arXiv subjects

Holger Dette

Publications and source records attributed to Holger Dette.

At least 199 records · Page 11Linked to original sources

arXiv2012

Nonparametric quantile regression for twice censored data

We consider the problem of nonparametric quantile regression for twice censored data. Two new estimates are presented, which are constructed by applying concepts of monotone rearrangements to estimates of the conditional distribution function. The proposed methods avoid the problem of crossing quantile curves. Weak uniform consistency and weak convergence is established for both estimates and their finite sample properties are investigated by means of a simulation study. As a by-product, we obtain a new result regarding the weak convergence of the Beran estimator for right censored data on the maximal possible domain, which is of its own interest.

stat.ME↗

arXiv2012

$T$-optimal designs for discrimination between two polynomial models

This paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree $n-2$ and $n$. In a fundamental paper, Atkinson and Fedorov [Biometrika 62 (1975a) 57--70] proposed the $T$-optimality criterion for this purpose. Recently, Atkinson [MODA 9, Advances in Model-Oriented Design and Analysis (2010) 9--16] determined $T$-optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide half of the circle into equal parts if the coefficient of $x^{n-1}$ in the polynomial of larger degree vanishes. In the present paper we give a strong justification of the conjecture and determine all $T$-optimal designs explicitly for any degree $n\in\mathbb{N}$. In particular, we show that there exists a one-dimensional class of $T$-optimal designs. Moreover, we also present a generalization to the case when the ratio between the coefficients of $x^{n-1}$ and $x^n$ is smaller than a certain critical value. Because of the complexity of the optimization problem, $T$-optimal designs have only been determined numerically so far, and this paper provides the first explicit solution of the $T$-optimal design problem since its introduction by Atkinson and Fedorov [Biometrika 62 (1975a) 57--70]. Finally, for the remaining cases (where the ratio of coefficients is larger than the critical value), we propose a numerical procedure to calculate the $T$-optimal designs. The results are also illustrated in an example.

math.ST↗

arXiv2012

Asymptotic optimal designs under long-range dependence error structure

We discuss the optimal design problem in regression models with long-range dependence error structure. Asymptotic optimal designs are derived and it is demonstrated that these designs depend only indirectly on the correlation function. Several examples are investigated to illustrate the theory. Finally, the optimal designs are compared with asymptotic optimal designs which were derived by Bickel and Herzberg [Ann. Statist. 7 (1979) 77--95] for regression models with short-range dependent error.

math.ST↗

arXiv2011

New estimators of the Pickands dependence function and a test for extreme-value dependence

We propose a new class of estimators for Pickands dependence function which is based on the concept of minimum distance estimation. An explicit integral representation of the function $A^*(t)$, which minimizes a weighted $L^2$-distance between the logarithm of the copula $C(y^{1-t},y^t)$ and functions of the form $A(t)\log(y)$ is derived. If the unknown copula is an extreme-value copula, the function $A^*(t)$ coincides with Pickands dependence function. Moreover, even if this is not the case, the function $A^*(t)$ always satisfies the boundary conditions of a Pickands dependence function. The estimators are obtained by replacing the unknown copula by its empirical counterpart and weak convergence of the corresponding process is shown. A comparison with the commonly used estimators is performed from a theoretical point of view and by means of a simulation study. Our asymptotic and numerical results indicate that some of the new estimators outperform the estimators, which were recently proposed by Genest and Segers [Ann. Statist. 37 (2009) 2990--3022]. As a by-product of our results, we obtain a simple test for the hypothesis of an extreme-value copula, which is consistent against all positive quadrant dependent alternatives satisfying weak differentiability assumptions of first order.

math.ST↗

arXiv2011

A test for Archimedeanity in bivariate copula models

We propose a new test for the hypothesis that a bivariate copula is an Archimedean copula. The test statistic is based on a combination of two measures resulting from the characterization of Archimedean copulas by the property of associativity and by a strict upper bound on the diagonal by the Fréchet-upper bound. We prove weak convergence of this statistic and show that the critical values of the corresponding test can be determined by the multiplier bootstrap method. The test is shown to be consistent against all departures from Archimedeanity if the copula satisfies weak smoothness assumptions. A simulation study is presented which illustrates the finite sample properties of the new test.

math.ST↗

arXiv2011

Response-adaptive dose-finding under model uncertainty

Dose-finding studies are frequently conducted to evaluate the effect of different doses or concentration levels of a compound on a response of interest. Applications include the investigation of a new medicinal drug, a herbicide or fertilizer, a molecular entity, an environmental toxin, or an industrial chemical. In pharmaceutical drug development, dose-finding studies are of critical importance because of regulatory requirements that marketed doses are safe and provide clinically relevant efficacy. Motivated by a dose-finding study in moderate persistent asthma, we propose response-adaptive designs addressing two major challenges in dose-finding studies: uncertainty about the dose-response models and large variability in parameter estimates. To allocate new cohorts of patients in an ongoing study, we use optimal designs that are robust under model uncertainty. In addition, we use a Bayesian shrinkage approach to stabilize the parameter estimates over the successive interim analyses used in the adaptations. This approach allows us to calculate updated parameter estimates and model probabilities that can then be used to calculate the optimal design for subsequent cohorts. The resulting designs are hence robust with respect to model misspecification and additionally can efficiently adapt to the information accrued in an ongoing study. We focus on adaptive designs for estimating the minimum effective dose, although alternative optimality criteria or mixtures thereof could be used, enabling the design to address multiple objectives.

stat.AP↗

arXiv2011

A note on the de la Garza phenomenon for locally optimal designs

The celebrated de la Garza phenomenon states that for a polynomial regression model of degree $p-1$ any optimal design can be based on at most $p$ design points. In a remarkable paper, Yang [Ann. Statist. 38 (2010) 2499--2524] showed that this phenomenon exists in many locally optimal design problems for nonlinear models. In the present note, we present a different view point on these findings using results about moment theory and Chebyshev systems. In particular, we show that this phenomenon occurs in an even larger class of models than considered so far.

math.ST↗

arXiv2011

Matrix measures, random moments and Gaussian ensembles

We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$ real or complex matrix measures defined on the interval $[0,1]$. The asymptotic properties of the first $k$ components of a uniformly distributed vector $(S_{1,n}, ..., S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n \to \infty$. In particular, it is shown that an appropriately centered and standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$ converges weakly to a vector of $k$ independent $p \times p$ Gaussian ensembles. For the proof of our results we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first $k$ canonical moments corresponding to the uniform distribution on the real or complex moment space $\mathcal{M}_n$ are independent multivariate Beta distributed random variables and that each of these random variables converge in distribution (if the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.

math.PR↗

arXiv2010

Optimal designs for discriminating between dose-response models in toxicology studies

We consider design issues for toxicology studies when we have a continuous response and the true mean response is only known to be a member of a class of nested models. This class of non-linear models was proposed by toxicologists who were concerned only with estimation problems. We develop robust and efficient designs for model discrimination and for estimating parameters in the selected model at the same time. In particular, we propose designs that maximize the minimum of $D$- or $D_1$-efficiencies over all models in the given class. We show that our optimal designs are efficient for determining an appropriate model from the postulated class, quite efficient for estimating model parameters in the identified model and also robust with respect to model misspecification. To facilitate the use of optimal design ideas in practice, we have also constructed a website that freely enables practitioners to generate a variety of optimal designs for a range of models and also enables them to evaluate the efficiency of any design.

math.ST↗

arXiv2010

Optimal designs for random effect models with correlated errors with applications in population pharmacokinetics

We consider the problem of constructing optimal designs for population pharmacokinetics which use random effect models. It is common practice in the design of experiments in such studies to assume uncorrelated errors for each subject. In the present paper a new approach is introduced to determine efficient designs for nonlinear least squares estimation which addresses the problem of correlation between observations corresponding to the same subject. We use asymptotic arguments to derive optimal design densities, and the designs for finite sample sizes are constructed from the quantiles of the corresponding optimal distribution function. It is demonstrated that compared to the optimal exact designs, whose determination is a hard numerical problem, these designs are very efficient. Alternatively, the designs derived from asymptotic theory could be used as starting designs for the numerical computation of exact optimal designs. Several examples of linear and nonlinear models are presented in order to illustrate the methodology. In particular, it is demonstrated that naively chosen equally spaced designs may lead to less accurate estimation.

stat.AP↗

arXiv2009

A geometric characterization of $c$-optimal designs for heteroscedastic regression

We consider the common nonlinear regression model where the variance, as well as the mean, is a parametric function of the explanatory variables. The $c$-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of $c$-optimal designs in this context is presented, which generalizes the classical result of Elfving [Ann. Math. Statist. 23 (1952) 255--262] for $c$-optimal designs. As in Elfving's famous characterization, $c$-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking, the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The $c$-optimal designs are characterized as representations of the points where the line in direction of the vector $c$ intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects.

math.ST↗

arXiv2009

Optimal discrimination designs

We consider the problem of constructing optimal designs for model discrimination between competing regression models. Various new properties of optimal designs with respect to the popular $T$-optimality criterion are derived, which in many circumstances allow an explicit determination of $T$-optimal designs. It is also demonstrated, that in nested linear models the number of support points of $T$-optimal designs is usually too small to estimate all parameters in the extended model. In many cases $T$-optimal designs are usually not unique, and in this situation we give a characterization of all $T$-optimal designs. Finally, $T$-optimal designs are compared with optimal discriminating designs with respect to alternative criteria by means of a small simulation study.

math.ST↗

arXiv2009

Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szegö recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [-1,1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.

math.CA↗

arXiv2009

Some asymptotic properties of the spectrum of the Jacobi ensemble

For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | λ_i - λ_j |^β\prod^n_{i=1} (2 - λ_i)^a (2 + λ_i)^b I_{(-2,2)} (λ_i) $$ $(a, b > -1, β> 0) $ a strong uniform approximation by the roots of the Jacobi polynomials is derived if the parameters $a, b,$ $β$ depend on $n$ and $n \to \infty$. Roughly speaking, the eigenvalues can be uniformly approximated by roots of Jacobi polynomials with parameters $((2a+2)/β-1, (2b+2)/β-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$. These results are used to investigate the asymptotic properties of the corresponding spectral distribution if $n \to \infty$ and the parameters $a, b$ and $β$ vary with $n$. We also discuss further applications in the context of multivariate random $F$-matrices.

math.PR↗

arXiv2009

Optimal designs for dose-finding experiments in toxicity studies

We construct optimal designs for estimating fetal malformation rate, prenatal death rate and an overall toxicity index in a toxicology study under a broad range of model assumptions. We use Weibull distributions to model these rates and assume that the number of implants depend on the dose level. We study properties of the optimal designs when the intra-litter correlation coefficient depends on the dose levels in different ways. Locally optimal designs are found, along with robustified versions of the designs that are less sensitive to misspecification in the initial values of the model parameters. We also report efficiencies of commonly used designs in toxicological experiments and efficiencies of the proposed optimal designs when the true rates have non-Weibull distributions. Optimal design strategies for finding multiple-objective designs in toxicology studies are outlined as well.

stat.ME↗

arXiv2008

Random block matrices and matrix orthogonal polynomials

In this paper we consider random block matrices, which generalize the general beta ensembles, which were recently investigated by Dumitriu and Edelmann (2002, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density, which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.

math.PR↗

arXiv2007

On the number of support points of maximin and Bayesian optimal designs

We consider maximin and Bayesian $D$-optimal designs for nonlinear regression models. The maximin criterion requires the specification of a region for the nonlinear parameters in the model, while the Bayesian optimality criterion assumes that a prior for these parameters is available. On interval parameter spaces, it was observed empirically by many authors that an increase of uncertainty in the prior information (i.e., a larger range for the parameter space in the maximin criterion or a larger variance of the prior in the Bayesian criterion) yields a larger number of support points of the corresponding optimal designs. In this paper, we present analytic tools which are used to prove this phenomenon in concrete situations. The proposed methodology can be used to explain many empirically observed results in the literature. Moreover, it explains why maximin $D$-optimal designs are usually supported at more points than Bayesian $D$-optimal designs.

math.ST↗

arXiv2006

Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors

We determine optimal designs for some regression models which are frequently used for describing three-dimensional shapes. These models are based on a Fourier expansion of a function defined on the unit sphere in terms of spherical harmonic basis functions. In particular, it is demonstrated that the uniform distribution on the sphere is optimal with respect to all $Φ_p$ criteria proposed by Kiefer in 1974 and also optimal with respect to a criterion which maximizes a $p$ mean of the $r$ smallest eigenvalues of the variance--covariance matrix. This criterion is related to principal component analysis, which is the common tool for analyzing this type of image data. Moreover, discrete designs on the sphere are derived, which yield the same information matrix in the spherical harmonic regression model as the uniform distribution and are therefore directly implementable in practice. It is demonstrated that the new designs are substantially more efficient than the commonly used designs in three-dimensional shape analysis.

math.ST↗