Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, we define \begin{equation} D_{\alpha,\beta}(f \| g) = 1 - \alpha \Pr[X \leq b \mid Y>\beta ] - ( 1- \alpha) \Pr[X > b \mid Y\leq\beta ] \:, \tag{1}\end{equation} as a measure of distance of $f$ from $g$ at point $\beta$. Two questions:

1. Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?

2. Is there any relationship between (1) and the well-known distance (or divergence) measures?

*Definition of statistical divergence: https://en.wikipedia.org/wiki/Divergence_(statistics)

*Definition of statistical distance: https://en.wikipedia.org/wiki/Statistical_distance

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:

1. Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?

2. 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

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, we define \begin{equation} D_{\alpha,\beta}(f \| g) = 1 - \alpha \Pr[X \leq b \mid Y>\beta ] - ( 1- \alpha) \Pr[X > b \mid Y\leq\beta ] \:. \end{equation} Two questions:

1. Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?

2. Is there any relationship between this divergence measure and the well-known measures?

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, we define \begin{equation} D_{\alpha,\beta}(f \| g) = 1 - \alpha \Pr[X \leq b \mid Y>\beta ] - ( 1- \alpha) \Pr[X > b \mid Y\leq\beta ] \:, \tag{1}\end{equation} as a measure of distance of $f$ from $g$ at point $\beta$. Two questions:

1. Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?

2. Is there any relationship between (1) and the well-known distance (or divergence) measures?

*Definition of statistical divergence: https://en.wikipedia.org/wiki/Divergence_(statistics)

*Definition of statistical distance: https://en.wikipedia.org/wiki/Statistical_distance

Let $X$ and $Y$ be two continuous random variables with PDF $f(x)$ and $g(y)$. For a constant $b$, define $P_{e1} = \Pr[X \leq b \mid Y>b ]$, and $P_{e2} = \Pr[X > b \mid Y \leq b ]$. For any constant $0 \leq \alpha \leq 1$, we define $P(f,g) = 1 - \alpha P_{e1} - ( 1- \alpha) P_{e2}$, respectively. Consider common support for $X$ and $Y$. Two questions:

1. Is $P^{\alpha}(f,g,b)$ a distance function of two distributions $f$ and $g$ at point $b$, where $b$ is within the support of $X$ and $Y$.

2. Is there any relationship between this distance measure and the well-known measures?

Regarding the first question, $P^{\alpha}(f,g,b)$ may not satisfy "symmetry" condition of a distance; in this case, it is enough to say distance of $f$ from $g$. This is similar to the Kullback–Leibler divergence also does not satisfy the symmetry condition, but it is a well-known distance metric for many applications.

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, we define \begin{equation} D_{\alpha,\beta}(f \| g) = 1 - \alpha \Pr[X \leq b \mid Y>\beta ] - ( 1- \alpha) \Pr[X > b \mid Y\leq\beta ] \:. \end{equation} Two questions:

1. Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?

2. Is there any relationship between this divergence measure and the well-known measures?