EXTREME VALUES ANDPROBABILITY DISTRIBUTION FUNCTONSON FINITE DIMENSIONAL SPACES

VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Do Dai Chi EXTREME VALUES AND PROBABILITY DISTRIBUTION FUNCTONS ON FINITE DIMENSIONAL SPACES Undergraduate Thesis Advanced Undergraduate Program in Mathematics Hanoi - 2012 VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Do Dai Chi EXTREME VALUES AND PROBABILITY DISTRIBUTION FUNCTONS ON FINITE DIMENSIONAL SPACES Undergraduate Thesis Advanced Undergraduate Program in Mathematics Thesis advisor: Assoc.Prof.Dr. Ho Dang Phuc Hanoi - 2012 Acknowledgments It would not have been possible to write this undergraduate thesis without the help, and support, of the kind people around me, to only some of whom it is possible to give particular mention here. This thesis would not have been possible without the help, support and patience of my advisor, Assoc.Prof.Dr. Ho Dang Phuc, not to mention his advice and unsur- passed knowledge of probability and statistic. The advice, support and friendship of his have been invaluable on both an academic and a personal level, for which I am extremely grateful. I would like to show my gratitude to my teachers at Faculty of Mathematics, Me- chanics and Informatics, University of Sciences, VietNam National University who equip me with important mathematics knowledge during first four years at the uni- versity. I would like to thank my parents for their personal support and great patience at all times. My parents have given me their unequivocal support throughout, as always, for which my mere expression of thanks likewise does not suffice. Last, but by no means least, I thank my friends in K53-Advanced Math for their support and encouragement throughout. i List of abbreviations and symbols Here is a glossary of miscellaneous symbols, in case you need a reference guide. ∼ f (x) ∼ g(x) as x → x 0 means that lim x→x 0 f (x) g(x) = 1 d → X n d → X : convergence in distribution. P → X n P → X convergence in probability. a.s → X n a.s → X almost surely convergence. v → µ n v → µ vague convergence. d = X d = Y: X and Y have the same distribution. o(1) f (x) = o(g(x)) as x → x 0 means that lim x→x 0 f (x) g(x) = 0. f ← The generalized inverse of a monotone function f defined by f ← (x) = inf{y : f ( y) ≥ x}. Λ(x) Gumbel distribution. Φ α (x) Fr ´ echet distribution. Ψ α (x) Weibull distribution. x F x F = sup{x ∈ R : F(x) < 1}. f (x−) f (x−) = lim y↑x f (y). [F > 0] means the set {x : F(x) > 0}. M + (E) The space of nonnegative Radon measures on E. C (f ) The points at which the function f is continuous. d.f. Distribution function. r.v. Random variable. DOA Domain of attraction. ii Contents Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i List of abbreviations and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1. Univariate Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1. Limit Probabilities for Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Maximum Domains of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1. Max-Stable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3. Extremal Value Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.1. Extremal Types Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2. Generalized Extreme Value Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4. Domain of Attraction Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.1. General Theory of Domains of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5. Condition for belonging to Extreme Value Domain . . . . . . . . . . . . . . . . . . . . 26 Chapter 2. Multivariate Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2. Limit Distributions of Multivariate Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1. Max-infinitely Divisible Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.2. Characterizing Max-id Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3. Multivariate Domain of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.1. Max-stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4. Basic Properties of Multivariate Extreme Value Distributions . . . . . . . . . . 41 2.5. Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 iii Chapter A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 A.1. Modes of Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 A.2. Inverses of Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 A.3. Some Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iv Introduction Extreme value theory developed from an interest in studying the behavior of the maximum or minimum (extremes) of independent and identically distributed ran- dom 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 scat- tered randomly on a straight line of some fixed length (Gumbel.1958 [15]). Extreme value theory provides important applications in finance, risk management, telecom- munication, environmental and pollution studies and other fields. In this thesis, we study the probabilistic approach to extreme value theory. The thesis is divided into the two chapters, namely, Chapter 1: Univariate Extreme Value Theory. Chapter 2: Multivariate Extreme Value Theory. Chapter 1 introduces the basic concepts related to Univariate Extreme Value Theory. This chapter concerns with the limit problem of determining the possible limits of sample extremes and the domain of attraction problem. Chapter 2 provides basic results in Multivariate Extreme Value Theory. We deal with the probabilistic aspects of multivariate extreme value theory by including the possible limits and their domain of attraction. The main materials of the thesis were taken from the books by M. R. Leadbetter, G. Lindgren, and H. Rootz ´ en [16], Resnick [18], Embrechts [12] and de Haan and Ana Ferreira [11]. We have also borrowed extensively from lecture notes of Bikramjit Dass [9]. v CHAPTER 1 Univariate Extreme Value Theory This chapter is primarily concerned with the central result of classical extreme value theory, the Extremal Types Theorem, which specifies the possible forms for the limit- ing distribution of maxima in sequences of independent and identically distributed (i.i.d.) random variables(r.v.s). In the derivation, the possible limiting distributions are identified with a class having a certain stability property, the so-called max-stable distributions. It is further shown that this class consists precisely of the three families known (loosely) as the three extreme value distributions. 1.1. Introduction The asymptotic theory of sample extremes has been developed in parallel with the central limit theory, and in fact the two theories bear some resemblance. Let X 1 , X 2 , . . . , X n be i.i.d. random variables. The central limit theory is concerned with the limit behavior of the partial sums S n = X 1 + X 2 + ··· + X n as n → ∞, whereas the theory of sample extremes is concerned with the limit behavior of the sample extremes max(X 1 , X 2 , . . . , X n ) or min(X 1 , . . . , X n ) as n → ∞. We consider some basic theory for sums of independent random variables. This in- cludes classical results such as the strong law of large numbers and the Central Limit Theorem. Throughout this chapter X 1 , X 2 , . . . is a sequence of i.i.d. non-degenerate real random variables defined on a probability space (Ω, F, P) with common distri- butions function(d.f.) F. We consider the partial sums S n = X 1 + ···+ X n , n ≥ 1. and of the sample means X n = n −1 S n = S n n , n ≥ 1. Let X be random variable and denote the expectation, the variance of X by E(X) = µ, Var(X) = σ 2 . Firstly, we assume that E(X) = µ < ∞. From the strong law of 1 large numbers, we get X n = n −1 S n a.s → µ. With the additional assumption of Var(X 1 ) = σ 2 < ∞, we get the Central Limit Theorem: S n −nµ √ nσ d → Z, Z ∼ N(0, 1). Hence for large n, we can approximate P(S n ≤ x) ≈ P  Z ≤ x −nµ √ nσ  . Taking an alternative approach, we can deal with the problem of finding possi- ble limit distributions for (say) sample maxima of independent and identically dis- tributed random variables. 1.1.1. Limit Probabilities for Maxima Whereas in above, we introduced ideas on partial sums, in this section we investi- gate the fluctuations of the sample maxima: M n = n  i=1 X i = max(X 1 , . . . , X n ), n ≥ 1. Remark 1.1. Corresponding results for minima can easily be obtained from those for maxima by using the identity min(X 1 , . . . , X n ) = −max(−X 1 , . . . , −X n ). We shall therefore only briefly discuss minima explicitly in this work, except where its joint distribution with M n is considered. We have the exact d.f of the maximum M n for x ∈ R, n ∈ N, P(M n ≤ x) = P(X 1 ≤ x, . . . , X n ≤ x) = n ∏ i=1 P(X i ≤ x) = F n (x). (1.1) Extreme events happen ’near’ the upper end of the support of the distribution. We denote the right endpoint of F by x F = sup{x ∈ R : F(x) < 1}. (1.2) That is, F(x) < 1 for all x < x F and F(x) = 1 for all x > x F . We immediately obtain  P(M n ≤ x) = F n (x) → 0, n → ∞ for all x < x F P(M n ≤ x) = F n (x) → 1, n → ∞ in the case x F < ∞, ∀x > x F 2 Therefore the limit distribution lim n→∞ F n (x) is degenerate. Thus M n P → x F as n → ∞ where x F < ∞. Since the sequence (M n ) is non decreasing in n, it converges almost surely(a.s), no matter whether it is finite or infinite and hence we conclude that M n a.s → x F , n → ∞. This result is quite uninformative for our purpose and does not answer the basic question in our mind. This difficulty is avoided by allowing a linear renormalization of the variable M n : M ∗ n = M n −b n a n , for sequences of constants {a n > 0} and {b n } ∈ R. Definition 1.1. A univariate distribution function F, belong to the maximum do- main of attraction of a distribution function G if 1. G is non degenerate distribution. 2. There exist real valued sequence a n > 0, b n ∈ R, such that P  M n −b n a n ≤ x  = F n (a n x + b n ) d → G(x). (1.3) Finding the limit distribution G(x) is called the Extremal Limit Problem. Finding the F(x) that have sequences of constants as described above leading to G(x) is called the Domain of Attraction Problem. For large n, we can approximate P(M n ≤ x) ≈ G( x−b n a n ). We denote F ∈ D(G). We often ignore the term ’maximum’ and abbreviate domain of attraction as DOA. 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 1.1? 4. Given a limit G, what properties should F have so that F ∈ D(G)? 5. How can we compute a n , b n ? The goal of the next section is to answer the above questions. 3 [...]... shown that: Class of Extreme Value distributions = Max-stable distributions = Distributions appearing as limits in Definition 1.1 Thus we have a characterization of the limit distributions appearing as limits in Definition 1.1, which answers question 3 1.3.2 Generalized Extreme Value Distributions Definition 1.5 (Generalized Extreme Value Distributions) For any γ ∈ R, defined the distribution 1 Gγ ( x... handled similarly In words, The extreme type theorems say that for a sequence of i.i.d random variables with suitable normalizing constants, the limiting distribution of maximum statistics, if it exists, follows one of three types of extreme value distributions that labeled I, II and III Collectively, these three classes of distribution are termed the extreme value distributions, with types I, II and... exponential distribution Then the distribution function of X is given by FX ( x ) = 1 − e− x , x > 0 If X1 , X2 , are i.i.d random variables with common distribution function F, then P( Mn ≤ x + log n) = 1 − e− x−log n n e− x n ) n → exp{−e−x } =: Λ( x ), = (1 − x ∈ R The limit distribution Λ( x ) is called the Gumbel distribution So we obtain that the Gumbel distribution is a possible limit distribution. .. an extreme value distribution, abbreviated by EVD The parameter γ is called the extreme value index Since (1 + γx )−1/γ → exp(− x ), as γ → 0 interpret for γ = 0, we have G0 ( x ) = exp{−e− x } The family of distributions Gγ ( x −µ σ ), for µ, γ ∈ R, σ > 0 is called the family of generalized extreme value distributions under von Mises or von MisesJenkins parametrization It shows that the limit distribution. .. continuous at x } If two r.v.s X and Y have the same distribution, we write d X = Y Definition 1.3 Two distribution functions U ( x ) and V ( x ) are of the same type if for some A > 0, B ∈ R V ( x ) = U ( Ax + B) for all x In terms of random variables, if X has distribution U and Y has distribution V, then d Y= X−B A Example 1.6 Let denote N (0, 1, x ) (normal distribution function with mean 0 and variance... the distribution is infinity Moreover, as x → ∞, 1 − Gγ ( x ) ∼ γ−1/γ x −1/γ i.e., the distribution has a rather heavy right tail We use Gγ ( x−1 ) and get with α = γ Φγ ( x ) = 0 > 0, x≤0 exp(− x −α ) 1 γ x>0 This class is often called the Fr´chet class of distributions e (b) For γ = 0 The distribution with γ = 0 G0 ( x ) = exp(−e− x ), for all x ∈ R, is called the double-exponential or Gumbel distribution. .. type called normal type If X0,1 has d N (0, 1, x ) as its distribution and Xµ,σ has N (µ, σ2 , x ) as its distribution, then Xµ,σ = σX0,1 + µ Now we state the theorem developed by Gnedenko and Khintchin Theorem 1.3 (Convergence to types theorem) (a) Suppose U ( x ) and V ( x ) are two nondegenerate distribution functions Suppose for n ≥ 1, Fn is a distribution, an ≥ 0, bn ∈ R, αn > 0, β n ∈ R and d d Fn... k ∈ N, n i =1 Yi a∗ k d Y1 = ∗ − bk which implies n d ∗ Yi = a∗ Y1 + bk k i =1 1.3 Extremal Value Distributions 1.3.1 Extremal Types Theorem The extreme type theorems play a central role of the study of extreme value theory In the literature, Fisher and Tippett (1928) were the first who discovered the extreme type theorems and later these results were proved in complete generality by Gnedenko (1943)... Hence, by virtue of Theorem 1.2 we can see that no non-degenerate distribution can be limit of normalized maxima taken from a sequence of random variables identically distributed as X 7 Example 1.4 (Geometric distribution) We consider the random variable X with geometric distribution: P ( X = k ) = p (1 − p ) k −1 , 0 < p < 1, k ∈ N For this distribution, we have F (k) = 1 − (1 − p ) k −1 F ( k − 1) ∞ ∑... relationships appear in the course of the proof of the theorem above Example 1.11 (Pareto distribution) As a simple example, we consider now the Pareto distribution F ( x ) = 1 − κx −α , α > 0, κ > 0, x ≥ κ 1/α We have (tx )−α 1 − F (tx ) = −α = x −α 1 − F (t) t so F belongs to DOA of a Type II extreme value distribution By setting n(1 − F (un )) = τ We have un = ( κn 1/α ) τ so that Theorem 1.1 gives