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There are many families of bivariate distributions with given marginals. Most families, such as the Farlie–Gumbel–Morgenstern (FGM) and the Ali–Mikhail–Haq (AMH), are absolutely continuous, with an ordinary probability density. In contrast, there are few families with a singular part or a positive mass on a curve. We define a general condition useful to detect the singular part of a distribution. By continuous extension of the bivariate diagonal expansion, we define and study a wide family containing these singular distributions, obtain the probability density, and find the canonical correlations and functions. The set of canonical correlations is described by a continuous function rather than a countable sequence. An application to statistical inference is given. [ABSTRACT FROM AUTHOR]

Some statistical models, quite different in the symbolic mathematical sense, may provide similar results. After commenting two probability examples, we comment and compare multiple factor analysis (MFA) with related metric scaling (RMDS), two multivariate procedures dealing with mixed data. Each data set can be quantitative, binary, qualitative or nominal, and has been observed on the same individuals but coming from several sources. Then MFA and RMDS are two approaches for representing the individuals. We study the analogies and differences between both methodologies to guide users interested in performing multidimensional representations of mixed-type data. Though in general MFA and RMDS provide similar results, we prove that RMDS takes into account the association between the different sets of variables, providing, in some cases, better and more coherent representations. We also propose a parametric RMDS which includes MFA as a particular case. Article in memory of John C. Gower (1930-2019). [ABSTRACT FROM AUTHOR]

The Farlie-Gumbel-Morgensten (FGM) family of bivariate distributions with given marginals, is frequently used in theory and applications and has been generalized in many ways. With the help of two auxiliary distributions, we propose another generalization and study its properties. After defining the rank of a distribution as the cardinal of the set of canonical correlations, we prove that some well-known distributions have practically rank two. Consequently we introduce several extended FGM families of rank two and study how to approximate any bivariate distribution to a simpler one belonging to this family. [ABSTRACT FROM AUTHOR]

Hoeffding’s lemma provides a representation of the covariance of two random variables in terms of the difference between the joint and marginal distributions. This article proposes a multivariate generalization of the covariance between functions of bounded variation in the semialgebra of rectangles on. Some applications include the covariance inequality among functions where the variables are positive orthant dependent. [ABSTRACT FROM PUBLISHER]