Pareto distribution
Topic of A first look at extremes, part of Stochastic processes
Pareto distribution
| Type | Topic |
| Taught in | A first look at extremes |
| Part | Stochastic processes |
| Questions using it | 2 |
A law with power-law tail \mathrm{P}(X>x)=x^{-\alpha} for x\ge1, with \alpha>0. Its mean is finite only when \alpha>1 and its variance only when \alpha>2. The maximum of n draws, divided by n^{1/\alpha}, converges to the Fréchet law.
In A first look at extremes
Suppose \mathrm{P}(X>x)=x^{-\alpha} for x\ge 1 and some \alpha>0, a Pareto tail. Set a_n=n^{1/\alpha} and b_n=0. For x>0,
\mathrm{P}(M_n/a_n\le x)=\bigl(1-(n^{1/\alpha}x)^{-\alpha}\bigr)^n=\Bigl(1-\frac{x^{-\alpha}}{n}\Bigr)^n\to e^{-x^{-\alpha}}.
Used in
- Taught in A first look at extremes
- Questions that use it (2) — A heavier sample and its maximum · Bayesian German tank problem
- All 3 questions on A first look at extremes
See also
Variance · Exponential tails, and the Gumbel law · Power-law tails, and the Fréchet law · A finite endpoint, and a Weibull extreme-value law · Extreme-value classification