Pareto Distribution

Overview¶

How quickly does the chance of a large observation fall as we raise the threshold? The Pareto law models a positive quantity above a lower cutoff. Its tail exponent $\alpha$ controls that rate: smaller exponents give more probability to very large observations. Doubling any threshold in its support multiplies the survival probability by $2^{-\alpha}$.

Basic probability and integration are enough for this page. A survival probability means $\mathbb P(X>x)$. The formulas below define the CDF $F$, density $f$, and quantile $Q$; the survival table illustrates the exponent.

Definition and formula summary¶

Let $X$ be a positive random variable with lower cutoff $x_m>0$ and tail exponent $\alpha>0$. The Pareto Type I distribution is defined by

For $x\ge x_m$, the CDF, density, and quantile function are

The exact Pareto tail is regularly varying with index $-\alpha$. Its raw moment of order $p>0$ exists exactly when $p<\alpha$:

In particular,

and

Thus the mean exists only for $\alpha>1$, and a finite variance exists only for $\alpha>2$. Here $p$ is the moment order and $q$ is the quantile level.

Shape and tail intuition¶

Above its lower cutoff, the Pareto law has no characteristic upper scale. Once both thresholds lie in the Pareto tail, multiplying the threshold by a fixed factor changes the exceedance probability by a fixed power:

This is why the exponent $\alpha$ is the central knob. Smaller $\alpha$ means that threshold doublings are punished less severely, so rare observations remain large enough to dominate sums, moments, and empirical estimates.

The density also shows how the distribution piles mass near the lower cutoff while leaving a long right tail. Its values and scaling make this explicit.

At the cutoff, $f(x_m)=\alpha/x_m$. For $x\ge x_m$,

This ratio describes the decay of the density without a numerical plot.

Larger $\alpha$ concentrates more mass near $x_m$ and makes the density decay faster. Smaller $\alpha$ keeps more visible mass far from the cutoff.

Power-law survival¶

Survival probabilities are usually more revealing than densities for fat-tail work. For an exact Pareto law,

so a log-log survival plot is a straight line with slope $-\alpha$:

For $x_m=1$, the survival values are powers of the threshold:

These survival functions trace straight lines on log-log axes because each is an exact power. The heaviest tail has the shallowest line.

The same scaling appears numerically. For a fixed multiplier $t>0$, the ratio $\bar F(tx)/\bar F(x)$ depends only on $t$, provided both $x\ge x_m$ and $tx\ge x_m$. The identity follows by canceling $(x_m/x)^\alpha$. A Lean formalization is planned; this page supplies an ordinary algebraic argument.

For example, with $\alpha=1.16$, doubling any threshold in the support multiplies survival by $2^{-1.16}\approx0.4475$. This is a direct evaluation of the scaling identity, not an empirical estimate.

Moment thresholds¶

For a positive moment order $p$, the exact Pareto formula is

and the moment is infinite for $p\ge\alpha$. The Pareto moment theorem gives the direct integration proof and the truncated-moment formulas and table. Truncation makes each truncated population moment finite: raising the upper cutoff reveals whether it approaches a limit, grows logarithmically at $p=\alpha$, or grows as a power for $p>\alpha$.

For a general regularly varying tail, $\bar F(x)=x^{-\alpha}L(x)$, the boundary $p=\alpha$ depends on $L$. Karamata’s theorem explains this distinction; the exact Pareto boundary always diverges.

The moment-existence thresholds give a concrete example:

With $x_m=1$ and $\alpha=1.16$, the mean is $1.16/(1.16-1)=7.25$ and the second raw moment is infinite; the variance is therefore infinite.

For simulation, inverse transform sampling gives

When $x_m$ is known and the observations are assumed to be exact Pareto draws, the maximum-likelihood estimator is

This estimator is not a license to fit an exact Pareto from the sample minimum: empirical tail thresholds are rarely known and must be diagnosed separately.

A separate discussion of the two-sided double Pareto model is planned.

Caveats¶

• Empirical data rarely follows an exact Pareto law from its minimum value. A tail model needs a threshold choice, diagnostic plots, and sensitivity checks.

• A finite theoretical mean can still be practically hard to estimate when $\alpha$ is close to 1. The issue is pre-asymptotic behavior, not merely the formal existence of $\mathbb E[X]$.

• The exact failure mode depends on the moment order. At $p=\alpha$, the cutoff moment grows like $\log b$; for $p>\alpha$, it grows like $b^{p-\alpha}$.

• Do not estimate high moments of heavy-tailed samples without checking whether those moments are implied by the fitted tail exponent.

References¶

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

• Resnick, Heavy-Tail Phenomena 2007.

Backlinks¶

• Generalizes to: Regular Variation; moment consequences are covered by Karamata’s theorem.

• Used by: Double Pareto Distribution (planned), Pareto Moment Existence, Pre-Asymptotic LLN Behavior (planned), Max-to-Sum Ratio (planned), and Hill Estimator.

Source and adaptation¶

Adapted from incerto-wiki, content/concepts/distributions/pareto.md, revision 9717c9c (2026-09-13 import). Copyright (c) 2023 xshi19. Licensed under MIT. Links and notation were adapted for this site; executable figures and simulations were replaced with static calculations. No upstream execution or formal-proof verification is claimed for this adaptation.