On Minimally-supported D-optimal Designs for Polynomial Regression with Log-concave Weight Function

Chang, Fu; Lin, Hung‐Ming

doi:10.1007/s00184-006-0072-9

Cited by 5 publications

(2 citation statements)

References 12 publications

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“…As a first application of Theorem 2.2, we consider the de la Garza phenomenon for the weighted heteroscedastic polynomial regression model on a compact interval, say [0, 1]. Optimal design problems in this model have found considerable attention in the literature (see, for example, Fang, 2002;Chang, 2005;Dette et al, 2005;Chang and Lin, 2006;Chang and Jiang, 2007;Sekido, 2012, among others). To be precise let X = [0, 1] (or any other compact interval on the non-negative line) and consider the vector of regression functions…”

Section: Optimal Designs and A Geometric Characterizationmentioning

confidence: 99%

Design admissibility and de la Garza phenomenon in multi-factor experiments

Dette¹,

Liu²,

Yue³

2020

Preprint

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The determination of an optimal design for a given regression problem is an intricate optimization problem, especially for models with multivariate predictors. Design admissibility and invariance are main tools to reduce the complexity of the optimization problem and have been successfully applied for models with univariate predictors. In particular several authors have developed sufficient conditions for the existence of saturated designs in univariate models, where the number of support points of the optimal design equals the number of parameters. These results generalize the celebrated de la Garza phenomenon (de la Garza, 1954) which states that for a polynomial regression model of degree p − 1 any optimal design can be based on at most p points.This paper provides -for the first time -extensions of these results for models with a multivariate predictor. In particular we study a geometric characterization of the support points of an optimal design to provide sufficient conditions for the occurrence of the de la Garza phenomenon in models with multivariate predictors and characterize properties of admissible designs in terms of admissibility of designs in conditional univariate regression models.

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Section: Optimal Designs and A Geometric Characterizationmentioning

confidence: 99%

Design admissibility and de la Garza phenomenon in multi-factor experiments

Dette¹,

Liu²,

Yue³

2020

Preprint

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“…The theory for the construction of Bayesian optimal design depends uniquely on the model and the criterion, and the mathematics required to solve the optimisation problem can be challenging even for linear models (Dette & Wong, ; ). In practice, Bayesian optimal designs are determined numerically using various types of algorithms such as those discussed in Fedorov (), Wynn (), Chaloner & Larntz (), Molchanov & Zuyev () and Chang & Lin (). For instance, Chaloner & Larntz () used the Nelder–Mead method, which is a simplex‐based approach to find Bayesian D‐optimal designs for the logistic model, and Molchanov & Zuyev () used a steepest‐ascent algorithm that guarantees convergence to the optimum but can become slow in its vicinity.…”

Section: Introductionmentioning

confidence: 99%

Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming‐Based Approach

Duarte

Wong

2014

Int Statistical Rev

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Summary This paper uses semidefinite programming (SDP) to construct Bayesian optimal design for nonlinear regression models. The setup here extends the formulation of the optimal designs problem as an SDP problem from linear to nonlinear models. Gaussian quadrature formulas (GQF) are used to compute the expectation in the Bayesian design criterion, such as D-, A- or E-optimality. As an illustrative example, we demonstrate the approach using the power-logistic model and compare results in the literature. Additionally, we investigate how the optimal design is impacted by different discretising schemes for the design space, different amounts of uncertainty in the parameter values, different choices of GQF and different prior distributions for the vector of model parameters, including normal priors with and without correlated components. Further applications to find Bayesian D-optimal designs with two regressors for a logistic model and a two-variable generalised linear model with a gamma distributed response are discussed, and some limitations of our approach are noted.

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The polytope of optimal approximate designs: extending the selection of informative experiments

Harman,

Filová,

Rosa

2024

Stat Comput

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Consider the problem of constructing an experimental design, optimal for estimating parameters of a given statistical model with respect to a chosen criterion. To address this problem, the literature usually provides a single solution. Often, however, there exists a rich set of optimal designs, and the knowledge of this set can lead to substantially greater freedom to select an appropriate experiment. In this paper, we demonstrate that the set of all optimal approximate designs generally corresponds to a polytope. Particularly important elements of the polytope are its vertices, which we call vertex optimal designs. We prove that the vertex optimal designs possess unique properties, such as small supports, and outline strategies for how they can facilitate the construction of suitable experiments. Moreover, we show that for a variety of situations it is possible to construct the vertex optimal designs with the assistance of a computer, by employing error-free rational-arithmetic calculations. In such cases the vertex optimal designs are exact, often closely related to known combinatorial designs. Using this approach, we were able to determine the polytope of optimal designs for some of the most common multifactor regression models, thereby extending the choice of informative experiments for a large variety of applications.

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On Minimally-supported D-optimal Designs for Polynomial Regression with Log-concave Weight Function

Abstract: Approximate D-optimal design, Cyclic exchange algorithm, Gershgorin Circle Theorem, Log-concave, Minimally-supported design, Weighted polynomial regression,

Cited by 5 publications

References 12 publications

Design admissibility and de la Garza phenomenon in multi-factor experiments

Design admissibility and de la Garza phenomenon in multi-factor experiments

Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming‐Based Approach

The polytope of optimal approximate designs: extending the selection of informative experiments

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