Uniform approximation of eigenvalues in Laguerre and Hermite beta-ensembles by roots of orthogonal polynomials

We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite beta-ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity. We also provide estimates of the rate of convergence. --Gaussian ensemble,random matrix,rate of convergence,Weyl?s inequality,Wishart matrix

Robust designs in non-inferiority three arm clinical trials with presence of heteroscedasticity

In this paper, we describe an adjusted method to facilitate a non-inferiority trial by a three-arm robust design. Because local optimal designs derived in Hasler et al. [2007] require knowledge about the ratios of the population variances and are not necessarily robust with respect to possible misspecifications, a maximin approach is adopted. This method requires only the specification of an interval for the of variance ratios and yields robust and efficient designs. We demonstrate that a maximin optimal design only depends on the boundary points specified for the intervals of the variance ratios and receive numerical and analytical solutions. The derived designs are robust and very efficient for statistical analysis in non inferiority three arm trials. --maximin design,robust design,non-inferiority,three arm design,gold design trials,randomized clinical trial

Quantile Correlations: Uncovering temporal dependencies in financial time series

We conduct an empirical study using the quantile-based correlation function to uncover the temporal dependencies in financial time series. The study uses intraday data for the S\&P 500 stocks from the New York Stock Exchange. After establishing an empirical overview we compare the quantile-based correlation function to stochastic processes from the GARCH family and find striking differences. This motivates us to propose the quantile-based correlation function as a powerful tool to assess the agreements between stochastic processes and empirical data

Bayesian and maximin optimal designs for heteroscedastic regression models

The problem of constructing standardized maximin D-optimal designs for weighted polynomial regression models is addressed. In particular it is shown that, by following the broad approach to the construction of maximin designs introduced recently by Dette, Haines and Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian Φq-optimal designs. The approach is illustrated for two specific weighted polynomial models and also for a particular growth model. --

Maximin and Bayesian optimal designs for regression models

For many problems of statistical inference in regression modelling, the Fisher information matrix depends on certain nuisance parameters which are unknown and which enter the model nonlinearly. A common strategy to deal with this problem within the context of design is to construct maximin optimal designs as those designs which maximize the minimum value of a real valued (standardized) function of the Fisher information matrix, where the minimum is taken over a specified range of the unknown parameters. The maximin criterion is not differentiable and the construction of the associated optimal designs is therefore difficult to achieve in practice. In the present paper the relationship between maximin optimal designs and a class of Bayesian optimal designs for which the associated criteria are differentiable is explored. In particular, a general methodology for determining maximin optimal designs is introduced based on the fact that in many cases these designs can be obtained as weak limits of appropriate Bayesian optimal designs. --maximin optimal designs,Bayesian optimal designs,nonlinear regression models,parameter estimation,least favourable prior

Optimal design for linear models with correlated observations

In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. For one parameter models this condition is also shown to be sufficient in many cases and for several models optimal designs can be identified explicitly. For the multi-parameter regression models a simple relation which allows verifying the necessary optimality condition is established. Moreover, it is proved that the arcsine distribution is universally optimal for the polynomial regression model with a correlation structure defined by the logarithmic potential. It is also shown that for models in which the regression functions are eigenfunctions of an integral operator induced by the correlation kernel of the error process, designs satisfying the necessary conditions of optimality can be found explicitly. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model

Robust designs in non-inferiority three arm clinical trials with presence of heteroscedasticity

In this paper, we describe an adjusted method to facilitate a non-inferiority trial by a three-arm robust design. Because local optimal designs derived in Hasler et al. [2007] require knowledge about the ratios of the population variances and are not necessarily robust with respect to possible misspecifications, a maximin approach is adopted. This method requires only the specification of an interval for the of variance ratios and yields robust and efficient designs. We demonstrate that a maximin optimal design only depends on the boundary points specified for the intervals of the variance ratios and receive numerical and analytical solutions. The derived designs are robust and very efficient for statistical analysis in non inferiority three arm trials

Bayesian and Maximum Optimal Designs for Heteroscedastic Regression Models

The problem of constructing standardized maximin D-optimal designs for weighted polynomial regression models is addressed. In particular it is shown that, by following the broad approach to the construction of maximin designs introduced recently by Dette, Haines and Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian Φ_q-optimal designs. The approach is illustrated for two specific weighted polynomial models and also for a particular growth model

A copula-based nonparametric measure of regression dependence

This paper presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO). In addition, we define a new nonparametric measure of regression dependence and study its properties. Beside being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. A consistent nonparametric estimator of the new measure is constructed and its asymptotic properties are investigated. Finally, the finite sample properties of the estimate are studied by means of small simulation study