I have a "real life" problem.
I am modelling a form of Operational risk at a Company. Each department of the company is asked for how a particular event will impact their department, in terms of cost.
Each department provides a "probability that the event occurs" (say, $p$). They then give an opinion of the severity of the event (amount of loss), given that the event has occurred, at different percentiles (the percentiles being $0, 0.5, 0.75, 0.9, 1$).
For this particular department, the frequency is modelled as $ N \sim Bin(1,p)$. The severity, $X$, is modelled as a piecewise distribution; the CDF increases linearly between the given loss amounts.
I.e. the severity can be modelled as a mixture of continuous uniform distributions:
$F(x) = 0.5 * U(a,b) + 0.25*U(b,c) + 0.15*U(c,d) + 0.1*U(d,e)$
Where $U(a,b)$ is the CDF of a continuous uniform distirubtion with parameters $a, b$. And the values $a, b, c, d, e$ are the severity amounts that the department has inputted.
This then implies an aggregate distribution $S_{1}$ for this department. For which the mean and variance can be calculated.
$\underline{\text{First Question}}$: What is the CDF of the aggregate distribution $S_{1}$?
Let's say that another department offers their view of how this event will affect them. They give a new "probability that the event occurs" (say, $q$).
They will also provide severities at the given percentiles. They will then have their own aggregate distribution $S_{2}$.
The way our modelling software works, we can only take one "probability of the event occuring". We take the higher of $q$ and $p$. Let's say this is $q$.
$\underline{\text{Second Question}}$: Is there a way to "edit" the severity distribution from the first department so that, given their probability is now $q$, the new, implied aggregate distribution is "as close as possible" to $S_{1}$?
Perhaps we can find the new values $a,b,c,d,e$ so that the 1st to 5th moments of $S_{1}$ are the same as the new aggregate distribution? I'm not sure how to do this - I can get as far as mean and variance.
I appreciate the second question is a little vague - I can clarify any point. I have also done some pre-lim calculations in Excel which I can provide (how do I attach this?). Thanks very much for your thoughts.
The aggregate distribution is a zero-inflated distribution proportional to the severity distribution, with the probability of no loss being equal to $1-p$; thus it has a mixed discrete-continuous density. For the sake of simplicity, suppose the severity distribution is $\operatorname{Exponential}(\mu)$ and the frequency distribution is $\operatorname{Bernoulli}(p)$; then the aggregate distribution is $$\Pr[S = 0] = 1-p, \quad f_S(s) = \frac{p}{\mu}e^{-s/\mu}, \quad s > 0;$$ alternatively we can characterize such a distribution by the CDF $$F_S(s) = 1 - pe^{-s/\mu}, \quad s \ge 0.$$ The same principle applies for other severity distributions.
This would also tell you that the answer to your second question is no, because if $q > p$, no choice of the parameters of $X$ will change $\Pr[S = 0]$. If your belief that the true frequency probability is $\max(p_1, p_2, \ldots p_n)$ for all departments where the $i^{\rm th}$ department's estimate is $p_i$, then the attempt to modify the severity distribution to somehow "preserve" the aggregate is misguided, since that distribution is, by your construction, the probability of observing a loss of a given size, for a range of sizes. Simply put, I don't see why you'd need to modify this.