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Design of experiments in nonlinear models : asymptotic normality, optimality criteria and small-sample properties / Luc Pronzato, Andrej Pázman.



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Publication date:
New York : Springer, c2013.
  • Book
  • xv, 399 pages : illustrations (some color) ; 23 cm.
Includes bibliographical references and indexes.
  • 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 parameter-dependent weights
  • Consistency and asymptotic normality of penalized WLS
  • Consistency and asymptotic normality of two-stage LS
  • Consistency and asymptotic normality of iteratively reweighted LS
  • Misspecification of the variance function
  • Different parameterizations for the mean and variance
  • Penalized WLS or two-stage 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 one-dimensional sufficient statistic
  • Models with a multidimensional sufficient statistic
  • The Cramer-Rao 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 c-Optimality 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
  • c-Optimum design in linearized nonlinear models
  • Elfving's theorem and related properties
  • c-Maximin efficiency and D-Optimality
  • A duality property for c-Optimality
  • Equivalence theorem for c-Optimality
  • Specific difficulties with c-Optimum design in presence of nonlinearity
  • Optimality criteria for asymptotic variance-covariance 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 small-sample 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
  • Higher-order approximations of optimality criteria
  • Approximate bias and mean-squared 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 p-point designs
  • Design of experiments for improving estimability
  • Extended (globalized) E-Optimality
  • Extended (globalized) c-Optimality
  • Maximum-entropy regularization of estimability criteria
  • Numerical examples
  • Remarks on estimability for estimators other than LS
  • Bibliographie notes and further remarks
  • Nonlocal optimum design
  • Average-optimum design
  • Properties
  • A Bayesian interpretation
  • Maximin-optimum design
  • Regularization of maximin criteria via average criteria
  • Regularization via Lq norms
  • Maximum-entropy regularization
  • Probability level and quantile criteria
  • Sequential design
  • Two-stage allocation
  • Full-sequential D-optimum design for LS estimation in nonlinear regression models
  • Algorithms : a survey
  • Maximizing a concave differentiable functional of a probability measure
  • Vertex-direction algorithms
  • Constrained gradient and gradient projection
  • Multiplicative algorithms
  • D-optimum design
  • Exact design
  • Exchange methods
  • Branch and bound
  • Maximin-optimum design
  • Non-differentiable optimization of a design measure
  • Maximin-optimum exact design
  • Average-optimum design
  • Average-optimal design measures and stochastic approximation
  • Average-optimum exact design
  • Two methods for convex programming
  • Principles for cutting strategies and interior-point methods
  • The ellipsoid method
  • The cutting-plane method
  • Subdifferentials and subgradients
  • Computation of derivatives through sensitivity functions
  • Proofs
  • Symbols and notation
  • List of labeled assumptions
  • References
  • Author index
  • Subject index.
Pronzato, Luc, 1959- Author
Pázman, Andrej. Author
Lecture notes in statistics, 0930-0325 ; 212
Lecture notes in statistics (Springer-Verlag) ; 212.

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