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For a random variable $X$ with pdf $f(x)$, the loss function* is defined as $$n(x) = \mathbb{E}[(X-x)^+] = \int_{x}^\infty (y-x)f(y)dy,$$ where $a^+ = \max\{a,0\}$. Or, for a discrete distribution, $$n(x) = \mathbb{E}[(X-x)^+] = \sum_{y=x}^\infty (y-x)f(y).$$ 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 ($\mathbb{E}[(X-x)^-]$) and second-order loss functions ($\frac12\mathbb{E}\left[\left([X-x]^+\right)^2\right]$)!

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    $\begingroup$ I think this question might be interesting for meta, to discuss where the line between statistics and or should be $\endgroup$ – Michael Feldmeier Jun 1 at 8:53
  • $\begingroup$ I've never heard about this kind of "Loss functions" in statistics ... $\endgroup$ – kjetil b halvorsen Jun 20 at 9:44
  • $\begingroup$ 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. $\endgroup$ – TheSimpliFire Jul 31 at 7:54
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There is indeed a paper titled Loss Distributions that provides the limited expected value functions $L(x)$ for several probability distributions (on page 15). It is directly related to the first-order loss function $n(x)$ through $$n(x)=\Bbb E(X)-L(x)\tag1$$ and notice that the loss function can also be written as $$n(x)=\int_x^\infty yf(y)\,dy-x(1-F(x))\tag2$$ after splitting the term $(y-x)$. The expressions for $L(x)$ and $\Bbb E(X)$ are tabulated below.

\begin{array}{c|c}\small\sf{Distribution}&L(x)&\Bbb E(X)\\\hline\small\sf{Log-Normal}&e^{\mu+\frac{\sigma^2}2}\Phi\left(\frac{\ln x-\mu-\sigma^2}{\sigma}\right)+x\left[1-\Phi\left(\frac{\ln x-\mu}{\sigma}\right)\right]&e^{\mu+\frac{\sigma^2}2}\\\hline\small\sf{Exponential}&\frac1\lambda\left(1-e^{-\lambda x}\right)&\frac1\lambda\\\hline\small\sf{Pareto}&\frac{\beta-\beta^\alpha(x+\beta)^{1-\alpha}}{\alpha-1}&\frac{\alpha\beta}{\alpha-1}\\\hline \small\sf{Burr}&\small\frac{\lambda^{1/\tau}\Gamma\left(\alpha-\frac1\tau\right)\Gamma\left(1+\frac1\tau\right)}{\Gamma(\alpha)}{\rm B}\left(1+\frac1\tau,\alpha-\frac1\tau;\frac{x^\tau}{\lambda+x^\tau}\right)+x\left(\frac\lambda{\lambda+x^\tau}\right)^\alpha&\frac{\lambda^{1/\tau}\Gamma\left(\alpha-\frac1\tau\right)\Gamma\left(1+\frac1\tau\right)}{\Gamma(\alpha)}\\\hline\small\sf{Weibull}&\frac{\Gamma\left(1+\frac1\tau\right)}{\beta^{1/\tau}}\Gamma\left(1+\frac1\tau,\beta x^\alpha\right)+xe^{-\beta x^\alpha}&\frac{\Gamma\left(1+\frac1\tau\right)}{\beta^{1/\tau}}\\\hline\small\sf{Gamma}&\frac\alpha\beta F(x,\alpha+1,\beta)+x(1-F(x,\alpha,\beta))&\frac\alpha\beta\end{array}

Notice how in the majority of cases, $\Bbb E(X)$ is the same as the starting coefficient of $L(x)$. Of course, $n(x)$ can be found using $(1)$.

In particular, for extensive details on the first-order loss function (and its complementary function) for the normal distribution, I highly recommend Piecewise linear approximations of the standard normal first order loss function.

For a more generic and heuristic approach on any type of distribution, there is a follow-up paper on Piecewise linearisation of the first order loss function for families of arbitrarily distributed random variables.


References

[1] Burnecki, K., Misiorek, A., Weron, R. (2010). Loss Distributions. MPRA Paper No. 22163. Available from: https://mpra.ub.uni-muenchen.de/22163/2/MPRA_paper_22163.pdf.

[2] Rossi, R., Tarim, S.A., Prestwich, S., Hnich, B. (2013). Piecewise linear approximations of the standard normal first order loss function. Available from: https://arxiv.org/pdf/1307.1708.pdf.

[3] Rossi, R., Hendrix, E.M.T. (2014). Piecewise linearisation of the first order loss function for families of arbitrarily distributed random variables. Proceedings of MAGO. pp. 1-4. Available from: https://gwr3n.github.io/chapters/Rossi_et_al_MAGO_2014_2.pdf.

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    $\begingroup$ Wow, this is fantastic. I use these functions all the time but had no idea about paper [1]. (I think I knew of [2] and [3].) Thanks also for that Herculean use of MathJax!! $\endgroup$ – LarrySnyder610 Jun 1 at 11:26
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    $\begingroup$ This is pure art $\endgroup$ – independentvariable Jun 1 at 13:06
  • $\begingroup$ @TheSimpliFire I’m just waiting a bit to see whether other answers get posted... :) $\endgroup$ – LarrySnyder610 Jun 1 at 13:09
  • $\begingroup$ Obv I upvoted. I’d upvote it twice if I could. :) $\endgroup$ – LarrySnyder610 Jun 1 at 13:12
  • $\begingroup$ @LarrySnyder610 Herculean is one allusion, Ceasarean is another. Reference. Reducing the font size (see here small?) and using 2 lines. $\endgroup$ – Rob Jun 14 at 1:00

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