Bachelor thesis of do dai chi

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Bachelor thesis of do dai chi

  1. 1. Introduction Univariate Extreme Value Theory Multivariate Extreme Value TheoryExtreme Values and Probability Distribution Functions on Finite Dimensional Spaces Do Dai Chi Thesis advisor: Assoc.Prof.Dr. Ho Dang Phuc K53 - Undergraduate Program in Mathematics Viet Nam National University - University of Science December 7, 2012 Do Dai Chi EVT and Probability D.Fs on F.D.S
  2. 2. Introduction Univariate Extreme Value Theory Multivariate Extreme Value TheoryOutline 1 Introduction Limit Probabilities for Maxima Maximum Domains of Attraction 2 Univariate Extreme Value Theory Max-Stable Distributions Extremal Value Distributions Domain of Attration Condition 3 Multivariate Extreme Value Theory Limit Distributions of Multivariate Maxima Multivariate Domain of Atrraction Do Dai Chi EVT and Probability D.Fs on F.D.S
  3. 3. Introduction Univariate Extreme Value Theory Multivariate Extreme Value TheoryOutline 1 Introduction Limit Probabilities for Maxima Maximum Domains of Attraction 2 Univariate Extreme Value Theory Max-Stable Distributions Extremal Value Distributions Domain of Attration Condition 3 Multivariate Extreme Value Theory Limit Distributions of Multivariate Maxima Multivariate Domain of Atrraction Do Dai Chi EVT and Probability D.Fs on F.D.S
  4. 4. Introduction Univariate Extreme Value Theory Multivariate Extreme Value TheoryOutline 1 Introduction Limit Probabilities for Maxima Maximum Domains of Attraction 2 Univariate Extreme Value Theory Max-Stable Distributions Extremal Value Distributions Domain of Attration Condition 3 Multivariate Extreme Value Theory Limit Distributions of Multivariate Maxima Multivariate Domain of Atrraction Do Dai Chi EVT and Probability D.Fs on F.D.S
  5. 5. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryMotivation Extreme value theory developed from an interest in studying the behavior of the extremes of i.i.d random variables. Historically, the study of extremes can be dated back to Nicholas Bernoulli who studied the mean largest distance from the origin to n points scattered randomly on a straight line of some fixed length. Our focus is on probabilistic aspects of univariate modelling and of the behaviour of extremes. Do Dai Chi EVT and Probability D.Fs on F.D.S
  6. 6. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryLimit Probabilities for Maxima Sample maxima: Mn = max(X1 , . . . , Xn ), n ≥ 1. (1) P(Mn ≤ x) = F n (x). (2) Renormalization : ∗ Mn − bn Mn = (3) an for {an > 0} and {bn } ∈ R. Do Dai Chi EVT and Probability D.Fs on F.D.S
  7. 7. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryLimit Probabilities for Maxima Definition A univariate distribution function F , belong to the maximum domain of attraction of a distribution function G if 1 G is non-degenerate distribution. 2 There exist real valued sequence an > 0, bn ∈ R, such that Mn − bn d P ≤x = F n (an x + bn ) → G (x). (4) an Extremal Limit Problem : Finding the limit distribution G (x). Domain of Attraction Problem: Finding the F (x) (F ∈ D(G )). Mn −bn P an ≤ x = P(Mn ≤ un ) where un = an x + bn . Do Dai Chi EVT and Probability D.Fs on F.D.S
  8. 8. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryLimit Probabilities for Maxima Example (standard exponential distribution) FX (x) = 1 − e −x , x > 0. (5) Taking an = 1 and bn = log n, we have Mn − bn P ≤x = F n (x + log n) = [1 − e −(x+log n) ]n an = [1 − n−1 e −x ]n → exp(−e −x ) (6) =: Λ(x), x ∈ R. Do Dai Chi EVT and Probability D.Fs on F.D.S
  9. 9. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryLimit Probabilities for Maxima Remark min(X1 , . . . , Xn ) = − max(−X1 , . . . , −Xn ). (7) Now we are faced with certain questions: 1 Given any F , does there exist G such that F ∈ D(G ) ? 2 Given any F , if G exist, is it unique ? 3 Can we characterize the class of all possible limits G according to definition definition #1 ? 4 Given a limit G , what properties should F have so that F ∈ D(G ) ? 5 How can we compute an , bn ? Do Dai Chi EVT and Probability D.Fs on F.D.S
  10. 10. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryMaximum Domains of Attraction Theorem (Poisson approximation) For given τ ∈ [0, ∞] and a sequence {un } of real numbers, the following two conditions are equivalent for F = 1 − F 1 nF (un ) → τ as n → ∞, 2 P(Mn ≤ un ) → e −τ as n → ∞. We denote f (x−) = limy ↑x f (y ) Theorem Let F be a d.f. with right endpoint xF ≤ ∞ and let τ ∈ (0, ∞). There exists a sequence (un ) satisfying nF (un ) → τ if and only if F (x) lim =1 (8) x↑xF F (x−) Do Dai Chi EVT and Probability D.Fs on F.D.S
  11. 11. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryExample (Geometric distribution) P(X = k) = p(1 − p)k−1 , 0 < p < 1, k ∈ N. (9) For this distribution, we have ∞ −1 F (k) k−1 r −1 = 1 − (1 − p) (1 − p) F (k − 1) r =k = 1 − p ∈ (0, 1). (10) No limit P(Mn ≤ un ) → ρ exists except for ρ = 0 or 1, that implies there is no non-degenerate limit distribution for the maxima in the geometric distribution case. Do Dai Chi EVT and Probability D.Fs on F.D.S
  12. 12. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryMaximum Domains of Attraction Definition Distribution functions U(x) and V (x) are of the same type if for some A > 0, B ∈ R V (x) = U(Ax + B) (11) d X −B Y = (12) A Example (Normal distribution function) x −µ N(µ, σ 2 , x) = N(0, 1, ) for σ > 0, µ ∈ R. (13) σ d Xµ,σ = σX0,1 + µ. (14) Do Dai Chi EVT and Probability D.Fs on F.D.S
  13. 13. Introduction Limit Probabilities for Maxima Univariate Extreme Value Theory Maximum Domains of Attraction Multivariate Extreme Value TheoryConvergence to types theorem Theorem (Convergence to types theorem) Suppose U(x) and V (x) are two non-degenerate d.f.’s . Suppose for n ≥ 1, Fn is a distribution, an ≥ 0, αn > 0, bn , βn ∈ R and d d Fn (an x + bn ) → U(x), Fn (αn x + βn ) → V (x). (15) Then as n → ∞ αn βn − bn → A > 0, → B ∈ R, (16) an an and V (x) = U(Ax + B) (17) Do Dai Chi EVT and Probability D.Fs on F.D.S
  14. 14. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionMax-Stable Distributions What are the possible (non-degenerate) limit laws for the maxima Mn when properly normalised and centred? Definition A non-degenerate random d.f. F is max-stable if for X1 , X2 , . . . , Xn i.i.d. F there exist an > 0, bn ∈ R such that d Mn = an X1 + bn . (18) Theorem (Limit property of max-stable laws) The class of all max-stable d.f.’s coincide with the class of all limit laws G for maxima of i.i.d. random variables. Do Dai Chi EVT and Probability D.Fs on F.D.S
  15. 15. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionExtremal Value Distributions Theorem (Extremal types theorem) Suppose there exist sequence {an > 0} and {bn ∈ R}, such that Mn − bn d →G an where G is non-degenerate, then G is of one the following three types: 1 Type I, Gumbel : Λ(x) = exp{−e −x }, x ∈ R. 0 if x < 0 2 Type II, Fr´chet : e Φα (x) = exp{−x −α } if x ≥ 0 for some α > 0. exp{−(−x)α } if x < 0 3 Type III, Weibull : Ψα (x) = 1 if x ≥ 0 for some α > 0 Do Dai Chi EVT and Probability D.Fs on F.D.S
  16. 16. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionExtremal Value Distributions Remark 1 Suppose X > 0, then 1 X ∼ Ψα ⇔ − ∼ Ψα ⇔ log X α ∼ Λ (19) X 2 Class of Extreme Value distributions = Max-stable distributions = Distributions appearing as limits in Definition definition #1 Do Dai Chi EVT and Probability D.Fs on F.D.S
  17. 17. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionExtremal Value Distributions Example (standard Fr´chet distribution) e 1 F (x) = exp(− ), x > 0. (20) x For an = n and bn = 0. M n − bn 1 n P ≤x = F n (nx) = [exp{− }] an nx n = exp(− ) = F (x) (21) nx Because of the max-stability of F - is also the standard Fr´chet distribution. e Do Dai Chi EVT and Probability D.Fs on F.D.S
  18. 18. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionExtremal Value Distributions Example (Uniform distribution) F (x) = x for 0 ≤ x ≤ 1. 1 For fixed x < 0, suppose n > −x and let an = n and bn = 1. Mn − bn P ≤x = F n (n−1 x + 1) an x n = 1+ → ex (22) n The limit distribution is of Weibull type, that means Weibull distribution are max-stable. Do Dai Chi EVT and Probability D.Fs on F.D.S
  19. 19. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionGeneralized Extreme Value Distributions Definition (Generalized Extreme Value Distributions) For any γ ∈ R, defined the distribution 1 exp(−(1 + γx) γ ), if 1 + γx > 0; Gγ (x) = (23) − exp{−e −x } if γ = 0. is an extreme value distribution. The parameter γ is called the extreme value index. 1 For γ > 0, we have Fr´chet class of distributions. e 2 For γ = 0, we have Gumbel class of distributions. 3 For γ < 0, we have Weibull class of distributions. Do Dai Chi EVT and Probability D.Fs on F.D.S
  20. 20. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionDomain of Attration Condition Theorem (von Mises’condition) Let F be a distribution function. Suppose F ”(x) exists and F (x) is positive for all x in some left neighborhood of xF . If 1 − F (t) lim (t) =γ (24) t↑xF F or equivalently (1 − F (t))F (t) lim = −γ − 1 (25) t↑xF (F (t))2 then F is in the domain of attraction of Gγ (F ∈ D(Gγ )). Do Dai Chi EVT and Probability D.Fs on F.D.S
  21. 21. Introduction Max-Stable Distributions Univariate Extreme Value Theory Extremal Value Distributions Multivariate Extreme Value Theory Domain of Attration ConditionDomain of Attration Condition Example (standard normal distribution) Let F (x) = N(x). We have 1 2 F (x) = n(x) = √ e −x /2 (26) 2π 1 2 F (x) = − √ xe −x /2 = −xn(x) (27) 2π Using Mills’ ratio, we have 1 − N(x) ∼ x −1 n(x). (1 − F (x))F (x) −x −1 n(x)xn(x) lim = lim = −1. (28) x→∞ (F (x))2 x→∞ (n(x))2 Then γ = 0 and F ∈ D(Λ) - Gumbel distribution. Do Dai Chi EVT and Probability D.Fs on F.D.S
  22. 22. Introduction Limit Distributions of Multivariate Maxima Univariate Extreme Value Theory Multivariate Domain of Atrraction Multivariate Extreme Value TheoryLimit Distributions of Multivariate Maxima For d-dimensional vectors x = (x (1) , . . . , x (d) ). Marginal ordering: x ≤ y means x (j) ≤ y (j) , j = 1, . . . , d. Component-wise maximum: x ∨ y := (x (1) ∨ y (1) , . . . , x (d) ∨ y (d) ) (29) Our approach for extreme value analysis will be based on the Componentwise maxima depending on Marginal ordering. (1) (d) If Xn = (Xn , . . . , Xn ), then n n (1) (d) (1) (d) Mn = ( Xi , . . . , Xi ) = (Mn , . . . , Mn ) (30) i=1 i=1 Do Dai Chi EVT and Probability D.Fs on F.D.S
  23. 23. Introduction Limit Distributions of Multivariate Maxima Univariate Extreme Value Theory Multivariate Domain of Atrraction Multivariate Extreme Value TheoryMax-infinitely Divisible Distributions Definition The d.f. F on Rd is max-infinitely divisible or max-id if for every n there exists a distribution Fn on Rd such that n F = Fn . (31) Theorem Suppose that for n ≥ 0, Fn are probability distribution functions on n d Rd . If Fn → F0 then F0 is max-id. Consequently, 1 F is max-id if and only if F t is a d.f. for all t > 0. 2 The class of max-id distributions is closed under weak d convergence: If Gn are max-id and Gn → G0 , then G0 is max-id. Do Dai Chi EVT and Probability D.Fs on F.D.S
  24. 24. Introduction Limit Distributions of Multivariate Maxima Univariate Extreme Value Theory Multivariate Domain of Atrraction Multivariate Extreme Value TheoryMultivariate Domain of Atrraction Definition A multivariate distribution function F is said to be in the domain of attraction of a multivariate distribution function G if 1 G has non-degenerate marginal distributions Gi , i = 1, . . . , d. (i) (i) 2 There exist sequence an > 0 and bn ∈ R, such that (i) (i) Mn − bn P (i) ≤ x (i) = F n (an x (1) + bn , . . . , an x (d) + bn ) (1) (1) (d) (d) an d → G (x) (32) Do Dai Chi EVT and Probability D.Fs on F.D.S
  25. 25. Introduction Limit Distributions of Multivariate Maxima Univariate Extreme Value Theory Multivariate Domain of Atrraction Multivariate Extreme Value TheoryMax-stability Definition (Max-stable distribution) A distribution G (x) is max-stable if for i = 1, . . . , d and every t > 0, there exist functions α(i) (t) > 0 , β (i) (t) such that G t (x) = G (α(1) (t)x (1) + β (1) (t), . . . , α(d) (t)x (d) + β (d) (t)). (33) Every max-stable distribution is max-id. Theorem The class of multivariate extreme value distributions is precisely the class of max-stable d.f.’s with non-degenerate marginals. Do Dai Chi EVT and Probability D.Fs on F.D.S
  26. 26. Introduction Limit Distributions of Multivariate Maxima Univariate Extreme Value Theory Multivariate Domain of Atrraction Multivariate Extreme Value TheoryConclusion Extreme value theory is concerned with distributional properties of the maximum Mn of n i.i.d. random variables. 1 Extremal Types Theorem, which exhibits the possible limiting forms for the distribution of Mn under linear normalizations. 2 A simple necessary and sufficient condition under which P{Mn ≤ un } converges, for a given sequence of constants {un }. The maximum of n multivariate observations is defined by the vector of componentwise maxima. The structure of the family of limiting distributions can be studied in terms of max-stable distributions. We discuss characterizations of the limiting multivariate extreme value distributions. Do Dai Chi EVT and Probability D.Fs on F.D.S
  27. 27. Introduction Limit Distributions of Multivariate Maxima Univariate Extreme Value Theory Multivariate Domain of Atrraction Multivariate Extreme Value TheoryBibliography [S. Resnick] Extreme Values, Regular Variation, and Point Processes (Springer, 1987) [de Haan, Laurens and Ferreira, Ana] Extreme Value Theory: An Introduction (Springer, 2006) [Leadbetter, M. R. and Lindgren, G. and Rootz´n, H. ] e Extremes and Related Properties of Random Sequences and Processes (Springer-Verlag, 1983) [Bikramjit Dass] A course in Multivariate Extremes (Spring-2010) Do Dai Chi EVT and Probability D.Fs on F.D.S
  28. 28. Introduction Limit Distributions of Multivariate Maxima Univariate Extreme Value Theory Multivariate Domain of Atrraction Multivariate Extreme Value TheoryThank you for listening Do Dai Chi EVT and Probability D.Fs on F.D.S

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