Survival Tail Ratio

Statement¶

For a right-tail survival function $\bar F(x)=\mathbb P(X>x)$, the survival tail-ratio diagnostic is

where $\bar F(2x)>0$. With iid copies $X_1$ and $X_2$, the numerator is

Thus $Q_F(x)$ compares a two-coordinate moderate-extreme event with a one-coordinate double-threshold event. It is a tail-geometry diagnostic, not an estimator by itself.

Here $x>0$ is the comparison threshold, $F$ is the CDF, and $\alpha>0$, $x_m>0$ are the Pareto exponent and lower cutoff. $Q_F$ is distinct from the regular-variation ratio $\bar F(tx)/\bar F(x)$.

Calibration cases¶

The diagnostic separates three useful geometries:

The Pareto calculation is exact once $x\ge x_m$. The normal calculation uses Mills’ ratio, $\bar\Phi(x)\sim e^{-x^2/2}/(\sqrt{2\pi}x)$, for the standard Gaussian tail. This also follows from the complementary-error function expansion in NIST DLMF, Section 7.12.

The regular variation connection is simple. If $\bar F\in RV_{-\alpha}$ with $\alpha>0$, then

Therefore

So regularly varying tails fall on the one-big-jump side of this diagnostic. That agrees with subexponentiality, though $Q_F(x)\to0$ is only a diagnostic ratio and not the full convolution-tail definition.

Static calibration¶

For an exact Pareto law with $x_m=1$ and $\alpha=1.5$, $Q_F(x)=(2/x)^{3/2}$ for $x\ge1$. The exponential comparison is exactly one at every positive threshold.

A heavy-tailed model can therefore have $Q_F(x)\ge1$ at finite thresholds; the asymptotic direction and the range inspected both matter. For the normal, substitution of Mills’ ratio gives

Student-t tails with positive degrees of freedom (including the Cauchy case) are regularly varying, so the preceding regular-variation argument gives $Q_F(x)\to0$. These are population calculations, not empirical fits.

What a borderline ratio does and does not establish¶

SCoFT v4 §§5.0.2–5.0.3, pp. 93–95 Taleb, 2025, motivates this event comparison with exponential and sigmoid survivals. We derive their values here and keep them separate from a convolution-tail criterion.

For centered logistic survival $\bar F(x)=1/(1+e^{2kx})$, $k>0$,

This law is light-tailed. Its density is bounded by a constant times $e^{-2k|x|}$, so its MGF is finite for $|\theta|<2k$ by comparison with an exponential integral on each half-line. The density is $\tfrac{k}{2}\operatorname{sech}^2(kx)$, the hypsecant2 helper’s law with loc=0, scale=1/k. The name identifies this squared-secant convention, not the ordinary hyperbolic-secant distribution.

For a centered Laplace law with scale $b>0$, the right survival is $\tfrac12e^{-x/b}$ on $x\ge0$, giving $Q_F(x)=1/2$. More generally, $\bar F(x)\sim c e^{-\lambda x}$ implies $Q_F(x)\to c$. Translating $X$ by $d$ changes that constant to $ce^{\lambda d}$. Thus the value one is not an invariant boundary between light and heavy tails. It reflects the tail prefactor as well as its decay.

For Weibull survival $e^{-x^\beta}$ on $x\ge0$,

It tends to $-\infty$, zero, or $+\infty$ as $\beta<1$, $\beta=1$, or $\beta>1$. The catalog supplies the separate subexponential classification for the first range. A vanishing $Q_F$ is not a proof of the full convolution condition.

Use pareto_type1, iid_two_sum_tail.

import numpy as np
from xmath.incerto.distributions import pareto_type1
from xmath.incerto.tail_diagnostics import iid_two_sum_tail

law = pareto_type1(1.5)
u = np.array([3., 10., 100.])
ratio = np.exp(2 * law.logsf(u) - law.logsf(2 * u))
print("Squared-survival ratio: [" + ", ".join(f"{v:.4g}" for v in ratio) + "]")
result = iid_two_sum_tail(law, u)
assert result["success"].all()
print("Two-sum ratio: [" + ", ".join(f"{v:.4g}" for v in result["ratio"]) + "]")

Output

Squared-survival ratio: [0.5443, 0.08944, 0.002828]
Two-sum ratio: [1.769, 1.391, 1.044]

The convolution ratio uses independent summands.

Figure regeneration: uv run python demos/incerto/power_laws.py. Log survival values prevent division of underflowed probabilities. The convolution helper uses one-dimensional adaptive quadrature; the demo stops on a failed error check. No random sampling or empirical fitting is involved.

Caveats¶

• $Q_F(x)$ is a distribution-level diagnostic. A finite empirical estimate can be dominated by sampling noise because both the numerator and denominator involve rare events.

• The iid interpretation of $\bar F(x)^2$ fails under dependence. Clustered extremes need their own dependence model before this ratio can be read as a joint-event probability.

• The ratio uses right tails. Two-sided returns, losses, and absolute values require an explicit transformation before the diagnostic is applied.

• $Q_F(x)\to0$ is compatible with one-big-jump geometry, but it is not a replacement for the convolution-tail definition of Subexponentiality.

References¶

• Taleb (2025), §§5.0.2–5.0.3, pp. 93–95: event-ratio context. The limiting prefactor and translation calculations are proved here.

• Feller, An Introduction to Probability Theory and Its Applications, Vol. II (2nd ed., Wiley, 1971).

• Embrechts, Klueppelberg, and Mikosch, Modelling Extremal Events 1997.

Backlinks¶

• Depends on: Regular Variation and Subexponentiality.

• Related catalog: Tail Class Catalog.

• Related diagnostic: Max-to-Sum Ratio.

• Used by: Iso-Density Tail Geometry.