Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any $0 \leq \alpha \leq 1$, and any constant $\beta$ within the support of $X$ and $Y$ such that $\Pr [Y>\beta ] >0$ and that $\Pr[Y\leq \beta ] > 0$, we define \begin{equation} D_{\alpha,\beta}(f \| g) = 1 - \alpha \Pr[X \leq \beta \mid Y>\beta ] - ( 1- \alpha) \Pr[X > \beta \mid Y\leq\beta ] \:, \tag{1}\end{equation} as a measure of distance of $f$ from $g$ at point $\beta$. Two questions:
Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?
Is there any relationship between (1) and the well-known distance (or divergence) measures?
Special case of $\alpha = \Pr[Y>\beta]$ is important. Moreover, for notational simplicity, I did not mention that $D_{\alpha,\beta}(f \| g)$ also depends on the joint distribution of $X$ and $Y$.
*Definition of statistical divergence: https://en.wikipedia.org/wiki/Divergence_(statistics)
*Definition of statistical distance: https://en.wikipedia.org/wiki/Statistical_distance