Shift function vs. shift distribution

Let’s say we have two distributions \(X\) and \(Y\), and we want to express the “absolute difference” between them. This abstract term could be expressed in various ways. My favorite approach is to build the Doksum’s shift function. In order to do this, for each quantile \(p\), we should calculate \(Q_Y(p)-Q_X(p)\) where \(Q\) is the quantile function. However, some people prefer using the shift distribution \(Y-X\). While both approaches may provide similar results for narrow non-overlapping di...

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