Loss functions for specific probability distributions?

For a random variable $X$ with pdf $f(x)$ , the loss function* is defined as $$n(x) = \mathbb[(X-x)^+] = \int_^\infty (y-x)f(y)dy,$$ where $a^+ = \max\$ . Or, for a discrete distribution, $$n(x) = \mathbb[(X-x)^+] = \sum_^\infty (y-x)f(y).$$ Loss functions are used frequently in inventory theory and other fields. *This is different from the "loss function" used in machine learning. For some well known probability distributions, there are explicit forms for the loss function, typically using the pdf/pmf and cdf. For example, if $X$ has a standard normal distribution, then $$n(x) = \phi(x) - x(1-\Phi(x)),$$ where $\phi(\cdot)$ and $\Phi(\cdot)$ are the standard normal pdf and cdf. And if $X$ has a Poisson( $\lambda$ ) distribution, then $$n(x) = -(x-\lambda)(1-F(x)) + \lambda f(x).$$ These explicit forms are nice because they can be calculated without performing numerical integration or computing long sums, using pdf/pmf and cdf functions that are built into nearly every programming language and mathematical software package. I have seen explicit forms for loss functions for a handful of distributions, but typically somewhat scattershot in the appendix of an inventory-theory textbook (e.g., Zipkin 2000). I've never found them nicely collated anywhere. Do you know of a resource to find explicit-form loss functions for more probability distributions? Bonus points if the resource also has complementary loss functions ( $\mathbb[(X-x)^-]$ ) and second-order loss functions ( $\frac12\mathbb\left[\left([X-x]^+\right)^2\right]$ )!

• inventory
• loss-functions
• probability-distributions

$\begingroup$ I think this question might be interesting for meta, to discuss where the line between statistics and or should be $\endgroup$

$\begingroup$ Revisiting this, the second-order loss function is equivalent to calculating $\frac12\left(\Bbb V[(X-x)^+]+n(x)^2\right)$ but the closest literature I can find for the variance is through James, G. M. under squared error loss. $\endgroup$