Higher Dimensions

Abstract

In a single dimension, unit shifts to either left or right at are unequivocal with reference to the single axis. In two or more dimensions, elementary shifts can be defined with reference to each dimension separately, and then recombined with weights according to the desired direction in the plane. The elementary shifts themselves can be obtained with Radon-Nikodym shift factors as the logs of the respective conditional distribution functions. Left and right smoothed moments can then be defined. A related context is co-smoothing, where the conditional expectation of a variable is taken over a progressive range of a covariate. The most important application is to the ordered mean difference in finance, which indentifies whether a given security is defensive or aggressive relative to a given benchmark return.