Compare two distributions based on various characteristics of their tails, including moments, ratios of moments, limiting tail behavior, hazard rate functions, and mean excess functions.

Two severities can share the same mean and still produce wildly different reinsurance prices. Tail comparison tools tell you which one will hurt.

Why tails matter. Premiums for excess layers, deductible savings, and reinsurance attachment all depend on $S(x)=P(X>x)$ far out in the right tail. Two distributions with identical means but different tails generate different layer prices.

Moment comparison. Compare $E[X^k]$ across distributions for increasing $k$. The distribution whose moments grow faster (or whose moments become infinite first) has the heavier tail.

• Exponential$(\theta )$: all moments finite, $E[X^k]={\theta }^{k}k!$.
• Pareto$(\alpha ,\theta )$: only moments with $k<\alpha$ are finite.
• Lognormal: all moments finite, but they grow as $\exp (k^2{\sigma }^2/2)$, faster than any polynomial.

KEY: "All moments finite" does not mean "thin tail." Lognormal has every moment finite yet is heavier-tailed than gamma.

Ratio of moments. A sharper diagnostic. Examine $E[X^{k+1}]/E[X^k]$ as $k$ grows. For light-tailed families this ratio grows slowly; for heavy-tailed families it explodes. Equivalently, look at $E[X^k{]}^{1/k}$.

Common mistakes

• Declaring a tail "light" because all moments are finite. Lognormal disproves this.
• Comparing only means. Means say nothing about $S(x)$ at $x=100,000$.
• Flipping the limit-ratio direction. $L=\infty$ means the numerator is heavier.

Bottom line

• Heavier tail = larger moments, faster-growing moment ratios, slower-decaying survival, decreasing or flatter hazard, increasing mean excess.
• Existence of moments: if $E[X^k]$ is infinite for some $k$, the tail is heavier than any distribution with all moments finite, yet lognormal has every moment finite and is still heavy-tailed.
• Limit ratio test: ${\lim }_{x\to \infty }{S}_A(x)/S_B(x)=\infty$ means $A$ is heavier; this asymptotic test overrules hazard or mean-excess disagreements at small $d$.
• Hazard rate $h(x)=f(x)/S(x)$, or $h(x)=-\frac{d}{dx}\ln S(x)$ when only $S$ is given; smaller or decreasing $h$ means heavier tail.

Exam shortcut

Polynomial-decay survival beats any exponential-decay survival. If one $S(x)$ is rational and the other has an $e^{-x}$ factor, the rational one is heavier. For mean-excess sorting: linear-increasing $e(d)$ is Pareto-like (heavy); flat is exponential; decreasing is Weibull $\tau >1$ or gamma $\alpha >1$ (thin). When stuck, take $\ln S(x)$ and compare the dominant terms as $x\to \infty$. The smaller magnitude is the heavier tail.

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Learning objectives

• 1d

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