On the preservation of copula structure under truncation

The author characterizes the copula associated with the bivariate survival model of Clayton (1978) as the only absolutely continuous copula that is preserved under bivariate truncation.     

A fast splitting procedure for classification trees

This paper provides a faster method to find the best split at each node when using the CART methodology. The predictability index is proposed as a splitting rule for growing the same classification tree as CART does when using the Gini index of heterogeneity as an impurity measure. A theorem is introduced to show a new property of the index : the for a given predictor has a value not lower than the for any split generated by the predictor. This property is used to make a substantial saving in the time required to generate a classification tree. Three simulation studies are presented in order to show the computational gain in terms of both the number of splits analysed at each node and the CPU time. The proposed splitting algorithm can prove computational efficiency in real data sets as shown in an example.     

Fitting competing risks data to bivariate Pareto models

This paper revisits two bivariate Pareto models for fitting competing risks data. The first model is the Frank copula model, and the second one is a bivariate Pareto model introduced by Sankaran and Nair (1993 Sankaran, P. G., and N. U. Nair. 1993. A bivariate Pareto model and its applications to reliability. Naval Research Logistics 40 (7):1013–1020. doi:10.1002/1520-6750(199312)40:7%3c1013::AID-NAV3220400711%3e3.0.CO;2-7.[Crossref], [Web of Science ®] , [Google Scholar]). We discuss the identifiability issues of these models and develop the maximum likelihood estimation procedures including their computational algorithms and model-diagnostic procedures. Simulations are conducted to examine the performance of the maximum likelihood estimation. Real data are analyzed for illustration.     

A class of weighted dependence measures for bivariate failure time data

This paper considers a class of summary measures of the dependence between a pair of failure time variables over a finite follow-up region. The class consists of measures that are weighted averages of local dependence measures, and includes the cross-ratio-measure and finite region version of Kendall's τ; recently proposed by the authors. Two new special cases are identified that can avoid the need to estimate the bivariate survivor function and that admit explicit variance estimators. Nonparametric estimators of such dependence measures are proposed and are shown to be consistent and asymptotically normal with variances that can be consistently estimated. Properties of selected estimators are evaluated in a simulation study, and the method is illustrated through an analysis of Australian Twin Study data.     

Finite normal mixture copulas for multivariate discrete data modeling

A new family of copulas is introduced that provides flexible dependence structure while being tractable and simple to use for multivariate discrete data modeling. The construction exploits finite mixtures of uncorrelated normal distributions. Accordingly, the cumulative distribution function is simply the product of univariate normal distributions. At the same time, however, the mixing operation introduces association. The properties of the new family of copulas are examined and a concrete application is used to show its applicability.     

Copula structure analysis

Summary.  We extend the standard approach of correlation structure analysis for dimension reduction of high dimensional statistical data. The classical assumption of a linear model for the distribution of a random vector is replaced by the weaker assumption of a model for the copula. For elliptical copulas a correlation-like structure remains, but different margins and non-existence of moments are possible. After introducing the new concept and deriving some theoretical results we observe in a simulation study the performance of the estimators: the theoretical asymptotic behaviour of the statistics can be observed even for small sample sizes. Finally, we show our method at work for a financial data set and explain differences between our copula-based approach and the classical approach. Our new method yielear models also.     

Four Correlation Coefficients with a Third Blocking Variable: Their Efficacy,Relative Efficiency,and Test Statistics

Abstract

The efficacy and the asymptotic relative efficiency (ARE) of a weighted sum of Kendall's taus, a weighted sum of Spearman's rhos, a weighted sum of Pearson's r's, and a weighted sum of z-transformation of the Fisher–Yates correlation coefficients, in the presence of a blocking variable, are discussed. The method of selecting the weighting constants that maximize the efficacy of these four correlation coefficients is proposed. The estimate, test statistics and confidence interval of the four correlation coefficients with weights are also developed. To compare the small-sample properties of the four tests, a simulation study is performed. The theoretical and simulated results all prefer the weighted sum of the Pearson correlation coefficients with the optimal weights, as well as the weighted sum of z-transformation of the Fisher–Yates correlation coefficients with the optimal weights.     

Large Differences between Partial Correlation Coefficients

A plot of each ranking of N objects in N-dimensional space is shown to provide geometric interpretations of Kendall's tau and Spearman's rho and also of the relationship of rho to a sum of inversion weights. The computation of rho from a sum of inversion weights is shown to allow sequential calculation of rho.     

Dependence Estimation Over a Finite Bivariate Failure Time Region

This articleconcerns nonparametric estimation of association between bivariatefailure times. In the presence of independent right censoring,the support for failure time variates may be restricted and measuresof dependence over a finite failure time region may be of particularinterest. To this end, the reciprocal cross ratio function, weightedby the bivariate failure time density, is proposed as a summarymeasure of dependence over a failure time region. This `relativerisk' estimator is shown to be consistent and asymptoticallynormally distributed, with consistent bootstrap variance estimator.A finite-region version of Kendall's tau, which is suitable forcensored failure time data, is also proposed, and correspondingasymptotic distribution theory is noted. The accuracy of theseasymptotic approximations is studied in simulations and an illustrationis provided.     

Catanova for Two-Way Contingency Tables with Ordinal Variables Using Orthogonal Polynomials

ABSTRACT

The analysis of variance of cross-classified (categorical) data (CATANOVA) is a technique designed to identify the variation between treatments of interest to the researcher. There are well-established links between CATANOVA and the Goodman and Kruskal tau statistic as well as the Light and Margolin R ² for the purposes of the graphical identification of this variation.

The aim of this article is to present a partition of the numerator of the tau statistic, or equivalently, the BSS measure in the CATANOVA framework, into location, dispersion, and higher order components. Even if a CATANOVA identifies an overall lack of variation, by considering this partition and calculations derived from them, it is possible to identify hidden, but statistically significant, sources of variation.