Holger Dette

The theory of canonical moments with applications in statistics, probability, and analysis

Canonical Moments. Orthogonal Polynomials. Continued Fractions and the Stieltjes Transform. Special Sequences of Canonical Moments. Canonical Moments and Optimal DesignFirst Applications. Discrimination and Model Robust Designs. Applications in Approximation Theory. Canonical Moments and Random Walks. The Circle and Trigonometric Functions. Further Applications. Bibliography. Indexes.

Box-Type Approximations in Nonparametric Factorial Designs

Abstract Linear rank statistics in nonparametric factorial designs are asymptotically normal and, in general, heteroscedastic. In a comprehensive simulation study, the asymptotic chi-squared law of the corresponding quadratic forms is shown to be a rather poor approximation of the finite-sample distribution. Motivated by this problem, we propose simple finite-sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error structure. These approximations are based on an F distribution with estimated degrees of freedom that generalizes ideas of Patnaik and Box. Simulation studies show that the nominal level is maintained with high accuracy and in most cases the power is comparable to the asymptotic maximin Wald test. Data-driven guidelines are given to select the most appropriate test procedure. These ideas are finally transferred to nonparametric factorial designs where the same quadratic forms as in the parametric case are applied to the vector ...

Estimating the variance in nonparametric regression—what is a reasonable choice?

The exact mean-squared error (MSE) of estimators of the variance in nonparametric regression based on quadratic forms is investigated. In particular, two classes of estimators are compared: Hall, Kay and Titteringtons optimal difference-based estimators and a class of ordinary difference-based estimators which generalize methods proposed by Rice and Gasser, Sroka and Jennen-Steinmetz. For small sample sizes the MSE of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function. It is shown that in many situations ordinary difference-based estimators are more appropriate for estimating the variance, because they control the bias much better and hence have a much better overall performance. It is also demonstrated that Rices estimator does not always behave well. Data-driven guidelines are given to select the estimator with the smallest MSE.

Testing heteroscedasticity in nonparametric regression

The importance of being able to detect heteroscedasticity in regression is widely recognized because efficient inference for the regression function requires that heteroscedasticity is taken into account. In this paper a simple consistent test for heteroscedasticity is proposed in a nonparametric regression set‐up. The test is based on an estimator for the best L2‐approximation of the variance function by a constant. Under mild assumptions asymptotic normality of the corresponding test statistic is established even under arbitrary fixed alternatives. Confidence intervals are obtained for a corresponding measure of heteroscedasticity. The finite sample performance and robustness of these procedures are investigated in a simulation study and Box‐type corrections are suggested for small sample sizes.

Optimal Designs for Dose-Finding Studies

Understanding and properly characterizing the dose–response relationship is a fundamental step in the investigation of a new compound, be it a herbicide or fertilizer, a molecular entity, an environmental toxin, or an industrial chemical. In this article we investigate the problem of deriving efficient designs for the estimation of target doses in the context of clinical dose finding. We propose methods to determine the appropriate number and actual levels of the doses to be administered to patients, as well as their relative sample size allocations. More specifically, we derive local optimal designs that minimize the asymptotic variance of the minimum effective dose estimate under a particular dose–response model. We investigate the small-sample properties of these designs, together with their sensitivity to a misspecification of the true parameter values and of the underlying dose–response model, through simulation. Finally, we demonstrate that the designs derived for a fixed model are rather sensitive with respect to this assumption and construct robust optimal designs that take into account a set of potential dose–response profiles within classes of models commonly used in drug development practice.

Robust and Efficient Designs for the Michaelis-Menten Model

For the Michaelis–Menten model, we determine designs that maximize the minimum of the D-efficiencies over a certain interval for the nonlinear parameter. The best two point designs can be found explicitly, and a characterization is given when these designs are optimal within the class of all designs. In most cases of practical interest, the determined designs are highly efficient and robust with respect to misspecification of the nonlinear parameter. The results are illustrated and applied in an example of a hormone receptor assay.

A consistent test for heteroscedasticity in nonparametric regression based on the kernel method

This paper presents a new residual based test for heteroscedasticity in nonparametric regression. The construction of the test statistic is motivated by the idea that the problem of testing heteroscedasticity is equivalent to the problem of testing pseudoresiduals for a constant mean. Asymptotic normality is established with different rates corresponding to the null hypothesis of homoscedasticity and the alternative. A Monte Carlo simulation is conducted in order to investigate the finite sample performance of a bootstrap version of the proposed test.

Optimal Design for Goodness-of-Fit of the Michaelis–Menten Enzyme Kinetic Function

We construct efficient designs for the Michaelis–Menten enzyme kinetic model capable of checking model assumptions. An extended model called EMAX is also considered for this purpose. This model is widely used in pharmacokinetics and reduces to the Michaelis–Menten model for a specific choice of parameter settings. Our strategy is to find efficient designs for estimating the parameters in the EMAX model and at the same time test the validity of the Michaelis–Menten model against the EMAX model by maximizing a minimum of the D or D1 efficiencies taken over a range of values for the nonlinear parameters. In particular, we show that such designs are (a) efficient for estimating parameters in the EMAX model, (b) about 70% efficient for estimating parameters in the Michaelis–Menten model, (c) efficient for testing the Michaelis–Menten model against the EMAX model, and (d) robust with respect to misspecification of the unknown parameters in the nonlinear model.

Bayesian D-optimal designs for exponential regression models

Abstract We consider the Bayesian D-optimal design problem for exponential growth models with one, two or three parameters. For the one-parameter model conditions on the shape of the density of the prior distribution and on the range of its support are given guaranteeing that a one-point design is also Bayesian D-optimal within the class of all designs. In the case of two parameters the best two-point designs are determined and for special prior distributions it is proved that these designs are Bayesian D-optimal. Finally, the exponential growth model with three parameters is investigated. The best three-point designs are characterized by a nonlinear equation. The global optimality of these designs cannot be proved analytically and it is demonstrated that these designs are also Bayesian D-optimal within the class of all designs if gamma-distributions are used as prior distributions.

A Note on Nonparametric Estimation of the Effective Dose in Quantal Bioassay

For the common binary response model, we propose a direct method for the nonparametric estimation of the effective dose level. The estimator is obtained by the composition of a nonparametric estimate of the quantile response curve and a classical density estimate. The new method yields a simple and reliable monotone estimate of the effective dose-level curve α → EDα and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series estimators, or smoothing splines. Moreover, it is computationally very efficient, because it does not require a numerical inversion of a monotonized estimate of the quantile dose-response curve. We prove asymptotic normality of the new estimate and compare it with an available alternative estimate (based on a monotonized nonparametric estimate of the dose-response curve and calculation of the inverse function) by means of a simulation study.