On robust and efficient designs for risk estimation in epidemiologic studies

We consider the design problem for the estimation of several scalar measures suggested in the epidemiological literature for comparing the success rate in two samples. The designs considered so far in the literature are local in the sense that they depend on the unknown probabilities of success in the two groups and are not necessarily robust with respect to their misspecification. A maximin approach is proposed to obtain efficient and robust designs for the estimation of the relative risk, attributable risk and odds ratio, whenever a range for the success rates can be specified by the experimenter. It is demonstrated that the designs obtained by this method are usually more efficient than the uniform design, which allocates equal sample sizes to the two groups. --two by two table,odds ratio,relativ risk,attributable risk,optimal design,efficient design

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.Comment: 25 page

Complete classes of designs for nonlinear regression models and principal representations of moment spaces

In a recent paper Yang and Stufken [Ann. Statist. 40 (2012a) 1665-1685] gave sufficient conditions for complete classes of designs for nonlinear regression models. In this note we demonstrate that there is an alternative way to validate this result. Our main argument utilizes the fact that boundary points of moment spaces generated by Chebyshev systems possess unique representations.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1108 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal discriminating designs for several competing regression models

The problem of constructing optimal discriminating designs for a class of regression models is considered. We investigate a version of the $T_p$-optimality criterion as introduced by Atkinson and Fedorov [Biometrika 62 (1975a) 289-303]. The numerical construction of optimal designs is very hard and challenging, if the number of pairwise comparisons is larger than 2. It is demonstrated that optimal designs with respect to this type of criteria can be obtained by solving (nonlinear) vector-valued approximation problems. We use a characterization of the best approximations to develop an efficient algorithm for the determination of the optimal discriminating designs. The new procedure is compared with the currently available methods in several numerical examples, and we demonstrate that the new method can find optimal discriminating designs in situations where the currently available procedures fail.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1103 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Detecting gradual changes in locally stationary processes

In a wide range of applications, the stochastic properties of the observed time series change over time. The changes often occur gradually rather than abruptly: the properties are (approximately) constant for some time and then slowly start to change. In many cases, it is of interest to locate the time point where the properties start to vary. In contrast to the analysis of abrupt changes, methods for detecting smooth or gradual change points are less developed and often require strong parametric assumptions. In this paper, we develop a fully nonparametric method to estimate a smooth change point in a locally stationary framework. We set up a general procedure which allows us to deal with a wide variety of stochastic properties including the mean, (auto)covariances and higher moments. The theoretical part of the paper establishes the convergence rate of the new estimator. In addition, we examine its finite sample performance by means of a simulation study and illustrate the methodology by two applications to financial return data.Comment: Published at http://dx.doi.org/10.1214/14-AOS1297 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org