Khare, Shree. 2024. “Order Statistic Exceedance Probability Sensitivities to Alternative Model Assumptions.” Variance 17 (1).
• Figure 1. A depiction of a timeline simulation for a given frequency $P_N(k)$ and severity $f_X(x)$.
• Figure 2. The upper left panel depicts the order $M=1$ order statistic exceedance probabilities, for $\mathbb{E}[N]=4$, with varying values of dispersion: $D=1$ (black), $D=4$ (red), $D=2$ (yellow), $D=0.5$ (magenta) and $D=0.25$ (blue). The upper right panel depicts the results for $M=2$, the lower left with $M=4$ and the lower right has $M=6$.
• Figure 3. Results analogous to Figure 2 except that $\mathbb{E}[N]=20$ and for orders $M=[1,10,20,30]$.
• Figure 4. Results analogous to Figure 2 except that $\mathbb{E}[N]=50$ and for orders $M=[1,25,50,75]$
• Figure 5. Results analogous to Figure 2 except that $\mathbb{E}[N]=10$ and for orders $M=[1,5,10,20]$.
• Figure 6. Results analogous to Figure 2 with results from the Generalized Poisson replaced with the CMP.
• Figure 7. The upper panel displays the exceedance probabilities associated with the order $M=1$ order statistic distributions $\left(1-F_{X_1}(l)\right)$ for a variety of dispersion values $D \in[0.1,10]$ and frequency distribution expectations $\mathbb{E}[N] \in[1,20]$, with fixed severity loss threshold such that $f=0.5=\int_0^l f_X(x) d x$. Results are displayed for over 29.5 million model configurations. A thin black horizontal line is provided for dispersion value 1 (the Poisson case). The bottom two panels displays the analogous results for the higher order order statistics $M=4$ and $M=8$. The above plots are what is referred to as the “Sensitivity Spaces” in the main text. Results are presented for the canonical frequency distribution. Note that the scale for the numerical values associated with the different heat map colours varies.
• Figure 8. Results in this figure are analogous to Figure 7, except that the chosen loss threshold corresponds to $f=0.8=\int_0^l f_X(x) d x$.
• Figure 9. Results in this figure are analogous to Figure 7, except that the chosen loss threshold corresponds to $f=0.9=\int_0^l f_X(x) d x$.
• Figure 10. Results in this figure are analogous to Figure 7, except that the chosen loss threshold corresponds to $f=0.99=\int_0^l f_X(x) d x$.

Abstract

Frequency and severity based models form the basis for risk quantification in reinsurance. We provide the mathematical formulation of the order statistics under random sample sizes drawn from a generic discrete frequency distribution, and the canonical distributions (Poisson; negative binomial; binomial). We show how our results can enable practitioners to understand the sensitivity of order statistic exceedance probabilities under varying model assumptions, yielding useful information about reinsurance pricing metrics. We also study the order statistics implied by two generalized frequency distributions (generalized Poisson; Conway-Maxwell-Poisson), pointing out some advantages over the commonly applied canonical distributions in the order statistics context.

Accepted: December 08, 2022 EDT