Compound designs for dose‐finding in the presence of nondesignable covariates

Compound optimal designs are considered where one component of the design criterion is a traditional optimality criterion such as the D‐optimality criterion, and the other component accounts for higher efficacy with low toxicity. With reference to the dose‐finding problem, we suggest the technique to choose weights for the two components that makes the optimization problem simpler than the traditional penalized design. We allow general bivariate responses for efficacy and toxicity. We then extend the procedure in the presence of nondesignable covariates such as age, sex, or other health conditions. A new breast cancer treatment is considered to illustrate the procedures. Copyright © 2013 John Wiley & Sons, Ltd.     

Convergence criteria for interval-valued inequality indices

In this paper we introduce an interval-valued inequality index for random intervals based on a convex function. We show that if this function does not grow faster than x ^p , then the inequality index is continuous on the space of random intervals with finite p-th moment. A bound for the distance between the inequality indices of two random intervals is also constructed. An example is presented to motivate and illustrate the developments in this paper.     

Generalized skew-normal models: properties and inference

In this article, we introduce a new family of asymmetric distributions, which depends on two parameters namely, α and β, and in the special case where β = 0, the skew-normal (SN) distribution considered by Azzallini [Azzalini, A., 1985, A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12, 171–178.] is obtained. Basic properties such as a stochastic representation and the derivation of maximum likelihood and moment estimators are studied. The asymptotic behaviour of both types of estimators is also investigated. Results of a small-scale simulation study is provided illustrating the usefulness of the new model. An application to a real data set is reported showing that it can present better fit than the SN distribution.     

Optimal experimental designs when an independent variable is potentially censored

This paper considers the problem of constructing optimal approximate designs when an independent variable might be censored. The problem is which design should be applied in practice to obtain the best approximate design when a censoring distribution is assumed known in advance. The approach for finite or continuous design spaces deserves different attention. In both cases, equivalent theorems and algorithms are provided in order to calculate optimal designs. Some examples illustrate this approach for D-optimality.     

Correcting the bias due to dependent censoring of the survival estimator by conditioning

We consider the conditional estimation of the survival function of the time T₂ to a second event as a function of the time T₁ to a first event when there is a censoring mechanism acting on their sum T₁+T₂. The problem has been motivated by a treatment interruption study aimed at improving the quality of life of HIV-infected patients. We base the analysis on the survival function of T₂ given that T₁∈I, where I represents a period of scientific interest (1 trimester, 1 year, 2 years, etc.) and propose a non-parametric estimator for the survival function of T₂ given that T₁∈I, which takes into account both the selection bias and the heterogeneity due to the dependent censoring. The proposed estimator for the survival function uses the risk group of T₂ conditioned on the categories of T₁ and corrects for the dependent censoring using weights defined by the observed values of T₁. The estimator, properly normalized, converges weakly to a zero-mean Gaussian process. We estimate the variance of the limiting process via a bootstrap methodology. Properties of the proposed estimator are illustrated by an extensive simulation study. The motivating data set is analysed by means of this new methodology.     

A Note on the Variances of the Tests of Kendall,Jonckheere, and Terpstra

Recently, some papers have been appearing that have noticed some mistakes in the calculations in the statistical software in simple and classical tests (Wilcoxon–Mann-Whitney and Log-Rank). This article examines the variance formulas for the Kendall test, the Jonckheere test, and the Terpstra test. We show in this article that these variances (in spite of what is said in the literature and in the statistical software), in the presence of ties and r ≥ 3 groups, are only equal approximately and are exactly equal for two groups and ties as we prove in the Appendix.     

Finding Relevant Linear Manifolds in Classification by Gaussian Mixtures

In this article, we present a strategy for producing low-dimensional projections that maximally separate the classes in Gaussian Mixture Model classification. The most revealing linear manifolds are those along which the classes are maximally separable. Here we consider a particular probability product kernel as a measure of similarity or affinity between the class-conditional distributions. It takes an appealing closed analytical form in the case of Gaussian mixture components. The performance of the proposed strategy has been evaluated on real data.     

Estimation for Stochastic Models Driven by Laplace Motion

Laplace motion is a Lévy process built upon Laplace distributions. Non Gaussian stochastic fields that are integrals with respect to this process are considered and methods for their model fitting are discussed. The proposed procedures allow for inference about the parameters of the underlying Laplace distributions. A fit of dependence structure is also addressed. The importance of a convenient parameterization that admits natural and consistent estimation for this class of models is emphasized. Several parameterizations are introduced and their advantages over one another discussed. The proposed estimation method targets the standard characteristics: mean, variance, skewness and kurtosis. Their sample equivalents are matched in the closest possible way as allowed by natural constraints within this class. A simulation study and an example of potential applications conclude the article.     

Gamma-Generalized Inverse Gaussian Class of Distributions with Applications

In this article, a new family of probability distributions with domain in ?⁺ is introduced. This class can be considered as a natural extension of the exponential-inverse Gaussian distribution in Bhattacharya and Kumar (1986 Bhattacharya , S. K. , Kumar , S. ( 1986 ). E-IG model in life testing . Calcutta Statist. Assoc. Bull. 35 : 85 – 90 . [Google Scholar]) and Frangos and Karlis (2004 Frangos , N. , Karlis , D. ( 2004 ). Modelling losses using an exponential-inverse Gaussian distribution . Insur. Math. Econo. 35 : 53 – 67 .[Crossref], [Web of Science ®] , [Google Scholar]). This new family is obtained through the mixture of gamma distribution with generalized inverse Gaussian distribution. We also show some important features such as expressions of probability density function, moments, etc. Special attention is paid to the mixture with the inverse Gaussian distribution, as a particular case of the generalized inverse Gaussian distribution. From the exponential-inverse Gaussian distribution two one-parameter family of distributions are obtained to derive risk measures and credibility expressions. The versatility of this family has been proven in numerical examples.