Outline Outline Gumbel Gumbel Asymptotic Distributions - PowerPoint PPT Presentation

Outline Outline � Gumbel � Gumbel Asymptotic Distributions Asymptotic Distributions � Type 1: Largest Type 1: Largest Gumbel Gumbel Distribution Distribution � � Type 1: Smallest � Type 1: Smallest Gumbel Gumbel Distribution Distribution � Type 2: Largest Type 2: Largest Weibull Weibull Distribution Distribution � � � Type 3: Smallest Distribution Type 3: Smallest Distribution � Simulation of Random Variable With Simulation of Random Variable With � Known Distribution Known Distribution G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics Type 1: (Largest) Type 1: (Largest) Gumbel Gumbel Distribution Distribution Let X be the largest of n independent Let X be the largest of n independent random variables Y 1 , Y 2 , …, Y Y n . Then random variables Y 1 , Y 2 , …, n . Then Distribution Distribution Density Density { } { } ( ) ( ) ( ) ( ) ( ) ( ) { } { } { } { } ( ) ( ) ( ) = − − α − = α exp − α − − − α − = ≤ = ≤ ≤ ≤ = x u x u F x exp e f x x u e F x P Y x P Y x P Y x ... P Y x F x F x ... F x 1 2 3 X i 1 2 n Y Y Y X X 1 2 n = i 1 , 2 ,..., n f X (x) − ∞ < < +∞ x 1 If Y i If Y i are identically distributed: are identically distributed: u=0.275 0 . 5772 µ = u + α =2.566 α X ( ) ( ( ) ) ( ) ( ( ) ) ( ) − = = n n 1 F x F x f x n F x f x π X Y X Y Y 2 σ = 2 X 6 α 2 -0.5 0.0 0.5 1.0 1.5 x G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 1

Type 1: (Smallest) Type 1: (Smallest) Gumbel Gumbel Distribution Distribution Type 2: (Largest) Type 2: (Largest) Weibull Weibull Distribution Distribution Distribution Distribution Density Density Distribution Distribution Density Density { } { } ( ) ( ) ( ) ( ) ( ) = − − α − = α exp α − − α − y u z u k + k ⎛ ⎞ ⎛ ⎞ F z 1 exp e f z z u e u k 1 u ⎛ ⎞ − ⎜ ⎟ − ⎜ ⎟ ( ) ( ) k u ≥ Z Z = = x 0 ⎝ ⎠ ⎜ ⎟ ⎝ ⎠ x x F x e f x e − ∞ < < +∞ X X ⎝ ⎠ z u x f Z (z) 0 . 5772 µ = u − 1 α f X (x) Z u=0.275 ⎛ − ⎞ 1 α =2.566 µ = Γ ⎜ ⎟ u 1 > k 1 X ⎝ ⎠ k π 2 σ = 2 6 α Z 2 ⎡ ⎤ ⎛ − ⎞ ⎛ − ⎞ 2 1 σ = Γ − Γ 2 2 ⎜ ⎟ 2 ⎜ ⎟ > ⎢ ⎥ u 1 1 k 2 X ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ k k z 2 4 6 x -0.5 0.0 0.5 1.0 1.5 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics Type 3: (Smallest) Extremal Type 3: (Smallest) Extremal Distribution Distribution Simulation of Random Variables with a Simulation of Random Variables with a Known Distribution Known Distribution F F Y Y (y (y) ) Distribution Density Distribution Density Given that U is Given that U is ≤ ≤ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ − 1 0 u 1 − k − k 1 − k ( ) ⎪ ⎛ ⎞ ⎪ ⎛ ⎞ ⎪ ⎛ ⎞ ⎪ l l l uniform random uniform random ( ) z ( ) k z z = ⎨ ⎬ = − − = ⎜ ⎟ − ⎜ ⎟ f U u ⎨ ⎜ ⎟ ⎬ ⎨ ⎬ exp F z 1 exp f z Z − Z − − − ⎩ 0 otherwise ⎭ ⎪ ⎝ ⎠ ⎪ l ⎝ l ⎠ ⎪ ⎝ l ⎠ ⎪ variable with l u u u variable with ⎩ u ⎭ ⎩ ⎭ > ≥ l l z z ( ) = − Then Then 1 Y F U Y ⎛ + ⎞ ⎡ ⎛ + ⎞ ⎛ + ⎞ ⎤ ( ) 1 ( ) 2 1 µ = + − Γ ⎜ ⎟ σ = − Γ − Γ l l 2 l 2 ⎜ ⎟ 2 ⎜ ⎟ u 1 u ⎢ 1 1 ⎥ Z ⎝ ⎠ Z ⎝ ⎠ ⎝ ⎠ k ⎣ ⎦ k k has the desired distribution function F F Y (y). ). has the desired distribution function Y (y G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 2

{ } [ ] ( ) ( ) ( ) Exponential Gumbel ( ) Exponential Gumbel ln − = λ − λ = − − α − y y u ln U f y e u y F y exp e = − Y u Y Y α ( ) ( ) ( ) ⎡ ⎤ − λ k = 1 − ⎛ ⎞ y ( ) ( ) F y e u y u = exp ⎢ − ⎜ ⎟ ⎥ F y u y Y ⎜ ⎟ u = Y ⎢ ⎥ ⎝ y ⎠ Y ⎣ ⎦ ( ) k 1 − ( ) ln U − ln 1 U ln U or or = − = − Y Y λ λ ⎛ ⎞ ⎛ ⎞ k ( ) y ( ) ⎜ ⎟ = − − ⎜ ⎟ [ ( ) ] k 1 F y 1 exp u y = − ln − ⎜ ⎟ Y u 1 U Y ⎝ ⎠ ⎝ u ⎠ Weibull ( ) ( ) Weibull β = αβ β − − α 1 y f y y e u y Gaussian Y Gaussian = − π 1 Y 2 ln U cos 2 U ⎛− ⎞ β 1 1 1 2 = ⎜ ⎟ ( ) ( ) ( ) Y ln U α ⎝ ⎠ β = 1 − − α y F y e u y = − π Y Y 2 ln U sin 2 U 2 1 2 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics Concluding Remarks Concluding Remarks � Type 1: Smallest � Type 1: Smallest Gumbel Gumbel � Type 1: Largest Type 1: Largest Gumbel Gumbel � � Type 2: Largest � Type 2: Largest Weibull Weibull � Type 3: Smallest Type 3: Smallest � � Simulation of Random Variable With � Simulation of Random Variable With Known Distribution Known Distribution G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 3