On Extending Stochastic Dominance Comparisons to Ordinal Variables and Generalising Hammond Dominance

Following the increasing use of discrete ordinal data for well-being analysis, this note builds on Hammond (H-) dominance concepts developed in Gravel et al. (2020) for discrete ordinal variables by observing and exploiting the fact that the coefficients associated with successive sums of cumulative distribution functions are Binomial coefficient functions of the order of dominance under consideration. Drawing first on notions of stochastic dominance relations for continuous variables to develop analogous concepts for discrete ordinal variables, it highlights the important limitation that increasing orders of dominance lead to loss of degrees of freedom which can be significant when the number of categories is low, as is common among ordered categorical variables, effectively bounding the maximum order of dominance. However, expanding on H- dominance by utilising the Binomial coefficients facilitates sequential consideration of higher orders of H- dominance without this loss, thereby surmounting the limitation.