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By Max Born, H. S. Green

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F. 5, considering different values of the shape parameter. 6 we show the quantile function zG˜ in the ˜ plane − ln − ln 1 − G z : thus plotted is the Gumbel (or reduced) variate for minima. 2 we show the domain of attraction of maxima and minima for some “parent” distributions frequently used in applications. 10 are quite advanced [83], and are beyond the scope of this book. 2. f. of the GEV law of maxima, in the canonical form (a′ = 0 and b′ = 1). The shape parameter takes on the values ′ = −2 −1 −1/2 0 1/2 1 2 will follow.

For n ≫ 1, z1−1/n ≈ ln n and z1−1/ ne ≈ ln n + 1. Consequently, the limit in Eq. 70) is e−z . Thus, the Gumbel distribution belongs to its own domain of attraction. Then, using Eq. 7. 2. Type II (Fréchet). Let us consider the Fréchet probability law as the “parent” distribution F . Then, if z > 0 and t ∈ 0 1 , F z = exp −z− and F −1 t = − ln t −1/ . Here F = + , and the limit in Eq. 71) is x− . Thus, the Fréchet distribution belongs to its own domain of attraction. Then, using Eq. 7. 25 univariate extreme value theory 3.

104) ′ Again, if Eq. 105) univariate extreme value theory 43 Thus, Eq. 105) shows that, provided the expectation (or the variance) exists, to estimate GEV it suffices to apply linear regressions to E Z (or V Z ) vs. r on a log-log plane for different durations ’s, and then calculate the slope of the fit. Alternatively, GEV can be estimated using Eq. 101), and exploiting the scaling ◭ of the parameters a′ and b′ . We now present an application to rainfall data [252]. 18 (Scaling properties of rainfall).

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A General Kinetic Theory of Liquids, Max Born and Green by Max Born, H. S. Green


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