Thicker-Tailed Comparison¶

Definition¶

For integrable losses of equal mean, $Y$ is directly thicker-tailed than $X$ if some threshold $t_0$ satisfies $F_Y(t)\ge F_X(t)$ for $t<t_0$ and $F_Y(t)\le F_X(t)$ for $t\ge t_0$ $t_{0}$ . This is the course's equal-mean comparison in which the two cumulative distribution functions (CDFs) cross once. At a jump, this convention assigns the crossing point to the upper interval. For continuous CDFs, the two inequalities force $F_Y(t_0)=F_X(t_0)$ $F_{X} (t_{0})$ ; if a CDF jumps at $t_0$ , only $F_Y(t_0)\le F_X(t_0)$ is required there, not equality.

Interpretation¶

Probability shifts towards both lower and higher loss values while keeping the mean fixed. This is a relative dispersion comparison; it does not assert a particular asymptotic heavy-tail class.

Step CDFs of X = 1 and of Y equal to 0 or 2 with equal probabilities, crossing once at t_0 = 1; the two shaded gaps on either side of the crossing both have area 1/2.

The figure draws the Examples case. Before $t_0=1$ the CDF of $Y$ lies above that of $X$ (mass moved down to 0); from $t_0$ up to 2 it lies below (mass moved up to 2), and from 2 on the two agree. The two shaded gaps have the same area $\tfrac{1}{2}$ , which reflects the equal means: for nonnegative losses $\mathbb{E}[X]-\mathbb{E}[Y]=\int_0^\infty(F_Y(t)-F_X(t))\,\mathrm{d}t$ $F_{X} (t)) d t$ , so the signed gaps must cancel.

Stochastic order compares CDFs in the same direction at every threshold. Here the CDF direction reverses once, so the relation compares equal-mean dispersion rather than uniformly larger loss size; stochastic order is not the definition.

Key Results¶

The direct single-crossing relation is not generally transitive. The notes use its transitive closure when they say indirectly thicker-tailed.
The direct comparison implies stop-loss order $X\le_{\mathrm{sl}}Y$ $Y$ .
For $0\le X\le b$ with mean $\mu$ , the constant $\mu$ is the least dispersed and the two-point law on $\{0,b\}$ with upper mass $\mu/b$ is maximally dispersed in convex order.
Unequal means do not meet the definition.

Examples¶

Let $X=1$ surely and let $Y$ be 0 or 2 with equal probabilities. Both means are 1. The CDF of $Y$ is above that of $X$ below 1 and below it between 1 and 2.

Prerequisites¶

Stochastic Order

Sources¶

Risk Insurance · Lecture 9

Aliases: single-crossing dispersion order, concentration and dispersion, thicker-tailed order, directly thicker-tailed Keywords: indirectly thicker-tailed, single crossing of CDFs, equal means, transitive closure, implies stop-loss order Symbols: infinity, integral, mu

Last updated: 17 October 2026 21:53