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2019 | OriginalPaper | Buchkapitel

4. Partial Orderings

verfasst von : Gordon Anderson

Erschienen in: Multilateral Wellbeing Comparison in a Many Dimensioned World

Verlag: Springer International Publishing

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Abstract

This chapter outlines the details of the “partial ordering” process that uses ideas from the stochastic dominance (SD) literature applied to probability distributions and Lorenz curves. It relates particular types of dominance ordering to particular classes of the criterion function that are used in the practice of wellbeing measurement. When such a process is successful, it yields an unambiguous ordering in the sense that all criterion functions within the relevant class agree on the ordering. This chapter also develops a stochastic dominance based Utopia-Dystopia family of indicators for ranking groups together with Gini-like indices for examining distributional differences between groups.

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Vorheriges Kapitel Complete Orderings: Index Types and the Ambiguity Problem

Nächstes Kapitel Comparing Latent Subgroups

The Schutz coefficient is the maximal vertical distance between L(p) and 45° line.

Note here the semantic similarity with the Paretian judgment that “no one should be worse of and at least someone should be better off”, but beware this is not at all the same thing given the anonymity axiom.

See also Atkinson and Bourguingnon (1987), Foster and Shorrocks (1988), Davies and Hoy (1994, 1995), Lambert and Ok (1999).

Indeed, Atkinson (1970) expressed some scepticism regarding its use given the frequency with which he observed a “no decision”.

A lemma, due to Davidson and Duclos (2000), provides a rationale for this practice. It reveals that, if a distribution dominates over a restricted lower range at a given order of dominance, it will dominate over the whole range at some higher order.

Suppose 1 intersection point at z where 0 < z < ∞ where \( {F}_a(x)\ge {F}_b(x)\kern0.33em for\kern0.33em x<z \)\( and\kern0.33em {F}_a(x)\kern1em <{F}_b(x)\kern0.33em for\kern0.33em x>z, \) then \( {\int}_0^M\left|{F}_i(x)-{F}_j(x)\right| dx={\int}_0^M\left({F}_a(x)-{F}_b(x)\right) dx-{\int}_a^M\left({F}_a(x)-{F}_b(x)\right) dx \)\( ={\int}_0^M\left({F}_a(x)-{F}_b(x)\right) dx-2{\int}_a^M\left({F}_i(x)-{F}_j(x)\right) dx>{\int}_0^M\left({F}_i(x)-{F}_j(x)\right) dx \) since \( 2{\int}_a^M\left({F}_i(x)-{F}_j(x)\right) dx \) is negative. When dominance prevails there will be no intersection point, \( {F}_a(x)\ge {F}_b(x) \) for all x and \( \mathrm{TR}=0.5{\int}_a^M\left({F}_a(x)-{F}_b(x)\right) dx \).

Here we think of U(x) as corresponding to some index that is a criterion of x that reflects individual wellbeing, the class of U(x) being entertained is referred to as the preference space and imposing restrictions on U can be interpreted as reducing the preference space.

These particular dominance ideas are used in the finance literature in the study of risk loving behavior.

For a multi distributional higher order extension of the Gini (1916) transvariation measure, see Anderson, Linton and Thomas (2017).

It is interesting to note that the Gini coefficient is subgroup decomposable under perfect segmentation of subgroups Mookherjee and Shorrocks (1982).

The Gini coefficient is due to Gini (1912), the Absolute Gini to Hey and Lambert (1980).

OV(i,j) = \( {\int}_0^{\infty}\min \left({f}_i(x),\kern0.5em {f}_j(x)\right) dx \) (Anderson, Linton and Whang 2012).

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Titel: Partial Orderings
verfasst von: Gordon Anderson
Verlag: Springer International Publishing
Buch: Multilateral Wellbeing Comparison in a Many Dimensioned World
Print ISBN: 978-3-030-21129-5

Electronic ISBN: 978-3-030-21130-1

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-21130-1_4