Consider estimating a functional $T(F)$ of an unknown distribution $F \in \mathbf{F}$ from data $X_1, \cdots, X_n$ i.i.d. $F$. Let $\omega(\varepsilon)$ denote the modulus of continuity of the functional $T$ over $\mathbf{F}$, computed with respect to Hellinger distance. For well-behaved loss functions $l(t)$, we show that $\inf_{T_n \sup_\mathbf{F}} E_Fl(T_n - T(F))$ is equivalent to $l(\omega(n^{-1/2}))$ to within constants, whenever $T$ is linear and $\mathbf{F}$ is convex. The same conclusion holds in three nonlinear cases: estimating the rate of decay of a density, estimating the mode and robust nonparametric regression. We study the difficulty of testing between the composite, infinite dimensional hypotheses $H_0: T(F) \leq t$ and $H_1: T(F) \geq t + \Delta$. Our results hold, in the cases studied, because the difficulty of the full infinite-dimensional composite testing problem is comparable to the difficulty of the hardest simple two-point testing subproblem.

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Geometrizing Rates of Convergence, III

We review major developments in the design of experiments, offer our thoughts on important directions for the future, and make specific recommendations for experimenters and statisticians who are students and teachers of experimental design, practitioners of experimental design, and researchers jointly exploring new frontiers. Specific topics covered are optimal design, computer-aided design, robust design, response surface design, mixture design, factorial design, block design, and designs for nonlinear models.

Experimental Design: Review and Comment

We introduce a new class of `standardized' optimality criteria which depend on `standardized' covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory. Besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces.

Designing Experiments with Respect to ‘Standardized’ Optimality Criteria

General Principles History and Overview of Design and Analysis of Experiments Klaus Hinkelmann Introduction to Linear Models Linda M. Haines Designs for Linear Models Blocking with Independent Responses John P. Morgan Crossover Designs Mausumi Bose and Aloke Dey Response Surface Experiments and Designs Andre I. Khuri and Siuli Mukhopadhyay Design for Linear Regression Models with Correlated Errors Holger Dette, Andrey Pepelyshev, and Anatoly Zhigljavsky Designs Accommodating Multiple Factor Regular Fractional Factorial Designs Robert Mee and Angela Dean Multistratum Fractional Factorial Designs Derek Bingham Nonregular Factorial and Supersaturated Designs Hongquan Xu Structures Defined by Factors R.A. Bailey Algebraic Method in Experimental Design Hugo Maruri-Aguilar and Henry P. Wynn Optimal Design for Nonlinear and Spatial Models Optimal Design for Nonlinear and Spatial Models: Introduction and Historical Overview Douglas P. Wiens Designs for Generalized Linear Models Anthony C. Atkinson and David C. Woods Designs for Selected Nonlinear Models Stefanie Biedermann and Min Yang Optimal Design for Spatial Models Zhengyuan Zhu and Evangelos Evangelou Computer Experiments Design of Computer Experiments: Introduction and Background Max Morris and Leslie Moore Latin Hypercubes and Space-Filling Designs C. Devon Lin and Boxin Tang Design for Sensitivity Analysis William Becker and Andrea Saltelli Expected Improvement Designs William I. Notz Cross-Cutting Issues Robustness of Design Douglas P. Wiens Algorithmic Searches for Optimal Designs Abhyuday Mandal, Weng Kee Wong, and Yaming Yu Design for Contemporary Applications Design for Discrete Choice Experiments Heiko Grossmann and Rainer Schwabe Plate Designs in High-Throughput Screening Experiments for Drug Discovery Xianggui Qu (Harvey) and Stanley Young Up-and-Down Designs for Dose-Finding Nancy Flournoy and Assaf P. Oron Optimal Design for Event-Related fMRI Studies Jason Ming-Hung Kao and John Stufken Index

Handbook of Design and Analysis of Experiments

Full Factorial Designs- to Full Factorial Designs with Two-Level Factors- Analysis of Full Factorial Experiments- Common Randomization Restrictions- More Full Factorial Design Examples- Fractional Factorial Designs- Fractional Factorial Designs: The Basics- Fractional Factorial Designs for Estimating Main Effects- Designs for Estimating Main Effects and Some Two-Factor Interactions- Resolution V Fractional Factorial Designs- Augmenting Fractional Factorial Designs- Fractional Factorial Designs with Randomization Restrictions- More Fractional Factorial Design Examples- Additional Topics- Response Surface Methods and Second-Order Designs- Special Topics Regarding the Design- Special Topics Regarding the Analysis- Appendices and Tables- Upper Percentiles of t Distributions, t- Upper Percentiles of F Distributions, F- Upper Percentiles for Lenth t Statistics, and- Computing Upper Percentiles for Maximum Studentized Residual- Orthogonal Blocking for Full 2 Factorial Designs- Column Labels of Generators for Regular Fractional Factorial Designs- Tables of Minimum Aberration Regular Fractional Factorial Designs- Minimum Aberration Blocking Schemes for Fractional Factorial Designs- Alias Matrix Derivation- Distinguishing Among Fractional Factorial Designs

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A Comprehensive Guide to Factorial Two-Level Experimentation

This article outlines existing knowledge on optimal designs for comparing test treatments with controls under 0-, 1- and 2-way elimination of heterogeneity models. The results are motivated through numerical examples.

/pdf/optimal-designs-for-comparing-test-treatments-with-controls-db0b57rot1.pdf

Optimal designs for comparing test treatments with controls

In experimental situations where a factorial design with all factors occurring at two levels is to be run in a time sequence, the usual advice given to the experimenter is that the order of runs should be randomized before the experiment is performed; however, randomization may lead to an undesirable run order. For example, in a factory experiment, there may be a certain learning process that occurs over time as a result of making level changes in the factors being studied, or there may be equipment wear-out. In either case the observations obtained will be affected by uncontrollable variables that are highly correlated with the time or position in which they occur, and randomization may lead to a run order in which the estimates of factor effects are adversely effected by the presence of the trend. Joiner and Campbell (1976) gave some more specific examples as to when trend effects can occur in sequential experiments. Therefore, it is important to consider systematic designs in which the estimat...

The construction of trend-free run orders of two-level factorial designs

: The authors introduce the problem with an example. How should we design an experiment to compare 4 test treatments with a control, using 18 experimental units? As a statistical question we will not be able to answer it unless it is asked in a more precise manner. To begin with we need to postulate a model for the response observed upon application of a treatment, test treatment or control, to an experimental unit. This paper shall consider three possible models: 1) O-way elimination of heterogeneity model in which all experimental units are homogeneous before application of treatments; 2) 1-way elimination of heterogeneity model in which experimental units can be divided into several homogeneous blocks; and 3) 2-way elimination of heterogeneity model in which the experimental units can be conceptually arranged according to rows and columns.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA185277&Location=U2&doc=GetTRDoc.pdf

Recent Discoveries on Optimal Designs for Comparing Test Treatments with Controls.

On the optimality of chemical balance weighing designs

SUMMARY This paper is an investigation into the E-optimality of regular graph designs within various classes of proper block designs. Several sufficient conditions are given for the existence of an E-optimal regular graph design within the classes considered. It is shown that when a regular graph design exists whose minimum non-zero eigenvalue is large enough, then an E-optimal regular graph design exists. Several examples are given to show how the derived sufficient conditions can be applied. LET 9 denote the class of all connected block designs having v treatments arranged in b blocks of size k. The designs in 9 are called proper because all blocks contain the same number of experimental units. It is assumed throughout this paper that bk/v = r is an integer. We will be considering the determination of E-optimal designs within various classes and subclasses of ?. Suppose d E -9. The structure of d is completely determined by its v x b incidence matrix Nd whose entries ndij give the number of times treatment i occurs in block j. When the entries of Nd assume only the values zero and one, the design is called binary. The ith row sum of Nd is denoted by rdi and represents the number of times treatment i is replicated in the design. If all row sums of Nd are equal, the design is said to be equireplicated. The matrix Nd N' is referred to as the concurrence matrix of the design and its entries are denoted by idij. If Nd N' has all of its diagonal elements equal and all of its off-diagonal elements differing by at most one, then d is called a regular graph design (RGD). If d is an RGD and its concurrence matrix has the additional property that all of its off-diagonal elements are equal, then d is called a balanced block design (BBD). Assuming the usual additive two-way classification model, it is well known that the coefficient matrix of the reduced normal equations for estimating the treatment effects is

Mike Jacroux

Papers

Optimal designs for comparing test treatments with controls

The construction of trend-free run orders of two-level factorial designs

Recent Discoveries on Optimal Designs for Comparing Test Treatments with Controls.

On the optimality of chemical balance weighing designs

On the E‐Optimality of Regular Graph Designs