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We study the problem of transforming a multi-way contingency table into an equivalent table with uniform margins and same dependence structure. Such a problem relates to recent developments in copula modeling for discrete random vectors. Here, we focus on three-way binary tables and show that, even in such a simple case, the situation is quite different than for two-way tables. Many more constraints are needed to ensure a unique solution to the problem. Therefore, the uniqueness of the transformed table is subject to arbitrary choices of the practitioner. We illustrate the theory through some examples, and conclude with a discussion on the topic and future research directions.

In this paper, we extend the work of Pimentel et al. (2015) and propose an adjusted estimator of Kendall's $\tau$ for bivariate zero-inflated count data. We provide achievable lower and upper bounds of our proposed estimator and show its relationship with current literature. In addition, we also suggest an estimator of the achievable bounds, thereby helping practitioners interpret the results while working with real data. The performance of the proposed estimator for Kendall's $\tau$ is unbiased with smaller mean squared errors compared to the unadjusted estimator of Pimentel et al. (2015). Our results also show that the bound estimator can be used when knowledge of the marginal distributions is lacking.

Multivariate distributions are fundamental to modeling. Discrete copulas can be used to construct diverse multivariate joint distributions over random variables from estimated univariate marginals. The space of discrete copulas admits a representation as a convex polytope which can be exploited in entropy-copula methods relevant to hydrology and climatology. To allow for an extensive use of such methods in a wide range of applied fields, it is important to have a geometric representation of discrete copulas with desirable stochastic properties. In this paper, we show that the families of ultramodular discrete copulas and their generalization to convex discrete quasi-copulas admit representations as polytopes. We draw connections to the prominent Birkhoff polytope, alternating sign matrix polytope, and their most extensive generalizations in the discrete geometry literature. In doing so, we generalize some well-known results on these polytopes from both the statistics literature and the discrete geometry literature.

Optimum experimental design theory has recently been extended for parameter estimation in copula models. However, the choice of the correct dependence structure still requires wider analyses. In this work the issue of copula selection is treated by using discrimination design techniques. The new proposed approach consists in the use of $D_s$-optimality following an extension of corresponding equivalence theory. We also present some examples and highlight the strength of such a criterion as a way to discriminate between various classes of dependences.

Copula modelling has in the past decade become a standard tool in many areas of applied statistics. However, a largely neglected aspect concerns the design of related experiments. Particularly the issue of whether the estimation of copula parameters can be enhanced by optimizing experimental conditions and how robust all the parameter estimates for the model are with respect to the type of copula employed. In this paper an equivalence theorem for (bivariate) copula models is provided that allows formulation of efficient design algorithms and quick checks of whether designs are optimal or at least efficient. Some examples illustrate that in practical situations considerable gains in design efficiency can be achieved. A natural comparison between different copula models with respect to design efficiency is provided as well., Comment: IFAS research report 2014-66