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Author: Moudar Soumaya
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Author: Moudar Soumaya
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Author: Martijn P.F. Berger
Publisher: John Wiley & Sons
ISBN: 9780470746929
Category : Mathematics
Languages : en
Pages : 346
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Book Description
The increasing cost of research means that scientists are in more urgent need of optimal design theory to increase the efficiency of parameter estimators and the statistical power of their tests. The objectives of a good design are to provide interpretable and accurate inference at minimal costs. Optimal design theory can help to identify a design with maximum power and maximum information for a statistical model and, at the same time, enable researchers to check on the model assumptions. This Book: Introduces optimal experimental design in an accessible format. Provides guidelines for practitioners to increase the efficiency of their designs, and demonstrates how optimal designs can reduce a study’s costs. Discusses the merits of optimal designs and compares them with commonly used designs. Takes the reader from simple linear regression models to advanced designs for multiple linear regression and nonlinear models in a systematic manner. Illustrates design techniques with practical examples from social and biomedical research to enhance the reader’s understanding. Researchers and students studying social, behavioural and biomedical sciences will find this book useful for understanding design issues and in putting optimal design ideas to practice.
Author: Valerii V. Fedorov
Publisher: CRC Press
ISBN: 1439821526
Category : Mathematics
Languages : en
Pages : 402
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Book Description
Optimal Design for Nonlinear Response Models discusses the theory and applications of model-based experimental design with a strong emphasis on biopharmaceutical studies. The book draws on the authors' many years of experience in academia and the pharmaceutical industry. While the focus is on nonlinear models, the book begins with an explanation of
Author: Shenghua Fan
Publisher:
ISBN:
Category :
Languages : en
Pages : 420
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Author: Friedrich Pukelsheim
Publisher: SIAM
ISBN: 0898716047
Category : Mathematics
Languages : en
Pages : 527
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Book Description
Optimal Design of Experiments offers a rare blend of linear algebra, convex analysis, and statistics. The optimal design for statistical experiments is first formulated as a concave matrix optimization problem. Using tools from convex analysis, the problem is solved generally for a wide class of optimality criteria such as D-, A-, or E-optimality. The book then offers a complementary approach that calls for the study of the symmetry properties of the design problem, exploiting such notions as matrix majorization and the Kiefer matrix ordering. The results are illustrated with optimal designs for polynomial fit models, Bayes designs, balanced incomplete block designs, exchangeable designs on the cube, rotatable designs on the sphere, and many other examples.
Author: B.K. Sinha
Publisher: Springer
ISBN: 8132217861
Category : Mathematics
Languages : en
Pages : 213
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Book Description
The book dwells mainly on the optimality aspects of mixture designs. As mixture models are a special case of regression models, a general discussion on regression designs has been presented, which includes topics like continuous designs, de la Garza phenomenon, Loewner order domination, Equivalence theorems for different optimality criteria and standard optimality results for single variable polynomial regression and multivariate linear and quadratic regression models. This is followed by a review of the available literature on estimation of parameters in mixture models. Based on recent research findings, the volume also introduces optimal mixture designs for estimation of optimum mixing proportions in different mixture models, which include Scheffé’s quadratic model, Darroch-Waller model, log- contrast model, mixture-amount models, random coefficient models and multi-response model. Robust mixture designs and mixture designs in blocks have been also reviewed. Moreover, some applications of mixture designs in areas like agriculture, pharmaceutics and food and beverages have been presented. Familiarity with the basic concepts of design and analysis of experiments, along with the concept of optimality criteria are desirable prerequisites for a clear understanding of the book. It is likely to be helpful to both theoreticians and practitioners working in the area of mixture experiments.
Author: Wim J. van der Linden
Publisher: Springer Science & Business Media
ISBN: 0387290540
Category : Social Science
Languages : en
Pages : 421
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Book Description
Wim van der Linden was just given a lifetime achievement award by the National Council on Measurement in Education. There is no one more prominent in the area of educational testing. There are hundreds of computer-based credentialing exams in areas such as accounting, real estate, nursing, and securities, as well as the well-known admissions exams for college, graduate school, medical school, and law school - there is great need on the theory of testing. This book presents the statistical theory and practice behind constructing good tests e.g., how is the first test item selected, how are the next items selected, and when do you have enough items.
Author: Hao Ji
Publisher:
ISBN: 9780355151312
Category :
Languages : en
Pages :
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Book Description
The thesis focuses on methodological research and associated statistical theories originated from the functional data analysis perspective, and consists of three chapters. In the first chapter, we propose novel optimal designs for longitudinal data for the common situation where the resources for longitudinal data collection are limited, by determining the optimal locations in time where measurements should be taken. As for all optimal designs, some prior information is needed to implement the proposed optimal designs. We demonstrate that this prior information may come from a pilot longitudinal study that has irregularly measured and noisy measurements, where for each subject one has available a small random number of repeated measurements that are randomly located on the domain. A second possibility of interest is that a pilot study consists of densely measured functional data and one intends to take only a few measurements at strategically placed locations in the domain for the future collection of similar data. We construct optimal designs by targeting two criteria:(a) Optimal designs to recover the unknown underlying smooth random trajectory for each subject from a few optimally placed measurements such that squared prediction errors are minimized;(b) Optimal designs that minimize prediction errors for functional linear regression with functional or longitudinal predictors and scalar responses, again from a few optimally placed measurements. The proposed optimal designs address the need for sparse data collection when planning longitudinal studies, by taking advantage of the close connections between longitudinal and functional data analysis. We demonstrate in simulations that the proposed designs perform considerably better than randomly chosen design points and include a motivating data example from the Baltimore longitudinal study of aging. The proposed designs are shown to have an asymptotic optimality property. In the second chapter, we connect the optimal design method with model selection in multivariate regression framework. Model selection in linear regression models with scalar response has been a popular topic because of its important role in a vast range of applications, including high dimensional data samples with sample size n smaller than the number of predictors p. Classic model selection methods such as Lasso and state of the art methods in machine learning for the most part rely on the assumption of independent/uncorrelated covariates and sparseness of the regression coefficient vector, where most regression coefficients are assumed to be zero. However, highly correlated covariates are commonly encountered in real applications and the sparsity assumption is often hard to verify. In this work, we propose a novel methodology for model selection using stringing and the perspective of functional data analysis. Instead of sparsity and uncorrelatedness of the predictors, the stringing approach has been established under the alternative assumption that observed predictor vectors are drawn from a square integrable and smooth stochastic process with subsequent random scrambling of the order of the coordinates where the functions are sampled. We make use of recent results for optimal designs in functional regression models to develop the high-dimensional predictor selection with stringing (HIPS) approach with optimal designs, which provides covariate selection. The HIPS approach consists of three steps, namely(1) recovering the assumed underlying order of the observed high-dimensional covariates, which are assumed to be the scrambled coordinates of an underlying smooth process (the stringing step); (2) selecting the optimal design points/covariates for the recovered functional trajectories, which will then provide the set of selected covariates (the optimal design step); (3) using the selected predictors to fit the linear model. We compare the prediction performance as well as the ability of capturing the correct covariates under different scenarios of HIPS and the relaxed Lasso in simulation studies and real data applications. In the last chapter, we develop a novel method under the context of residual demography, which is a recent concept that has proved to be a useful tool to gain insights about the age distributions of wild populations, especially insects. We develop an operator equation that permits the derivation of functionals of the age distribution in wild populations, such as mean age, within the framework of residual demography. Our method combines information from an observed captive cohort, which consists of subjects that are sampled from the wild with unknown ages and then raised in the laboratory until death, and from a reference cohort that consists of subjects raised in the laboratory since birth of the same population. Targeting functionals such as the mean of the wild age distribution has the advantage of avoiding strong assumptions such as stationarity and stability of the population that one would need when targeting the entire survival distribution in thewild. Our main result characterizes the existence of a solution of the operator equation that yields the functional of interest. The proposed method also enjoys straightforward and easy implementation. A data example is included illustrating an application, whereone aims to attain the mean age of mosquitoes in the wild, based on seasonal captive cohorts from Greece and a simulated reference cohort, separately for various summer and fall months.
Author: Giorgio Celant
Publisher: John Wiley & Sons
ISBN: 1786300540
Category : Mathematics
Languages : en
Pages : 324
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Book Description
This book considers various extensions of the topics treated in the first volume of this series, in relation to the class of models and the type of criterion for optimality. The regressors are supposed to belong to a generic finite dimensional Haar linear space, which substitutes for the classical polynomial case. The estimation pertains to a general linear form of the coefficients of the model, extending the interpolation and extrapolation framework; the errors in the model may be correlated, and the model may be heteroscedastic. Non-linear models, as well as multivariate ones, are briefly discussed. The book focuses to a large extent on criteria for optimality, and an entire chapter presents algorithms leading to optimal designs in multivariate models. Elfving’s theory and the theorem of equivalence are presented extensively. The volume presents an account of the theory of the approximation of real valued functions, which makes it self-consistent.
Author: Jean-Bernard Lasserre
Publisher: World Scientific
ISBN: 1848164467
Category : Mathematics
Languages : en
Pages : 384
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Book Description
1. The generalized moment problem. 1.1. Formulations. 1.2. Duality theory. 1.3. Computational complexity. 1.4. Summary. 1.5. Exercises. 1.6. Notes and sources -- 2. Positive polynomials. 2.1. Sum of squares representations and semi-definite optimization. 2.2. Nonnegative versus s.o.s. polynomials. 2.3. Representation theorems : univariate case. 2.4. Representation theorems : mutivariate case. 2.5. Polynomials positive on a compact basic semi-algebraic set. 2.6. Polynomials nonnegative on real varieties. 2.7. Representations with sparsity properties. 2.8. Representation of convex polynomials. 2.9. Summary. 2.10. Exercises. 2.11. Notes and sources -- 3. Moments. 3.1. The one-dimensional moment problem. 3.2. The multi-dimensional moment problem. 3.3. The K-moment problem. 3.4. Moment conditions for bounded density. 3.5. Summary. 3.6. Exercises. 3.7. Notes and sources -- 4. Algorithms for moment problems. 4.1. The overall approach. 4.2. Semidefinite relaxations. 4.3. Extraction of solutions. 4.4. Linear relaxations. 4.5. Extensions. 4.6. Exploiting sparsity. 4.7. Summary. 4.8. Exercises. 4.9. Notes and sources. 4.10. Proofs -- 5. Global optimization over polynomials. 5.1. The primal and dual perspectives. 5.2. Unconstrained polynomial optimization. 5.3. Constrained polynomial optimization : semidefinite relaxations. 5.4. Linear programming relaxations. 5.5. Global optimality conditions. 5.6. Convex polynomial programs. 5.7. Discrete optimization. 5.8. Global minimization of a rational function. 5.9. Exploiting symmetry. 5.10. Summary. 5.11. Exercises. 5.12. Notes and sources -- 6. Systems of polynomial equations. 6.1. Introduction. 6.2. Finding a real solution to systems of polynomial equations. 6.3. Finding all complex and/or all real solutions : a unified treatment. 6.4. Summary. 6.5. Exercises. 6.6. Notes and sources -- 7. Applications in probability. 7.1. Upper bounds on measures with moment conditions. 7.2. Measuring basic semi-algebraic sets. 7.3. Measures with given marginals. 7.4. Summary. 7.5. Exercises. 7.6. Notes and sources -- 8. Markov chains applications. 8.1. Bounds on invariant measures. 8.2. Evaluation of ergodic criteria. 8.3. Summary. 8.4. Exercises. 8.5. Notes and sources -- 9. Application in mathematical finance. 9.1. Option pricing with moment information. 9.2. Option pricing with a dynamic model. 9.3. Summary. 9.4. Notes and sources -- 10. Application in control. 10.1. Introduction. 10.2. Weak formulation of optimal control problems. 10.3. Semidefinite relaxations for the OCP. 10.4. Summary. 10.5. Notes and sources -- 11. Convex envelope and representation of convex sets. 11.1. The convex envelope of a rational function. 11.2. Semidefinite representation of convex sets. 11.3. Algebraic certificates of convexity. 11.4. Summary. 11.5. Exercises. 11.6. Notes and sources -- 12. Multivariate integration 12.1. Integration of a rational function. 12.2. Integration of exponentials of polynomials. 12.3. Maximum entropy estimation. 12.4. Summary. 12.5. Exercises. 12.6. Notes and sources -- 13. Min-max problems and Nash equilibria. 13.1. Robust polynomial optimization. 13.2. Minimizing the sup of finitely many rational cunctions. 13.3. Application to Nash equilibria. 13.4. Exercises. 13.5. Notes and sources -- 14. Bounds on linear PDE. 14.1. Linear partial differential equations. 14.2. Notes and sources
