13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- 5. Generalized extreme values probability Let us consider a random variable x with the distribution function ( ) F x X . The n extreme realizations in n samples of the random variable can be defined as: ( )n n X X X Y ,..., , max 1 2 = (1) The distribution function of nY is defined as: ( ) ( ) ( ) FyPYyPXyX y X y n n Y n ≤ ≡ ≤ = ≤ ≤ ,..., , 2 1 (2) According to the Fisher-Tippet theorem, if there exist two real normalizing sequences ( ) 1≥ n n a , ( ) 1≥ n n b and a non-degenerated distribution (not reduced to a point) G so that: ( ) ( ) x F a x b G x a Y b P n n n n n n n → →+∞ + = ≤ − (3) G is necessarily one of the three types of distributions: Fréchet, Weibull or Gumbel. Jenkinson [13] combined the three limit distributions in a single parametric form called Generalized Extreme Value (GEV) distribution depending on a single parameter ξ: ( ) ( ) ( ) ( ) = − − ≠ ∀ + > − + = − 0 exp exp 0 0, /1 exp 1 1 ξ ξ ξ ξ ξ ξ si x x x si x G x (4) The ξ parameter is called extreme index. Its sign indicates the type of asymptotic distribution: Weibull ( 0<ξ ), Gumbel ( 0=ξ ) or Fréchet ( 0>ξ ). The variable ( ) n n n Y b a − is called normalized maximum of the random variable x.The parameters na and nb are also called shape factors of the distribution. Figure 5. Probability density and cumulative probability determined using the maximum likelihood method from the extreme values of Crossland FIP for tensile loading ( 1 =− ∑R )
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