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1995 | Buch

Optimal Designs for Time—Dependent Responses Kapitel lesen Erstes Kapitel lesen

MODA4 — Advances in Model-Oriented Data Analysis

Proceedings of the 4th International Workshop in Spetses, Greece June 5–9, 1995

herausgegeben von: Dr. Christos P. Kitsos, Dr. Werner G. Müller

Verlag: Physica-Verlag HD

Buchreihe : Contributions to Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

This volume is the proceedings of the 4th International Workshop on Model-Oriented Data Analysis. This series of events originated in 1987 at a meeting in Eisenach, that successfully brought together scientists from numerous countries of the 'East ' and 'West'. Now that this distinction is obsolete dialogue has been greatly facilitated, providing opportunities for this dialogue, however, is as vital as ever. The present meeting at Spetses, Greece from 5th to 9th of June 1995 again assembles statisticians from all over the world as this book documents. The hospitality offered by the University of Economics of Athens and the Korgialenios School made it possible to organize this workshop. The editors are also grateful to Intracom (Greece), the Ionian Bank and the Procter & Gamble Company (USA) for their generous support. We would particularly like to mention Dr. Michael Meredith, who being our contact person at Procter & Gamble, enabled us to publish these proceedings. Further thanks go to Dr. Peter Schuster from Physica Verlag Heidelberg for his continuing support of the project. The contributions to this volume were carefully selected from the submissions by the editors after a one stage refereeing process. We would like to thank the members of the MODA committee, A.C. Atkinson, R.D. Cook, V.V. Fedorov, P.Hackl, H. Lauter, B.Torsney, LN. Vuchkov, H.P.Wynn,and A.A. Zhigljavsky, who not only defined the main topics of the workshop, but also served as the referees.

Inhaltsverzeichnis

Frontmatter

Optimal Design

Frontmatter
Optimal Designs for Time—Dependent Responses
Abstract
The purpose of this paper is to investigate methods for the design of optimal dynamic experiments. In Section 2, we introduce notation and a determinant criterion for dynamic experiments. In Section 3, optimal dynamic designs are constructed analytically in a number of simple cases. As will be seen, the dimension of the design problem increases with the dimension of the control vectors (equivalently, the response vector). In Section 4, we discuss the use of suitable parameterizations of the control variable trajectories to reduce the dimensionality of the optimization problem. The relationship between the design of optimal dynamic experiments and the results developed for marginally restricted designs is considered in Section 5. Numerical methods are discussed in Section 6. We close, in Section 7, with an extension of the methodology to the case where one or more linear combinations of the response trajectory is observed for each control trajectory.
Valery Fedorov, Chris Nachtsheim
Robust Optimal Designs with Constraints
Abstract
A researcher wants to design an experiment from which he can investigate the relationship between a specified response variable y and several controllable independent variables. The model, notations, and design problem are the same as in Fedorov et. al. (1994), and therefore our introductory part will be very sketchy.
Grace Montepiedra
Bayesian Designs for Approximate Normality
Abstract
In many experimental design problems, the primary interest is in estimating functions of the parameters and a design is selected according to some optimality criterion. The assumption that parameter estimates are approximately normally distributed is often used to find optimal designs, as well as simplify data analysis. How well this approximation holds for small to moderate sample sizes depends on the intrinsic and parameter-effects curvatures. These measures depend on both the parameterization used as well as the experimental design. For a particular parameterization of interest, these curvatures can be reduced by the choice of the experimental design. A Bayesian approach is taken to find designs that optimize the primary design criterion subject to satisfying constraints based on these curvature measures, with the goal of improving normal approximations. The constrained designs depend on the sample size, but as the sample size increases the constraints are satisfied. A nonlinear regression example is used to illustrate the approach.
Merlise A. Clyde
Simulation Approach to One-Stage and Sequential Optimal Design Problems
Abstract
In this presentation we introduce an algorithm for Bayesian optimal design based on smoothing a scatterplot of observed losses (or utilities) for a Monte Carlo sample of simulated experiments. Denote with d, θ and y the design parameters, the parameter vector and the data, respectively. The Bayesian optimal design problem is to find the design d* which maximizes the pre-posterior expected utility u(d) = ∫ U(d, y, θ)dp d (θ, y), where p d (θ, y) is the joint distribution under design d on parameter and sample space, and U(d, y, θ) is the relevant payoff when the data y is observed under design d and the parameter θ. For example, we might want to estimate the parameters θ in a regression model under squared error loss.
Giovanni Parmigiani, Peter Müller
One Bound for the Mean Duration of Sequential Testing Homogeneity
Abstract
A lower bound is proved for the mean duration of any sequential strategy for testing homogeneity under the alternative formulated in terms of maximal distance in variation between m ≥ 2 populations.
Leonid I. Galtchouk, Michail B. Maljutov
MV-optimization in Simple Linear Regression
Abstract
MV-optimality is a potentially difficult criterion because of its nondifferentiability at equal variance designs. However in many cases such designs can be easily determined. In this paper MV-optimum designs for simple linear regression are found. The equivalence theorem of and the directional derivative of the MV-criterion derived by Ford I., (3), have been used for this purpose. It turns out that for simple linear regression there exist an MV-optimal design with a support of at most two points. Such designs could be of a wide ranging practical value.
Ben Torsney, Jesus López-Fidalgo
On the Support Points of D-Optimal Nonlinear Experimental Designs for Chemical Kinetics
Abstract
In principle the nonlinear experiment design problem is distinguished from the linear one on the fact that the design points depend on the unknown parameter under investigation. Moreovere if the sequential principle is adopted there are not general convergence theorems neither for the sequence of estimators nor for the sequence of design measures. Fortunately, in practice, the nonlinear models involved are not too complicated and the support points can be evaluated, for the particular class of nonlinear models from chemical kinetics, so that the experimenter can built the local optimum experiment design.
Christos P. Kitsos
Designing Experiments for Additive Nonlinear Models
Abstract
According to a general result by Schwabe and Wierich (1993) product designs are D-optimal in additive linear models. In the present note this result is extended to a nonlinear setting. Local optimality is considered as well as minimax approaches and weight functions on the parameters. In particular, optimal designs can be constructed as a product of those designs which are optimal in the corresponding single-factor models. The results are obtained for the whole parameter vector and for the parameters associated with the single factors.
Rainer Schwabe
D-Optimal Designs for Generalized Linear Models
Abstract
This paper develops some simple methods for obtaining D-optimal designs for generalized linear models with multiple design variables. In some important cases the numerical complexity can be reduced to that of the two parameter case regardless of the original dimension. The form and properties of the obtained D-optimal designs are illustrated and discussed through a few interesting examples.
Randy R. Sitter, Ben Torsney
Further Results on Optimal Designs for Generalized Tic Polynomials on the Simplex
Abstract
In a recent paper Hilgers and Bauer (1994) discussed D-optimal designs for (generalized) tic polynomials in q variables on the simplex. The simplex is an appropriate factor space in mixture experiments, where the response also depends on the total amount of the components. These results are extended in two different aspects. Firstly the change of the optimal design under variation of the optimality criterion and secondly the exclusion of some of the regression functions are studied. In particular the support of the D-optimal extended simplex centroid design is also φ p -optimal. On the other hand, the deletion of the linear terms in the regression model seems to result in the deletion of the design points belonging to the corresponding regression functions.
Ralf-Dieter Hilgers
Relations between Spring and Chemical Balance Weighing Designs with the Diagonal Covariance Matrix of Errors
Abstract
The paper deals with the problem of estimating the individual weights of objects with minimum variances by using a weighing design with the diagonal covariance matrix of errors in the model. The necessary and sufficient conditions for optimum biased spring balance weighing designs with the diagonal covariance matrix of errors and for optimum chemical balance weighing designs with the diagonal covariance matrix of errors are given and the relations between these designs are investigated. Also new optimum weighing designs are found.
Bronisław Ceranka, Krystyna Katulska
Optimal Design for Experiments with Potentially Failing Trials
Abstract
We discuss the problem of optimal allocation of the design points of an experiment for the case where the trials may fail with non-zero probability. Numerical results for D-optimal designs are given for estimating the coefficients of a polynomial regression. For small sample sizes these designs may deviate substantially from the corresponding designs in the case of certain response. They can be less efficient, but are less affected by failing trials.
Peter Hackl
Regression Design for One-Dimensional Subspaces
Abstract
Consider a regression problem with univariate response y and p × 1 vector of design variables x, and assume that the cumulative distribution function F(y|x) depends on x only through the linear combination θ T × so that F(y|x) = F(y|θ T x) for all x in the design space. When the form of F is unknown, θ  is not estimable. However, under certain conditions the subspace S(θ) of R P spanned by θ is estimable. The goal of this paper is to begin investigating how to design an experiment so that standard methods of estimation may yield useful estimates of S(θ) when the family F(y|θ T x) is unknown. This may provide a baseline for assessing the robustness of designs based on an assumed family F, in addition to allowing insight into model robust design.
Abdelouafi Ibrahimy, R. Dennis Cook
D-Optimal First Order Saturated Designs with n ≡ 2mod4 Observations
Abstract
In experimental situations where n two-level factors are involved and n observations are taken, then the D-optimal first order saturated design is an n × n ±1 matrix with the maximum determinant. In this paper we discuss this problem for n ≡ 2mod4, we summarize all the known results, and we give some new D-optimal designs.
Christos Koukouvinos
On Information Matrices for Fixed and Random Parameters in Generally Balanced Experimental Block Designs
Abstract
Information matrices are arguments of most of optimality criteria defined under fixed linear models and also for fixed effects in mixed linear models. However, in the context of mixed models interest often lies on variances of random effects as well as on fixed effects. In the paper the forms and some properties of the information matrix for fixed treatment parameters and for strata variances, in case of generally balanced block designs, are shown. A short discussion on optimality criteria is also presented.
Barbara Bogacka
Estimation of Parameters in Factorial Triallel Analysis for BIB Design — the Mixed Model
Abstract
The paper presents the estimation some genetic parameters concerning hybrids obtained in factorial triallel crossing system. The hybrids are compared in a balanced incomplete block design in which the block effects are treated as random. The statistical analysis includes estimation (intra-block, inter-block and combining) of general line effects and two-line and three-line specific effects.
Bronisław Ceranka, Zygmunt Kaczmarek
On the Optimality of Certain Nested Block Designs under a Mixed Effects Model
Abstract
Some optimal statistical properties of C-designs in certain nested block designs under a mixed model are characterized.
Stanisław Mejza, Sanpei Kageyama
Construction of A-Optimum Cross-Over Designs
Abstract
We describe an algorithmic approach to the construction of A-optimum repeated measurements designs. The algorithm is very flexible and can search for designs in non-standard situations. Some illustrative examples are given.
Alexander N. Donev, Byron Jones
An Algorithm for Sampling Optimization for Semivariogram Estimation
Abstract
This paper describes an algorithm for the optimal selection of sampling locations for semivariogram estimation. We assume that the semivariogram is estimated by fitting a parametric function of separation distance between observation sites to a selected subset of the squared differences of original observations (thereby restricting ourselves to isotropic fields). We apply standard regression design theory to construct an optimal configuration of distances in the lag space, which is then mapped into the site space in such a way that dependence among the observations is minimized.
Werner G. Müller, Dale L. Zimmerman

Estimation and Optimization

Frontmatter
Multivariate Transformations, Regression Diagnostics and Seemingly Unrelated Regression
Abstract
The assumption of multivariate normality provides the customary powerful and convenient way of analysing multivariate data: if data are not normal, the analysis may often be simplified by an appropriate transformation. The paper derives deletion diagnostics for the effect of individual observations on the estimated transformation to normality, using the parametric family of power transformations of Box and Cox. The likelihood ratio test is compared with a seemingly unrelated regression test using constructed variables. The examples include both unstructured data and multivariate multiple regression. They indicate that the likelihood ratio test is more informative in the presence of appreciable correlation in the data than the seemingly unrelated regression test. Numerical results are given for the effect of deletion in the two main stages in the construction of deletion diagnostics for seemingly unrelated regression models.
Anthony C. Atkinson
Regression Rank Scores: Asymptotic Linearity and RR-Estimators
Abstract
The uniform asymptotic linearity of regression rank scores process, proved by the author in 1992, is extended to a broad class of distributions of the errors including the Cauchy. This extends the applicability of RR-estimators in the linear model and has various other applications.
Jana Jurečková
The Asymptotic Distribution of Regression Parameters
Abstract
The regression analysis is undoubtly one of the most popular methods of mathematical statistics.
Anne-Mai Parring
Some Simulation Results on Cross-Validation and Competitors for Model Choice
Abstract
The behaviour of model selection procedures based on different criteria such as cross-validation is investigated in a simulation study. Emphasis is on the relationship to the problem of estimating the prediction quality of a model.
Bernd Droge
Robust Estimation of Non-linear Aspects
Abstract
As a first step for dealing with efficient robust estimation in non-linear models, we regard the problem of efficient robust estimation of non-linear aspects (functions) φ(β) of the unknown parameter β of a linear model. For robust estimation of a general non-linear aspect we propose estimators which are based on one-step-M-estimators and derive their asymptotic behaviour at the contaminated linear model, where the errors have contaminated normal distributions. The asymptotic behaviour provides criteria for robustness and optimality of the estimators and the corresponding designs. Because it is impossible to find globally optimal robust estimators and designs locally optimal solutions are used for efficiency comparisons. Simple formulas for the efficiency rates are given for the general case. Using these results the efficiency rates for estimating robustly the relative variation of a circadian rhythm are calculated. These efficiency rates are very similar to those for non-robust estimation although on principle there is an important difference.
Christos P. Kitsos, Christine H. Müller
Robust Minimax Adaptive M-Estimators of Regression Parameters
Abstract
The minimax robust M-estimators of regression parameters designed over the classes with a bounded variance of a distribution are obtained. The properties of these new estimators and their adaptive versions are studied in asymptotics and in a finite sample size case.
Georgiy L. Shevlyakov, Nikita O. Vil’chevskiy
Modeling Heterogeneity and Extraneous Variation Using Weighted Distributions
Abstract
In one-parameter exponential families, the variance is a function of the mean. One powerful method of modeling heterogeneity and overdispersion in an exponential family is to use a parametrized weighted distribution. In this paper we interpret such a weighted distribution model as an overdispersed generalized linear model by introducing covariates and forming a very general class of models. Here, such models are fit from a Bayesian perspective, using non-informative priors in order to let the data (likelihood) drive the inference. Bayesian calculations are carried out using a Metropolis-within-Gibbs sampling algorithm. An illustrative example using a previously analyzed data set is presented with emphasis on model comparison.
Dipak K. Dey, Fengchun Peng, Daniel Larose
Gibbs Sampling for ARCH Models in Finance
Abstract
The paper develops a simple estimation procedure for Bayesian ARCH models: The Gibbs-importance algorithm (also called independence chain) is applied for the simulation step involving the ARCH parameters. We demonstrate this approach to model the volatility between the Dollar, the DM and the Yen. An extension of the model to multivariate VAR-VARCH models is proposed.
Wolfgang Polasek, Peter Müller
A Class of Recursive Algorithms Using Non-parametric Methods with Constant Step Size and Window Width: A Numerical Study
Abstract
Motivated by a wide range of applications in chemical engineering problems, recursive algorithms using nonparametric methods combined with stochastic approximation procedures are developed in this work. Loosely speaking, the problems are to find roots of nonlinear functions provided that only noisy measurements are available. In addition, in the systems under consideration, not only the outputs but also the inputs are noise corrupted. Thus the conventional stochastic approximation algorithms become inapplicable. Algorithms using a kernel function with constant step size and constant window width are proposed and analyzed. After presenting the convergence result of the algorithms under general conditions, effort is directed to the numerical studies. Simulation results and numerical experiments are also given.
George Yin, Kewen Yin
Robust Design of Products Depending on Both Qualitative and Quantitative Factors
Abstract
The paper considers a model-based approach to quality improvement of products or processes depending on both qualitative and quantitative factors. Models with dummy variables are used for this purpose. A procedure for model structure selection is proposed. Models of the mean value and var7iance of the performance characteristic in mass production are obtained. Some specific features of the optimization procedures for quality improvement for models with qualitative and quantitative variables are considered. An example concerning the quality of resistors is given.
Ivan N. Vuchkov, L. N. Boyadjieva
Improving on Golden-Section Optimisation for Locally Symmetric Functions
Abstract
We consider the minimisation of a uniextremal function f(.) on [0,1] using a “second-order” algorithm. At each iteration the current feasible region is resealed to [0,1], so that the optimizing value x* in the initial [0,1]-interval varies from iteration to iteration, which defines a dynamic system. Many line-search algorithms exhibit chaotic behaviour when resealing is applied. If f(.)is symmetric around x*, the associated dynamic system is time-homogeneous and often possesses an invariant density. In a first part, we show that the asymptotic behaviour of the classical Golden-Section algorithm is the same for locally symmetric functions as for pure symmetric functions. We believe that this property is also true for other line-search algorithms, with sometimes a better ergodic rate than the Golden-Section algorithm. In a second part, we consider the case where the number of iterations is fixed a priori, with a dynamic-programming approach, using a uniform prior density on [0,1] for x*.
Luc Pronzato, Henry P. Wynn, Anatoly A. Zhigljavsky
Metadaten
Titel
MODA4 — Advances in Model-Oriented Data Analysis
herausgegeben von
Dr. Christos P. Kitsos
Dr. Werner G. Müller
Copyright-Jahr
1995
Verlag
Physica-Verlag HD
Electronic ISBN
978-3-662-12516-8
Print ISBN
978-3-7908-0864-3
DOI
https://doi.org/10.1007/978-3-662-12516-8