Optimal discrimination designs for exponential regression models

We investigate optimal designs for discriminating between exponential regression models of different complexity, which are widely used in the biological sciences; see, e.g., Landaw [1995. Robust sampling designs for compartmental models under large prior eigenvalue uncertainties. Math. Comput. Biomed. Appl. 181–187] or Gibaldi and Perrier [1982. Pharmacokinetics. Marcel Dekker, New York]. We discuss different approaches for the construction of appropriate optimality criteria, and find sharper upper bounds on the number of support points of locally optimal discrimination designs than those given by Caratheodory's Theorem. These results greatly facilitate the numerical construction of optimal designs. Various examples of optimal designs are then presented and compared to different other designs. Moreover, to protect the experiment against misspecifications of the nonlinear model parameters, we adapt the design criteria such that the resulting designs are robust with respect to such misspecifications and, again, provide several examples, which demonstrate the advantages of our approach.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

Experimental designs for binary data in switching measurements on superconducting Josephson junctions

Summary.  We study the optimal design of switching measurements of small Josephsonjunction circuits which operate in the macroscopic quantum tunnelling regime. In the experiment, sequences of current pulses are applied to the Josephson junction sample, while the voltage over the structure is monitored. The appearance of a voltage pulse to a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Starting from the D -optimality criterion we derive the optimal design for the estimation of the unknown parameters of the underlying Gumbel-type distribution. As a practical method for the measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The design presented has immediate applications in the area of superconducting electronics, implying faster data acquisition. The experimental results presented confirm the usefulness of the method.     

Some robust design strategies for percentile estimation in binary response models

For the problem of percentile estimation of a quantal response curve, the authors determine multiobjective designs which are robust with respect to misspecifications of the model assumptions. They propose a maximin approach based on efficiencies which leads to designs that are simultaneously efficient with respect to various choices of link functions and parameter regions. Furthermore, the authors deal with the problems of designing model and percentile robust experiments. They give various examples of such designs, which are calculated numerically.     

Estimation of the quantile function using Bernstein–Durrmeyer polynomials

This paper studies quantile estimation using Bernstein–Durrmeyer polynomials in terms of its mean squared error and integrated mean squared error including rates of convergence as well as its asymptotic distribution. Whereas the rates of convergence are achieved for i.i.d. samples, we also show that the consistency more or less directly follows from the consistency of the sample quantiles, such that our proposal can also be applied for risk measurement in finance and insurance. Furthermore, an improved estimator based on an error-correction approach is proposed for which a general consistency result is established. A crucial issue is how to select the degree of Bernstein–Durrmeyer polynomials. We propose a novel data-adaptive approach that controls the number of modes of the corresponding density estimator. Its consistency including an uniform error bound as well as its limiting distribution in the sense of a general invariance principle are established. The finite sample properties are investigated by a Monte Carlo study. Finally, the results are illustrated by an application to photovoltaic energy research.     

Efficient design of experiments in the Monod model

Summary. Estimation and experimental design in a non-linear regression model that is used in microbiology are studied. The Monod model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, water research, pharmacokinetics and plant physiology. It is proved that least squares estimates are asymptotically unbiased and normally distributed. The asymptotic covariance matrix of the estimator is the basis for the construction of efficient designs of experiments. In particular locally D -, E - and c -optimal designs are determined and their properties are studied theoretically and by simulation. If certain intervals for the non-linear parameters can be specified, locally optimal designs can be constructed which are robust with respect to a misspecification of the initial parameters and which allow efficient parameter estimation. Parameter variances can be decreased by a factor of 2 by simply sampling at optimal times during the experiment.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Uniform coverage designs for mixture experiments

We investigate an optimization problem for mixture experiments. We consider the case when a large number of ingredients are available but mixtures can contain only a few number of ingredients. These conditions are held in experiments for self assembling molecular systems. First, we introduce a concept of uniform coverage design specialized for the situation. Next, we propose to use the stepwise technique for estimating coefficients of third-order Scheffe model which describes a response surface. After that, we propose a method of adding new mixtures for a movement to an extremum region. By this method, additional mixtures of experiments are extremum points of current estimated model and points which lead to more accurate estimation of current model prediction. This methodology is studied numerically for a model constructed from real data.