13.4 Estimation in EV Models

Different parametric estimation procedures are provided for estimating the shape, scale and location parameter of an extreme value distribution.


13.4.1 Linear Combination of Ratios of Spacings (LRS)


{gamma, mu, sigma} = 31584 lrseev (x)
applies the LRS estimator to the vector x and returns the estimated shape, location and scale parameter of an EV distribution

Let

$\displaystyle \widehat\gamma_i = {\log(\widehat r_i) \over -\log(c)}\,,
$

where

$\displaystyle \widehat r_i = {x_{[nq_2]:n} - x_{[nq_1]:n} \over x_{[nq_1]:n} - x_{[nq_0]:n}}
$

with

$\displaystyle q_0 = i/n,\quad q_1 = q_0^c, \quad q_2=(n-i)/n
$

and

$\displaystyle c = \left( \log((n-i)/n) /
\ \log(i/n)
\ \right)^{1/2}.
$

The previous construction becomes plausible by noting that $ x_{[nq]:n}$ is an estimator of the $ q$-quantile $ G^{-1}_{\gamma,\mu,\sigma}(q)$, and therefore the relation

$\displaystyle \widehat r_i \approx {G_\gamma^{-1}(q_2) - G_\gamma^{-1}(q_1) \over
\ G_\gamma^{-1}(q_1) - G_\gamma^{-1}(q_0)} =
\ \left({1 \over c}\right)^\gamma
$

holds. A natural estimator of $ \gamma$ is given by the linear combination

$\displaystyle \widehat\gamma = {1 \over [n/4]} \sum_{i=1}^{[n/4]} \widehat\gamma_i.
$

Estimates of the location and scale parameters are derived from a least square line $ x \to ax+b$ fitted to the QQ-plot

$\displaystyle \left(G_{\widehat\gamma}^{-1}\left({i\over n-1}\right),
\ x_{i:n}\right), \quad i=1,\dots,n.
$

One obtains $ \widehat\mu=b$ and $ \widehat\sigma=a$ as estimates.


13.4.2 ML Estimator in the EV Model


{gamma, mu, sigma} = 31646 mleev (x, k)
applies the ML estimator in the EV model to the vector x and returns the estimated shape, location and scale parameter of an EV distribution

The maximum likelihood estimator in the EV model is numerically evaluated by using an iteration procedure. The LRS estimator described in Subsection 13.4.1 serves as an initial value.

Note that the maximum likelihood estimator fails for $ \gamma \le -1$ because in that case no global maximum of the likelihood function exists. Yet it seems to be that a local maximum close to the initial value is attained.


13.4.3 ML Estimator in the Gumbel Model


{mu, sigma} = 31682 mleev0 (x)
applies the ML estimator in the Gumbel (EV0) model to the vector x and returns the estimated location and scale parameters

The ML estimator in the Gumbel (EV0) model must be evaluated numerically. A certain moment estimator is utilized as the initial value in the iteration procedure.