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On the centennial anniversary of Gini’s theory of statistical relations

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Abstract

The formulation of a unitary methodology for the analysis of relationships between statistical characters stands as a landmark in the Gini contribution to theoretical statistics. On the occasion of the centennial anniversary of the appearance of his most original and important contributions to the subject, this paper aims at highlighting Gini’s proposals, by reformulating them in a language which suits the modern treatments of mathematical statistics. An effort is made to stress the precursory value of those proposals, and to mention in a precise way some of their most significant developments, in particular as far as metrics for spaces of probability distributions and studies about monotone dependence are concerned.

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Notes

  1. This is a translation of Gini’s original \(\ll \)Se non vi è connessione tra le modalità di due caratteri, non vi può essere tra queste né concordanza, né discordanza; essendovi invece connessione, potrà esservi concordanza rispetto a tutte le modalità, oppure discordanza rispetto a tutte le modalità [...] L’esame della concordanza [...] appare dunque come una ricerca subordinata all’esame della connessione\(\gg \).

  2. As explained in Chapter 20 of [33], it is the graph of the sub-differential of a convex function. This fact deserves attention with a view to extending Gini’s methodology to \(\mathbb {R}^d\) with \(d > 2\).

  3. The characterization of this property (indifference) would be worthy of a more careful analysis. In fact, one could propose a more restrictive definition by stating that the coordinates of the random point P, distributed according to \(\varphi \in \mathcal {F}(\varphi _1,\varphi _2)\), are mutually indifferent, with respect to g, if the random numbers \(\sigma _g(P, \underline{\Gamma })\) and \(\sigma _g(P, \overline{\Gamma })\) turn out to be identically distributed.

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Authors and Affiliations

  1. Università “L. Bocconi”, Milano, Italy

    Donato Michele Cifarelli

  2. Dipartimento di Matematica “F. Casorati”, Universit à degli Studi di Pavia, Pavia, Italy

    Eugenio Regazzini

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  1. Donato Michele Cifarelli
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  2. Eugenio Regazzini
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Correspondence to Eugenio Regazzini.

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Dedicated to the memory of Delia

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Cifarelli, D.M., Regazzini, E. On the centennial anniversary of Gini’s theory of statistical relations. METRON 75, 227–242 (2017). https://doi.org/10.1007/s40300-017-0108-0

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