Multilateral Wellbeing Comparison in a Many Dimensioned World

Here, the various ways that the wellbeing of a collection of individuals has been construed is outlined and briefly discussed. Some have doubted that it can be measured at all; others have put forth the sorts of assumptions that are necessary in order for it to be measured. Refinements, particular aspects influencing aggregate wellbeing such as the extent of inequality, the degree of polarization and the notion of equality of opportunity are introduced, essentially providing a background for later chapters.

This chapter presents a hopefully concise summary of the nature and properties of, and relationships between, statistical concepts that are used in later analysis. It is not intended as a rigorous statistical discourse on such matters; for that the reader should consult the more detailed and thorough analyses in Linton (Probability, Statistics and Econometrics. London: Academic Press, 2017), McLachlan and Peel (Finite Mixture Models. New York: Wiley, 2000), Poirier (Intermediate Statistics and Econometrics: A Comparative Approach. Cambridge, MA: MIT Press, 1995), Rao (Linear Statistical Inference and Its Applications. New York: Wiley, 1973) and Whang (Econometric Analysis of Stochastic Dominance: Concepts, Methods, Tools, and Applications. Cambridge: Cambridge University Press, 2019, Silverman (Density Estimation for Statistics and Data Analysis, London Chapman and Hall 1986)). The properties of probability distributions, the instruments that describe the allocation of an individuals’ characteristic (e.g. their income) across a population are outlined. Particular attention is paid to Kernel estimation techniques, the way probability distribution functions are estimated when there is no information regarding their parametric structure. Finally, the notion of stochastic dominance, a comparison technique for comparing distributions is discussed.

Here, the various forms of indices used for summarizing aspects of distributions are discussed. Univariate and multivariate measures of location, dispersion and polarization are outlined as is the way they may be combined to reflect the collective wellbeing of individual groups in a society to provide a complete ordering of those groups. One of the problems with such an ordering is that it can be ambiguous; this is illustrated at the end of the chapter with an example comparing a collection of nations in the European Union.

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.

This chapter considers in detail the issue of ambiguity associated with a collection of indicators. By considering the circumstances under which an ordering would be unambiguous, it develops measures of the extent to which a collection of indicators could be ambiguous in a given situation. A methodology for dividing a collection of distributions into sets of “unambiguous subgroups” is also proposed. This chapter ends with an empirical example of ambiguity in ordering the income distributions of nations in the Eurozone.

The need to compare a collection of distributions multilaterally can arise in a variety of situations. Here in studies of distributional differences, the diversity of application in the literature is illustrated in univariate and multivariate paradigms. Five examples are reported: studies covering differences in income levels in Canadian Aboriginal–non-Aboriginal and gender based groupings; the effects of German education reforms on the educational attainments of students; investment portfolio performance comparisons; land allocation and crop returns in Sub Saharan Africa irrigation schemes; and a trivariate national income-education-health comparison.