Minimal Number Of Support Points Research Articles

Excess of locally D-optimal designs for Cobb–Douglas model

In this paper we study the problem of homothety’s influence on the number of optimal design support points under fixed values of a regression model’s parameters. The Cobb–Douglas two-dimensional nonlinear in parameters model used in microeconomics is considered. There exist two types of optimal designs: saturated (i.e. design with the number support points equal to the number of parameters) and excess design (i.e. design with greater number of support points). The optimal designs with the minimal number of support points are constructed explicitly. Numerical methods for constructing designs with greater number of points are used.

Excess of locally D-optimal designs and homothetic transformations

The paper is devoted to the study of homothety’s influence on the number of optimal design support points under fixed values of a regression model’s parameters. The Ayen–Peters two-dimensional nonlinear in parameters model used in analytical chemistry is considered. It is shown that the number of optimal design support points must be greater than or equal to the number of parameters depending on certain conditions. The optimal designs with the minimal number of support points are constructed explicitly. Some numerical methods for constructing designs with greater number of points (we suggest to call them excess designs) are used.

Optimal Designs for a Probit Model With a Quadratic Term

This article studies optimal designs to analyze dose-response functions with a downturn. Two interesting challenges are estimating the entire dose-response curve and estimating the ED50. Here, I obtain and compare optimal designs for these objectives, separately and together in a two-stage design. I adopt a probit model with a quadratic term to describe the dose-response. Under the probit model, Yang's method is used to obtain the minimal number of support points that maximize any concave function of the Fisher information matrix. Optimal designs are obtained based on the minimal number of support points, and their efficiencies are compared.

Identifying locally optimal designs for nonlinear models: A simple extension with profound consequences

We extend the approach in [Ann. Statist. 38 (2010) 2499–2524] for identifying locally optimal designs for nonlinear models. Conceptually the extension is relatively simple, but the consequences in terms of applications are profound. As we will demonstrate, we can obtain results for locally optimal designs under many optimality criteria and for a larger class of models than has been done hitherto. In many cases the results lead to optimal designs with the minimal number of support points.

Some results on D-optimal designs for nonlinear models with applications

Sufficient conditions are established for the locally D$-optimal design for a nonlinear model to have a minimal number of support points. The conditions are applied to obtain locally D-optimal designs for a one-compartment pharmacokinetic model and a Poisson regression model.

Cost-effective experimental design to support modeling of concentration–response functions

Modeling concentration–response function became extremely popular in ecotoxicology during the last decade. Indeed, modeling allows determining the total response pattern of a given substance. However, reliable modeling is consuming in term of data, which is in contradiction with the current trend in ecotoxicology, which aims to reduce, for cost and ethical reasons, the number of data produced during an experiment. It is therefore crucial to determine experimental design in a cost-effective manner. In this paper, we propose to use the theory of locally D-optimal designs to determine the set of concentrations to be tested so that the parameters of the concentration–response function can be estimated with high precision. We illustrated this approach by determining the locally D-optimal designs to estimate the toxicity of the herbicide dinoseb on daphnids and algae. The results show that the number of concentrations to be tested is often equal to the number of parameters and often related to the their meaning, i.e. they are located close to the parameters. Furthermore, the results show that the locally D-optimal design often has the minimal number of support points and is not much sensitive to small changes in nominal values of the parameters. In order to reduce the experimental cost and the use of test organisms, especially in case of long-term studies, reliable nominal values may therefore be fixed based on prior knowledge and literature research instead of on preliminary experiments.

On s-convex approximations

Let ℬs([a,b];μ1,μ2,…,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,…,μs-1 as their first s-1 moments. In this paper we examine some aspects of the structure of ℬs([a,b];μ1,μ2,…,μs-1) and of the s-convex stochastic extrema in it. Using representation results of moment matrices à la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in ℬs([a,b];μ1,μ2,…,μs-1) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in ℬs([a,b];μ1,μ2,…,μs-1) by one of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.

On s -convex approximations

Let B-s([a, b]; mu (1), mu (2),...,mu (s-1)) be the class of all distribution functions of random variables with support in [a, b] having mu (1), mu (2),..., mu (s-1) as their first s - 1 moments. In this paper we examine some aspects of the structure of B-s ([a, b]; mu (1), mu (2),..., mu (s-1)) and of the s -convex stochastic extrema in it. Using representation results of moment matrices Li la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in B-s ([a, b]; mu (1), mu (2),..., mu (s-1)) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in B-s ([a, b]; mu (1), mu (2),..., mu (s-1)) by One of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.

Optimal Bayesian designs for models with partially specified heteroscedastic structure

We consider the problem of finding a nonsequential optimal design for estimating parameters in a generalized exponential growth model. This problem is solved by first considering polynomial regression models with error variances that depend on the covariate value and unknown parameters. A Bayesian approach is adopted, and optimal Bayesian designs supported on a minimal number of support points for estimating the coefficients in the polynomial model are found analytically. For some criteria, the optimal Bayesian designs depend only on the expectation of the prior, but generally their dependence includes the derivative of the logarithm of the Laplace transform of a measure induced by the prior. The optimal design for the generalized exponential growth model is then determined from these optimal Bayesian designs.

Minimal number of support points for mini-max optimal designs

Conditions are given for a class of mini-max optimal designs to have a minimum number of support points. The result is applied to construct a G-optimal design when the error variance depends on the independent variable exponentially.