Crossing Criteria for Risk Orders¶

Proposition¶

For absolutely continuous laws, suppose $f_X(t)-f_Y(t)$ is nonnegative before a threshold and nonpositive afterwards, with both densities integrating to one. Then $F_X(t)\ge F_Y(t)$ for all $t$ , so $X\le_{\mathrm{st}}Y$ $Y$ . The crossing criterion is sufficient; it is not a necessary requirement for stochastic order.

Interpretation¶

Integrating the signed density difference accumulates a nonnegative cumulative distribution function (CDF) difference before the crossing and then brings it back to zero. The direction of the signs matters more than merely counting intersections.

Left: densities f_X(t) = 2(1-t) and f_Y(t) = 2t crossing at t = 1/2, with the sign of f_X - f_Y marked; right: the CDF 2t - t^2 lying above t^2 on (0,1).

The figure draws the Examples case. On the left, $f_X-f_Y$ is positive before the crossing at $1/2$ (blue region) and negative after it (red region). On the right, the CDF difference $F_X(t)-F_Y(t)=2t-2t^2$ accumulates these signed areas: it grows until $t=1/2$ and then shrinks back to zero at $t=1$ without becoming negative, so $F_X\ge F_Y$ everywhere and $X\le_{\mathrm{st}}Y$ $Y$ .

Key Results¶

With equal finite means, the pattern $f_Y-f_X\ge0$ $0$ , then $\le0$ $0$ , then $\ge0$ on three consecutive intervals gives the direct thicker-tailed comparison.
Multiple density crossings do not by themselves decide stochastic order.
Density comparisons are almost-everywhere comparisons; changing a density at one point changes no probabilities.
For discrete laws, use cumulative masses rather than derivatives.

Examples¶

On $(0,1)$ , $f_X(t)=2(1-t)$ and $f_Y(t)=2t$ cross at $1/2$ . Their CDFs are $2t-t^2$ and $t^2$ , confirming $F_X(t)\ge F_Y(t)$ $F_{Y} (t)$ .

Prerequisites¶

Stop-Loss Order

Sources¶

Risk Insurance · Lecture 9

Aliases: single density crossing criterion, two-crossing dispersion criterion, density crossing criterion, single-crossing criterion Keywords: sign change of f_X - f_Y, sufficient condition for stochastic order, three-interval sign pattern, densities crossing once

Last updated: 16 October 2026 22:28