Design of experiments in nonlinear models : asymptotic normality, optimality criteria and smallsample properties
 Language
 English.
 Imprint
 New York : Springer, c2013.
 Physical description
 xv, 399 pages : illustrations (some color) ; 23 cm.
 Series
 Lecture notes in statistics (SpringerVerlag) 212.
Access
Contributors
 Contributor
 Pronzato, Luc, 1959 Author
 Pázman, Andrej. Author
Contents/Summary
 Bibliography
 Includes bibliographical references and indexes.
 Contents

 Introduction
 Experiments and their designs
 Models
 Parameters
 Information and design criteria
 Asymptotic designs and uniform convergence
 Asymptotic designs
 Uniform convergence
 Bibliographie notes and further remarks
 Asymptotic properties of the LS estimator
 Asymptotic properties of the LS estimator in regression models
 Consistency
 Consistency under a weaker LS estimability condition
 Asymptotic normality
 Asymptotic normality of a scalar function of the LS estimator
 Asymptotic properties of functions of the LS estimator under singular designs
 Singular designs in linear models
 Singular designs in nonlinear models
 LS estimation with parameterized variance
 Inconsistency of WLS with parameterdependent weights
 Consistency and asymptotic normality of penalized WLS
 Consistency and asymptotic normality of twostage LS
 Consistency and asymptotic normality of iteratively reweighted LS
 Misspecification of the variance function
 Different parameterizations for the mean and variance
 Penalized WLS or twostage LS?
 Variance stabilization
 LS estimation with model error
 LS estimation with equality constraints
 Bibliographie notes and further remarks
 Asymptotic properties of M, ML, and maximum a Posteriori estimators
 M estimators in regression models
 The maximum likelihood estimator
 Regression models
 General situation
 Generalized linear models and exponential families
 Models with a onedimensional sufficient statistic
 Models with a multidimensional sufficient statistic
 The CramerRao inequality : efficiency of estimators
 Efficiency
 Asymptotic efficiency
 The maximum A Posteriori estimator
 Bibliographie notes and further remarks
 Local optimality criteria based on asymptotic normality
 Design criteria and their properties
 Ellipsoid of concentration
 Classical design criteria
 Positive homogeneity, concavity, and isotonicity
 Equivalence between criteria
 Concavity and isotonicity of classical criteria
 Classification into global and partial optimality criteria
 The upper semicontinuity of the cOptimality criterion
 Efficiency
 Combining criteria
 Design with a cost constraint
 Derivatives and conditions for optimality of designs
 Derivatives
 The equivalence theorem
 Number of support points
 Elfving's set and some duality properties
 cOptimum design in linearized nonlinear models
 Elfving's theorem and related properties
 cMaximin efficiency and DOptimality
 A duality property for cOptimality
 Equivalence theorem for cOptimality
 Specific difficulties with cOptimum design in presence of nonlinearity
 Optimality criteria for asymptotic variancecovariance matrices in product form
 The WLS estimator
 The penalized WLS estimator
 The LS estimator with model error
 The M estimator
 Bibliographie notes and further remarks
 criteria based on the smallsample precision of the LS estimator
 The geometry of the regression model
 Basic notions
 A classification of nonlinear regression models
 Avoiding failures of LS estimation
 The probability density of the LS estimator in nonlinear models with normal errors
 Intrinsically linear models
 Models with dim(Q) = 1
 Fiat models
 Models with Riemannian curvature tensor R(0)
 Density of the penalized LS estimator
 Marginal densities of the LS estimator
 Optimality criteria based on the p.d.f. of the LS estimator
 Higherorder approximations of optimality criteria
 Approximate bias and meansquared error
 Approximate entropy of the p.d.f. of the LS estimator
 Bibliographie notes and further remarks
 Identifiability, estimability, and extended optimality criteria
 Identifiability
 LS estimability of regression models
 Numerical issues related to estimability in regression models
 Estimability function
 Definition
 Properties
 Replications and design measures
 Estimability for parametric functions
 An extended measure of intrinsic nonlinearity
 Advantages and drawbacks of using ppoint designs
 Design of experiments for improving estimability
 Extended (globalized) EOptimality
 Extended (globalized) cOptimality
 Maximumentropy regularization of estimability criteria
 Numerical examples
 Remarks on estimability for estimators other than LS
 Bibliographie notes and further remarks
 Nonlocal optimum design
 Averageoptimum design
 Properties
 A Bayesian interpretation
 Maximinoptimum design
 Regularization of maximin criteria via average criteria
 Regularization via Lq norms
 Maximumentropy regularization
 Probability level and quantile criteria
 Sequential design
 Twostage allocation
 Fullsequential Doptimum design for LS estimation in nonlinear regression models
 Algorithms : a survey
 Maximizing a concave differentiable functional of a probability measure
 Vertexdirection algorithms
 Constrained gradient and gradient projection
 Multiplicative algorithms
 Doptimum design
 Exact design
 Exchange methods
 Branch and bound
 Maximinoptimum design
 Nondifferentiable optimization of a design measure
 Maximinoptimum exact design
 Averageoptimum design
 Averageoptimal design measures and stochastic approximation
 Averageoptimum exact design
 Two methods for convex programming
 Principles for cutting strategies and interiorpoint methods
 The ellipsoid method
 The cuttingplane method
 Subdifferentials and subgradients
 Computation of derivatives through sensitivity functions
 Proofs
 Symbols and notation
 List of labeled assumptions
 References
 Author index
 Subject index.
 Publisher's Summary
 Design of Experiments in Nonlinear Models: Asymptotic Normality, Optimality Criteria and SmallSample Properties provides a comprehensive coverage of the various aspects of experimental design for nonlinear models. The book contains original contributions to the theory of optimal experiments that will interest students and researchers in the field. Practitionners motivated by applications will find valuable tools to help them designing their experiments. The first three chapters expose the connections between the asymptotic properties of estimators in parametric models and experimental design, with more emphasis than usual on some particular aspects like the estimation of a nonlinear function of the model parameters, models with heteroscedastic errors, etc. Classical optimality criteria based on those asymptotic properties are then presented thoroughly in a special chapter. Three chapters are dedicated to specific issues raised by nonlinear models. The construction of design criteria derived from nonasymptotic considerations (smallsample situation) is detailed. The connection between design and identifiability/estimability issues is investigated. Several approaches are presented to face the problem caused by the dependence of an optimal design on the value of the parameters to be estimated. A survey of algorithmic methods for the construction of optimal designs is provided.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2013
 Responsibility
 Luc Pronzato, Andrej Pázman.
 Series
 Lecture notes in statistics, 09300325 ; 212
 ISBN
 9781461463627
 1461463629