## Extreme Value Distributions

The largest, or smallest, observation in a sample has one of three possible limiting distributions. This is another example of * convergence in distribution*.

The *average* of \(n\) samples taken from *any* distribution with finite mean and variance will have a *normal* distribution for large \(n\). This is the CLT.

The *largest* member of a sample of size \(n\) has a **LEV, Type I largest extreme value distribution**, also called **Gumbel** distribution, regardless of the parent population, * IF* the parent has an unbounded tail that decreases at least as fast as an exponential function, and has finite moments (as does the normal, for example).

The **LEV**, has pdf given by

\[f(x \lvert \theta_1, \theta_2) =\frac{1}{\theta_2} \exp\big(-z – \exp(-z) \big)\]

where \(z = (x-\theta_1)/\theta_2 \text{ and } \theta_1, \theta_2\) are location and scale* parameters, respectively and \(\theta_2 \gt 0\).

Similar sampling of the smallest member of a sample of size \(n\) produces an **SEV, Type I smallest extreme value distribution**, with density

\[f(x \lvert \theta_1, \theta_2) =\frac{1}{\theta_2} \exp\big(z-\exp(z) \big)\]

as \(n\) increases.

There are two other extreme value distributions. If not all moments exist for the initial distribution, the largest observation follows a **Type II** or **Frechet distribution**. If the parent density has a bounded tail, the smallest observation in a sample of size \(n\), has a **Type III**, or **Weibull distribution** of minima, as \(n\) increases. Examples are smallest samples taken from *lognormal*, *Gamma, Beta* or *Weibull* distributions.

The **Weibull distribution** is most easily described by its cdf:

\[F(x\vert \eta, \beta) = 1-\exp\Big(-(x/\eta)^{\beta}\Big)\]

where \(\eta\) is the scale (not location) parameter, and \(\beta\) is the shape parameter. **Weibull** is not a location, scale density*.

Notice that if \(x\) has a Weibull distribution, then \(\log_e(x) \) is SEV, so SEV is to Weibull, as normal is to lognormal. In other words, SEV is log(Weibull).

Type I and Type III limiting distributions are useful in describing physical phenomena where the outcome is determined by the behavior of the best, or worst, in the sample.

###### The Weibull Distribution Has Considerable Flexibility.

##### Notes:

* A probability density is a location, scale density if it can take the form of

\(Prob (X \le x) = F(x \lvert \theta_1, \theta_2) = \Phi\big((x-\theta_1)/\theta_2\big)\)

where \(\Phi\) is a proper density and does not depend on any unknown parameters. \(\theta_1\) is the location parameter and \(\theta_2\) is the scale parameter. The Normal, or Gaussian, is a special case with \(\theta_1 = \mu\) is the mean, and \(\theta_2 =\sigma\) is the standard deviation. Although location, scale densities are sometimes written using \(\mu, \sigma\) as generic parameters, these do not, in general, refer to the mean and standard deviation of a location, scale density. Note again that not all densities are location, scale. A proper density is one for which \(f(x) \gt 0\) and \(\int f(x)dx = 1\).