Comparing Lognormal, Weibull, and Pareto Tails

Why Compare Survival Tails?

For a nonnegative random variable \(X\), the survival function \(S(x)=P(X>x)\) gives the probability of exceeding a threshold \(x\). Comparing survival functions is useful when large values matter: for example, when describing extreme losses, lifetimes, or concentrations. Two models can fit the middle of a data set similarly yet assign very different probabilities to sufficiently large observations.

Here we compare three right-skewed families: the lognormal, Weibull, and Pareto Type I distributions. Their tail ranking has a simple asymptotic form, but the parameter values and the range of \(x\) matter in practice. A plot over observed values may therefore suggest a different ranking from the one that holds as \(x\) tends to infinity.

Survival Functions and Tail-Plot Coordinates

Use the following parameterizations, with all scale, shape, and tail-index parameters positive. A lognormal variable satisfies \(X=e^Y\), where \(Y\sim N(\mu,\sigma^2)\), with \(\sigma>0\). A Weibull variable has scale \(\lambda>0\) and shape \(k>0\). A Pareto Type I variable has minimum value \(x_m>0\) and tail index \(\alpha>0\).

The choice of plot changes which features are easiest to see. A plot of \(\log S(x)\) against \(x\) shows log-survival decay directly. On a log-log survival plot, the horizontal coordinate is \(\log x\) and the vertical coordinate is \(\log S(x)\). Pareto tails are straight lines on this plot, while the other two families are curved. A third useful plot for a Weibull variable is \(\log[-\log S(x)]\) against \(\log x\); its straight-line form is derived below.

Three Worked Comparisons

Asymptotic Tail Ordering

The key lognormal fact follows from the standard Mills bounds for a standard normal upper tail. For \(z>0\), \(\frac{z}{1+z^2}\phi(z)\leq\bar{\Phi}(z)\leq\frac{\phi(z)}{z}\), where \(\phi(z)=(2\pi)^{-1/2}e^{-z^2/2}\). Dividing both bounds by \(\phi(z)/z\) shows that \(\bar{\Phi}(z)/(\phi(z)/z)\) is between \(z^2/(1+z^2)\) and \(1\), so this ratio tends to \(1\) as \(z\to\infty\). Taking logarithms gives \(\log\bar{\Phi}(z)=-z^2/2-\log z-\tfrac12\log(2\pi)+o(1)\).

Proof. Put \(t=\ln x\). For all sufficiently large \(x\), \(z=(t-\mu)/\sigma>0\). Applying the normal-tail result above yields \(\log S_{\mathrm{LN}}(x)=-(t-\mu)^2/(2\sigma^2)-\log((t-\mu)/\sigma)-\tfrac12\log(2\pi)+o(1)\). In particular, its leading term is \(-t^2/(2\sigma^2)\), with the remaining terms of order at most \(t\) as \(t\to\infty\).

For \(x\geq x_m\), \(\log S_{\mathrm{P}}(x)=\alpha\log x_m-\alpha t\). Hence \(\log(S_{\mathrm{P}}/S_{\mathrm{LN}})=t^2/(2\sigma^2)+O(t)\to\infty\), which proves \(S_{\mathrm{P}}/S_{\mathrm{LN}}\to\infty\).

Also, \(\log S_{\mathrm{W}}(x)=-(x/\lambda)^k\), so \(\log(S_{\mathrm{LN}}/S_{\mathrm{W}})=(x/\lambda)^k-t^2/(2\sigma^2)+O(t)\to\infty\). The limit holds because \(x^k=e^{kt}\) grows faster than any quadratic polynomial in \(t\) for \(k>0\). Thus \(S_{\mathrm{LN}}/S_{\mathrm{W}}\to\infty\), proving the stated ordering. \(\square\)

The same reasoning also gives a direct Pareto–Weibull comparison: \(\log(S_{\mathrm{P}}/S_{\mathrm{W}})=\alpha\log x_m-\alpha\log x+(x/\lambda)^k\to\infty\). Thus the Pareto survival eventually exceeds the Weibull survival as well. Importantly, the ranking requires only that the parameters be in their valid ranges; their particular positive values do not change the asymptotic result.

What the Tail Plots Show

The log-log survival plot makes the Pareto form especially transparent. With horizontal coordinate \(t=\log x\), its equation is \(\log S_{\mathrm{P}}(x)=\alpha\log x_m-\alpha t\), an exact straight line of slope \(-\alpha\). A larger Pareto tail index \(\alpha\) means a steeper downward line and faster decay within the Pareto family.

For a lognormal variable, the Mills-bound calculation gives, as \(t=\log x\to\infty\), \(\log S_{\mathrm{LN}}(x)=-(t-\mu)^2/(2\sigma^2)-\log((t-\mu)/\sigma)-\tfrac12\log(2\pi)+o(1)\). Its leading term is quadratic in \(\log x\), rather than linear. This is why the lognormal survival eventually falls below every fixed Pareto survival, despite its slower-than-exponential decay on the original \(x\) scale.

For a Weibull variable, \(\log[-\log S_{\mathrm{W}}(x)]=\log((x/\lambda)^k)=k\log x-k\log\lambda\). This is an exact straight line in the Weibull plot, with slope \(k\) and intercept \(-k\log\lambda\). On the ordinary log-log survival plot, however, its log-survival is \(-(x/\lambda)^k\), which eventually falls much faster than the lognormal log-survival.

Common Pitfall: “Heavy-Tailed” Does Not Mean Heaviest

A frequent mistake is to reason that the lognormal must have a heavier tail than the Pareto because it is often called heavy-tailed and decays more slowly than an exponential. That comparison uses the wrong reference family: slower than exponential does not mean slower than every power law. In fact, for fixed \(\alpha>0\), the lognormal survival eventually falls below \(x^{-\alpha}\), as the quadratic term in \(\log x\) eventually dominates the linear term.

Another mistake is to declare the family with the largest survival at one measured threshold the asymptotically heaviest. Parameters can move crossover points far from the data range, and a finite sample cannot establish an infinite-threshold ordering by itself. For the parameter ranges considered here, the eventual ranking is always Pareto, then lognormal, then Weibull, even though finite-range values can appear in a different order.