2. Estimation of failure probability in higher-dimensional spaces Ana Ferreira, UTL, Lisbon, Portugal Laurens de Haan, UL, Lisbon Portugal and EUR, Rotterdam, NL Tao Lin, Xiamen University, China Research partially supported by Fundação Calouste Gulbenkian FCT/POCTI/FEDER – ERAS project

3. 2 A simple example Take r.v.’s (R, Ф), independent, and (X,Y) : = (R cos Ф, R sin Ф). Take a Borel set A  with positive distance to the origin. Write a A : = {a x : x  A}. Clearly

4. 3 Suppose: probability distribution of Ф unknown. We have i.i.d. observations (X 1,Y 1 ),... (X n,Y n ), and a failure set A away from the observations in the NE corner. To estimate P{A} we may use a {a A} where is the empirical measure. This is the main idea of estimation of failure set probability.

5. 4

6. 5 Some device can fail under the combined influence of extreme behaviour of two random forces X and Y. For example: rain and wind. “Failure set” C: if (X, Y) falls into C, then failure takes place. “Extreme failure set”: none of the observations we have from the past falls into C. There has never been a failure. Estimate the probability of “extreme failure” The problem:

7. 6

8. 7 A bit more formal Suppose we have n i.i.d. observations (X 1,Y 1 ), (X 2,Y 2 ),... (X n,Y n ), with distribution function F and a failure set C. The fact “none of the n observations is in C” can be reflected in the theoretical assumption P(C) < 1 / n. Hence C can not be fixed, we have C = C n and P(C n ) = O (1/n) as n → ∞. i.e. when n increases the set C moves, say, to the NE corner.

9. 8 Domain of attraction condition EVT There exist Functions a 1, a 2 >0, b 1, b 2 real Parameters  1 and  2 A measure on the positive quadrant [0, ∞ ] 2 \ {(0,0)} with (a A) = a -1 (A) ⑴ for each Borel set A, such that for each Borel set A ⊂ with positive distance to the origin.

10. 9 Remark Relation ⑴ is as in the example. But here we have the marginal transformations on top of that.

11. 10

12. 11

13. 12

14. 13 Hence two steps: 1)Transformation of marginal distributions 2)Use of homogeneity property of υ when pulling back the failure set.

15. 14 Conditions 1) Domain of attraction: 2) We need estimators with for i = 1,2 with k  k(n) →∞, k/n → 0, n →∞.

16. 15 3) C n is open and there exists (v n, w n ) ∈  C n such that (x, y) ∈ C n ⇒ x > v n or y > w n. 4) (stability condition on C n ) The set in does not depend on n where ⑵

17. 16 Further : S has positive distance from the origin.

18. 17

19. 18

20. 19 Before we go on, we simplify notation: Notation Note that

21. 20 With this notation we can write Cond. 1' : Cond. 4' :

22. 21 Then: Condition 5 Sharpening of cond.1: Condition 6  1,  2 > 1 / 2 and for i = 1,2 where

23. 22 The Estimator The Estimator Note that Hence we propose the estimator and we shall prove Then

24. 23 More formally: Write: p n :  P {C n }. Our estimator is Where

25. 24 Theorem Under our conditions as n →∞ provided (S) > 0.

26. 25 For the proof note that by Cond. 5 and Hence it is sufficient to prove and For both we need the following fundamental Lemma.

27. 26 Lemma For all real γ and x > 0, if γ n → γ (n →∞ ) and c n ≥ c>0, provided

28. 27 Proposition Proof Recall and Combining the two we get

29. 28 The Lemma gives Similarly Hence Ɯ

30. 29 Finally we need to prove We do this in 3 steps. Proposition 1 Proposition 1 Define We have

31. 30 Proof Just calculate the characteristic function and apply Condition 1. Proposition 2 Proposition 2 Define we have Next apply Lebesgue’s dominated convergence Theorem. Proof By the Lemma → identity.

32. 31 Proposition 3 The result follows by using statement and proof of Proposition 2 Proof Proof The left hand side is By the Lemma → identity. end of finite-dimensional case

33. 32 Similar result in function space Example: Example: During surgery the blood pressure of the patient is monitored continuously. It should not go below a certain level and it has never been in previous similar operations in the past. What is the probability that it happens during surgery of this kind?

34. 33 EVT in C [0,1] 1. Definition of maximum: Let X 1, X 2,... be i.i.d. in C [0,1]. We consider as an element of C [0,1]. 2. Domain of attraction. For each Borel set A ∈ C + [0,1] with we have

35. 34 where for 0 ≤ s ≤ 1 we define and is a homogeneous measure of degree –1.

36. 35 Conditions Cond. 1. Domain of attraction. Cond. 2. Need estimators such that Cond. 3. Failure set C n is open in C[0,1] and there exists h n ∈ ∂C n such that

37. 36 Cond. 4 with a fixed set (does not depend on n) and Further:

38. 37 and Cond. 5 Cond. 6

39. 38 Now the estimator for p n :  P{C n } : where and

40. 39 Theorem Under our conditions as n →∞ provided (S) > 0.