DISPERSION OF CATEGORICAL VARIABLES AND PENALTY-FUNCTIONS - DERIVATION, ESTIMATION, AND COMPARABILITY

Measures of dispersion for categorical random variables based on penal ty functions play a central role in establishing relevant measures of association between such variables. The literature concerning these me asures provides little systematic treatment of such aspects of these m easures as comparability, efficient estimation, and large-sample prope rties. This article provides a systematic and rigorous construction of dispersion measures based on penalty functions. Efficient estimation procedures and asymptotic properties of estimates are examined. Condit ions from majorization theory that ensure a meaningful comparability o f dispersion measures based on penalty functions are discussed. A larg e class of familiar dispersion measures is then given a new interpreta tion using these conditions.