We focus on a question that has been long addressed in economics, namely, of one distribution being better than another according to a normative criterion. Our criterion distinguishes between interdependence and behaviour in the margins. Many economics contexts concern interdependence only e.g. complementarities in production function, intergenerational mobility, social gradient in health. We compare bivariate discrete distributions and measure interdependence via a most general measure, namely, a copula (Schweizer and Wolff 1981). For discrete distributions we need to overcome a problem of many copulas associated with a given distribution. Drawing on a copula theory (Carley 2002, Genest andNeslehova 2007) we solve this problem, chose a method to compare copulas which together with first-order stochastic dominance of marginal distributions gives the ordering to compare distributions. We provide a type of Hardy-Littlewood-Pólya result (Hardy et al. 1934), that is, we give implementable characterizations of this ordering (Theorems 1 -3). As an application, we show how this ordering can be used to measure several phenomena that use either ordinal data (e.g. education-health gradient, bidimensional welfare) or simply discrete distributions (e.g. percentile income distributions of fathers and sons for intergenerational mobility). Welfare measures are easily decomposable into attributes and interdependence.3 Specifying the bounds of the treatment effects distribution is related to a copula theory (Frank et al. 1987). In particular, for discrete distributions of outcomes (e.g. life satisfaction and psychological indicators in assessing Moving to Opportunity (Ludwig et al. 2013)) these bounds can be improved using the results of Carley (2002) which we use a lot in the paper. 4 We use dependence/association/concordance interchangeably, although they are all different concepts (Nelsen 2006); in our setting, however, we do not need a detailed differentiation.
We consider a special situation of the Hess-Appelrot case of the Euler-Poisson system which describes the dynamics of a rigid body about a fixed point. One has an equilibrium point of saddle type with coinciding stable and unstable invariant 2-dimensional separatrices. We show rigorously that, after a suitable perturbation of the Hess-Appelrot case, the separatrix connection is split such that only finite number of 1-dimensional homoclinic trajectories remain and that such situation leads to a chaotic dynamics with positive entropy and to the non-existence of any additional first integral.
The dominant approach to evaluating distributional features of ordinal variables (e.g. selfreported health status) has been the Allison-Foster bipolarization ordering (henceforth AF). It has not yet been extended to a multidimensional setting. Here we fill this gap. A multidimensional extension of the AF relation is characterized by a sequence of median-preserving spreads on each dimension and association-changing switches. This extension does not pay attention to the dimensions' association. We then offer one that does and characterize it in terms of classes of polarization measures and welfare functions. Based on these two orderings we construct polarization indices and develop statistical inference for them. We measure bidimensional polarization in educational attainment and life satisfaction across OECD members. Dependence does not affect whether or not countries dominate each other bidimensionally.
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