Introduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT

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1 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT

2 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Sample extreme: Let X, X, X, be idepedet radom variables, all with 2 3 he same distributio fuctio F. Cosider ( ) Y : = max X, X,, X = X for =,2, 2, Probability distributio fuctio of Y : { x} = P{ X x, X2 x,, X x} idep. P{ X x} P{ X x} P{ X x} PY = 2 same = distr. F ( x). 2

3 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Limit theory: what ca we say about { x} PY as? If F( x ) <, the PY { x} = F ( x) 0 If F( x ) =, the PY { x} =. Hece we get a degeerate limit (adopts oly two values) which is ot very iterestig. Hece we put Y o the right scale ad locatio i.e. we cosider Y a b 3

4 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 with b some sequece of real umbers (locatio correctio) ad a some positive umbers (scale correctio). The Y b = + = + { } P x P Y ax b F ax b a We try to fid sequeces { b } ad { } F ( ax+ b) exists =: lim a such that G x () where G is a o-degeerate distributio fuctio i.e. G adopts at least 3 values (extreme value coditio).. 4

5 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 We are goig to fid all possibilities for G! I fact we look at 2 questios:. What probability distributio fuctios G ca occur as a limit i ()? 2. For each of the G foud i (): what are the coditios o the origial distributio fuctio F such that () holds with this give G? (F is i the domai of attractio of G, F ( G) D ) 5

6 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Prelimiary calculatios: ( + ) (for all x with G( x) F ax b G x log F a x b logg x :0< < ) () ( + ) (for x:0 logg( x) This ca hold oly if F( ax b) Now recall the limit log( s) log + 0. lim = s 0 s s : = F ax+ b. log F ax+ b F a x+ b ad apply with We get < < ) 6

7 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 hece F a x+ b logg x,. (2 ) ( ) With some effort it ca be proved that this also holds whe we replace by a cotiuous parameter t: ( ( () )) log t F a t x b t G x +, t, t real. (2) Hece ( 2). I wat to derive a third equivalet form for the covergece. This goes via the iverse fuctio 7

8 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Lemma Suppose Cosider f ( x) f ( =, 2, ). f x is o-decreasig i x for all., the iverse fuctio of Suppose f ( x) = g( x) for all x ( ab, ) lim The f ( x) = g ( x) for all x g( a), g( b) lim where g is the iverse fuctio of g. (picture) 8

9 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 We apply this to ad f ( x) : = g x F a x+ b : = logg x. Accordig to (2 ) we have f ( x) g( x) Hece f ( x) g ( x) What are for all x. for all x. f x : f ad g i this case? First 9

10 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Hece y = f x y = F a x+ b y = F ( ax + b) = F a x+ b y F b y ax+ b = F x. y = a f ( x) = F x a b 0

11 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Simpler otatio : U( x) : = F x. = F equivaletly U( x) : ( x) This was the iverse of ( ) y. f x. Now about the iverse of g: y = g( x) y = G( x) = e logg x x= G e. y Coclusio: () (2)

12 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 U x lim a iteger b ( e ) = G for x 0 x > (3 ) U tx lim t a t b t ( e ) = G for x 0 x >. (3 ) x = ) cotiuous variable (subtract the same with 0 at () U tx U t lim t ( ) ( ) x = G e G e for 0 x >. (3) 2

13 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Theorem Equivalet are: ) F ( ax+ b) = G( x) lim 2) 3) ( ( + )) = lim t F b t xa t logg x t U( t) U tx lim t at ( ) ( = G e G e ) () x Soo we shall see the use of this theorem. We proceed to idetify the limit G( x ). The complete class of possible limit distributios G is give i the ext theorem. 3

14 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Theorem (Fisher ad Tippett 928, Gedeko 943) Suppose that for some distributio fuctio F we have F ( ax + b) G( x), o-degeerate, for all cotiuity poits x. The G( x) G ( ax b) = + for some a > 0 ad b where { } : = exp ( + ) G x x for all x with + x > 0 ad where the parameter ca have ay real value (for = 0 read the formula as e x ). exp{ } 4

15 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Remark There are 3 parameters,, a, b but is the oly importat oe, the other two just represet scale ad locatio. They are arbitrary sice by chagig the sequeces { } a ad { } b, oe ca get ay a > 0 ad b. 5

16 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Proof: We foud G( x) U( t) U tx at F a x b G e ( ) ( G e ) : D( x) () x + = Note : D = 0. Take x, y > 0 ad write the idetity () ( t ) () () U tyx U t U tyx U ty a ty U ty U t = + at aty at at D( xy ) D( x ) A ( y) > 0 D( y ) ( say) 6

17 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Hece D( xy) D( x) A ( y) D( y) = + for all, 0 We have to solve this fuctioal equatio. We write ( s+ t) ( s) ( t) ( t De De A e De) x y >. = + for all real s, t. Itroduce ( s ) : & (): ( t H s De A t A e) = =. The or H( t s) H( s) A( t) H( t) H = D = A = + = + ts, & 0 0, 0 H( t+ s) H( t) = H( s) A( t) real 7

18 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Write this as ( + ) ( 0) H t s H t H s H = At (). s s Now H is mootoe hece t where H' The equality above shows that '0 exists for all t. t exists. H exists hece H' () t Coclusio H' t = H' 0 A t. 8

19 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Sice H caot be costat, this implies H '0 > 0. Write Q() : H ( t) H' ( 0) t =. Note Q ( 0) = 0, Q '0 =, Q' ( t) A( t) =. We kow H( t+ s) H( t) = H( s) A( t) hece ( + ) = = ' Qt s Qt QsAt QsQ t 9

20 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Write agai Qt ( + s) Qt = QsQ '( t) ad, equivaletly, Qt ( s) Qs QtQ '( s) Subtract, the i.e. + =. () = ' ' Qt Qs QtQ s QsQ t Q' s Q s Q s Q 0 Q() t = ( Q' () t ) = ( Q' () t ). s s s Hece ( s 0) () QtQ'' 0 = Q' 0 Q' t = Q' t 20

21 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 We kow that equatio ad get Q ' exists hece we differetiate the = Q' t Q'' 0 Q'' t hece ( t) () Q'' log Q' ' t = = Q'' ( 0 ) : Q' t = for all t. Now we just work backwards. 2

22 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Sice Q '0 =, by itegratio we get log Q' t = t i.e. Q' t = e t ad (sice Q ( 0) = 0) agai by itegratio () Qt t s eds 0 = = (but if 0 e t. = we get Qt = t). 22

23 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 We go through the trasformatios Q H D G G I order to idetify the fuctio G. Q H: Note that ( 0) 0 H =. Write a: H' ( 0) =. ad (H D) def. t e () '0 () H t = H Q t = a D def. t () ( log ) t = H t = a. 23

24 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 D G : goig further back recall that Dt G e G e t () ( = ) hece (write ( b: G e ) = ) t = + G e b a t. 24

25 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 G G : apply G to both sides: exp { } Replace t by t G b a = t +. a x b +. We get x b exp + = G x a, Quod erat demostradum. 25

26 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa,

27 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Cosider the graphs of G. Note that if < 0 G ( x) = for x. That meas that o value beyod is possible. Defie i geeral for a prob. dist. fuctio F x = x F : = max x: F x <. { } 27

28 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Note that for G : If F ( ax b) G( x) ( G ) > 0 x = < 0 x ( G ) < = 0 x ( G ) = +, for F we have similar behaviour Hece : 0 x ( F) < <. ( F) ( F) > 0 x = < 0 x < = 0 : ca be both. 28

29 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 We cosider the cases, > 0, = 0, < 0 separately. 29

30 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 ) 0 = : G ( x) ( e x ) = exp. 0 0 G x Note that < 0 < for all x hece the distributio has o lower or upper boud (all real values are possible). Also, sice y e lim y 0 y G0 ( x) lim =. x x e x =, we have with y e = : Hece the tail of the distributio G ( x) ( = ) goes 0 dow to zero very quickly. This meas for example that all momets exists (are fiite). We say that the distributio is light tailed. 30

31 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 2) 0 upper boud. > : Note that G ( x) < for all x hece there is o Also, we see G ( x) lim 0 x x = > hece the tail is approximately a power fuctio x. This meas that G ( x ) goes to zero much more slowly tha i the case = 0. I particular some momets are ot fiite. We say that i this case the distributio is heavy tailed. Note: ofte i fiace we have this case > 0. 3

32 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 3) 0 < : Note that G ( x) = for all x. Hece o values larger tha are possible. We say that the distributio is short tailed. Note: I evirometal data we ofte fid close to zero. I fiacial data we ofte fid positive. 32

33 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 I some cases we ca simplify the formula for G : ) > 0: I the formula { } = exp + ( + ) G x ax b we ca choose a = ad b =. The G x = exp x I this case oe simplifies by writig α for ad we get (traditioally) Φ x = exp x α for x > 0 (ad = 0 for 0 α x ). I this form it is referred to as the Fréchet class of extreme value distributios ( 0) α >. 33

34 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 2) < 0: Take a i the formula = ad b = { } = exp + ( + ) G x ax b ad write α (agai!) for The we get ( x exp x α ). Ψ α = for 0 x < (ad = for x ). I this form it is referred to as the reverse-weibull class of distributios ( 0) α >. 34

35 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 3) 0 = : ( x ) G x = exp{ e }. This oe is sometimes called the Gumbel distributio. We are ow able to reformulate the Theorem: 35

36 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Theorem For R the followig statemets are equivalet: ) There exist real costats a > 0 ad b real, such that ( ) ( ) lim F a x+ b G x = exp + x, (4) for all x with x 0 + >. 36

37 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 2) There exists a positive fuctio a such that for x > 0 at () U tx U t x lim t =, (5) where for = 0 the right-had side is iterpreted as log x. 3) There exists a positive fuctio a such that ( ) ( ) limt F a t x+ U t = + x, (6) t for all x with x 0 + >. 37

38 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 4) There exist a positive fuctio f such that xf ( t) () F( t+ ) lim = + t x F t for all x which x 0 ( x), (7) + >, where x x F( x) = sup : <. { } Moreover (4) holds with b : = U ad a : a holds with f () t = a ( F( t) ). =. Also (7) 38

39 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Remark: We say that F DG if the coditios of the Theorem hold for F. The parameter is called the extreme value idex. The class of distributios satisfyig the coditio is very wide. The coditio reflects a property of the far tail of F. Let us look at three cases: > 0, = 0 ad < 0. 39

40 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 > 0 It ca be proved that i that case oe ca take f ( t) i (7). = t Hece F with > 0 if ad oly if DG F tx lim t F t () = x for x > 0 ( F has regularly varyig tail ). 40

41 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Such distributio fuctio is called heavy tailed sice E ( max ( X,0) ) α < if a < = = if a >. Hece ot all momets exist. 4

42 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Sufficiet coditio: xf' x lim = x F x. Examples: Cauchy s distributio Ay Studet distributio Pareto distributio F x = x, x> 42

43 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 = 0 Sufficiet coditio: ( ) F'' ( x) F( x) lim = x x where x x F( x) { } ( F' ( x) ) 2 : = sup <. Light tailed sice E max ( X,0) Examples: Normal distributio Expoetial distributio Ay Gamma distributio Logormal distributio x F x = + e for x < 0 α < a > 0 43

44 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 < 0 The the probability distributio has a upper boud: F x = for x some x = < for x < x. f t = x t. It ca be proved that oe ca take Leads to a simple criterio: lim t 0 F x ( tx) ( t) F x (is agai a kid of regular variatio coditio) Short tailed Examples: uiform distributio ay Beta distributio = x for x > 0 44

45 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 A sufficiet coditio valid for all domai of attractio: If ( ) F'' ( x) F( x) lim t x F' () t 2 =, the F DG. A ecessary ad sufficiet coditio (provided that 0 x > ) is : If x x 2 F t F x x dx dy t y 2 x 2 2 ( ()) + if > 0 lim = t x if 0, t ( F( x) ) x dx 2 t the F DG. 45

46 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 There are probability distributios that are ot i ay domai of attractio. Examples: geometric distributio F ( x) Poisso distributio F( x) [ x] = e for x > 0 e λ x = λ for x 0 = 0! vo Mises' example x si F x e x = for x 0 46

47 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Remark Let X be a r.v. with distributio fuctio F. Relatio (7) ca be reformulated as follows: X t P > x X > t + x t f () t ( ) for 0 x >. (Geeralized Pareto distributio) (model for residual life time) 47

48 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 View towards applicatios observatios, t large t 48

49 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 The overshoots of t are i.i.d. observatios ad they follow approximately a geeralized Pareto distributio + x,. They ca be used to estimate the parameter of the Pareto distributio. The we ca use the fitted Pareto distributio to estimate the distributio fuctio beyod the observatios. I fact we take t to be oe of the observatios say, the k th highest observatio X. k, 49

50 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 We should choose k i such way, that k depeds o, k = k (allowig the use of CLT) k 0 (implies stayig i the tail). The we use oly X X X,,, k, k+,, for estimatig the parameter of the Pareto distributio ad also for estimatig the probability of extreme evets beyod the rage of the sample. 50

51 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 5

52 Itroductio to Extreme Value Theory Laures de Haa, ISM Japa,