used to select the best configuration from a set of alternatives, and the set typically has fewer than 500 alternatives. However, there are many R&S or simulation optimization problems having thousands to tens of thousands alternatives. In this paper we discuss how to solve these problems using cloud computing. In particular, we discuss how cloud computing changes the paradigm that is currently used to design R&S procedures, and show a specific procedure that works efficiently under cloud computing. We demonstrate the practical usefulness of our procedure on a simulation optimization problem with more than 2000 feasible solutions using a small-scale cloud of CPUs created by us.

http://www.informs-sim.org/wsc11papers/360.pdf

L. Jeff Hong

Papers

Robust selection of the best

Monte Carlo estimation of value-at-risk, conditional value-at-risk and their sensitivities

Estimating the mean of a non-linear function of conditional expectation

Learning-based Robust Optimization: Procedures and Statistical Guarantees

Large-scale ranking and selection using cloud computing