## ExP[ie xout p a1i e

[exp|- exp[- (xout - p)/a]j (e = 0), where 1 + e (xout - p) /a > 0, -to < p < to, a > 0 and -to < e < to. The parameters p and a identify location and scale, respectively, while the shape parameter, e, determines the tail behaviour of FGEV(xout).

The importance of the GEV distribution lies in the fact that it is the limiting distribution of the block maximum (for k large). Under mild conditions, nearly irrespective of what the common, but generally unknown distributional shape of the individual variables X(i) is, the distribution of Xout(j) approaches the GEV (Fig. 6.1). This is in essence the extreme value analogue of the central limit theorem (Coles 2001b). Figure 6.1. Distribution of the maximum of k independent standard normal variates. The plotted distribution functions, Fmax(x), are labelled with k. For k = 1, the symmetric form of the standard normal distribution, FN(x) (Eq. 3.49), appears. In general, Fmax(x) = [FN(x)]fc. Letting k increase has three effects: the location (average) is shifted to the right, the scale (standard deviation) is decreased and the right-skewness (shape parameter) is increased. With increasing k, Fmax(x) approaches Fgev(x). This is a theoretical example, with prescribed FN (x) and exactly determined Fmax(x). In a practical setting, with distribution and parameters of the independent variables unknown, Fmax(x) can still be approximated by Fgev(x).

Figure 6.1. Distribution of the maximum of k independent standard normal variates. The plotted distribution functions, Fmax(x), are labelled with k. For k = 1, the symmetric form of the standard normal distribution, FN(x) (Eq. 3.49), appears. In general, Fmax(x) = [FN(x)]fc. Letting k increase has three effects: the location (average) is shifted to the right, the scale (standard deviation) is decreased and the right-skewness (shape parameter) is increased. With increasing k, Fmax(x) approaches Fgev(x). This is a theoretical example, with prescribed FN (x) and exactly determined Fmax(x). In a practical setting, with distribution and parameters of the independent variables unknown, Fmax(x) can still be approximated by Fgev(x).

### 6.2.1.2 Maximum likelihood estimation

Assume that the approximation is perfect and the block maxima {xout(j)}j'L1 do come from a GEV distribution (Eq. 6.5). Assume further that e = 0. Adopting the maximum likelihood principle (Section 2.6, p. 58) requires then to maximize the (logarithm of the) likelihood function (Coles 2001b), ln [L(^ a, £)] = -m ln (a) - (1 + 1/£) £ ln [y(j)] - £ y(j), (6.6)