Multivariate Ranks-Based Concordance Indexes Emanuela Raffinetti and Paolo Giudici Abstract The theoretical contributions to a “good” taxation have put the attention on the relations between the efficiency and the vertical equity without considering the “horizontal equity” notion: only recently, measures connected to equity (iniq- uity) of a taxation have been introduced in literature. The taxation problem is limited to the study of two quantitative characters: however the concordance problem can be extended in a more general context as we present in the following sections. In particular, the aim of this contribution consists in defining concordance indexes, as dependence measures, in a multivariate context. For this reason a k-variate .k > 2/ concordance index is provided recurring to statistical tools such as ranks-based approach and multiple linear regression function. All the theoretical topics involved are shown through a practical example. 1 An Introduction to Concordance Index Problem The issue of defining a concordance index often recurs in the statistical and econom- ical literature. Although the presentation is general we will refer, for sake of clarity, to the taxation example throughout: in particular, the concordance index is strictly connected to the “horizontal equity” topic according to which people who own the same income level have to be taxed for the same amount (see e.g. Musgrave 1959). The analysis is focused on considering n pairs of ordered real values, .x i ;y i /, i D 1;2;:::;n, whose components describe measures of two quantitative variables referred to each element of a statistical population: let us denote by X and Y the income amount before taxation and the income amount after taxation. Our interest is in defining the i -th individual rank with respect to variable X (denoted by r.x i /) E. Raffinetti ()  P. Giudici University of Pavia, Via Strada Nuova 65, Italy e-mail: emanuela.raffinetti@unipv.it; giudici@unipv.it A. Di Ciaccio et al. (eds.), Advanced Statistical Methods for the Analysis of Large Data-Sets, Studies in Theoretical and Applied Statistics, DOI 10.1007/978-3-642-21037-2 42, © Springer-Verlag Berlin Heidelberg 2012 465