A finite endpoint, and a Weibull extreme-value law
Section of A first look at extremes, part of Stochastic processes
A finite endpoint, and a Weibull extreme-value law
| Type | Section |
| Taught in | A first look at extremes |
| Part | Stochastic processes |
Let X_i be uniform on (0,1), so the upper endpoint is 1. Set Y_n=n(1-M_n). For y\ge 0 and n>y,
\mathrm{P}(Y_n>y)=\mathrm{P}(M_n<1-y/n)=\Bigl(1-\frac{y}{n}\Bigr)^n\to e^{-y}.
Equivalently, n(M_n-1) converges in distribution to a reversed exponential law supported on (-\infty,0]. This is an example of the Weibull extreme-value type. The name is the extreme-value type, not the Weibull distribution used there as a lifetime model. A finite endpoint changes both the centring, which approaches the endpoint, and the support of the limit, which lies on one side of 0.
The three limits correspond to three upper-tail structures: a finite endpoint, a light unbounded tail, and a heavy unbounded tail.
Used in
See also
Weibull distribution · Exponential tails, and the Gumbel law · Power-law tails, and the Fréchet law · Extreme-value classification · Pareto distribution