Estimation of failure probability in higher-dimensional spaces

1. 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

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

3. Suppose: probability distribution of Ф unknown. • We have i.i.d. observations (X1,Y1), ... (Xn,Yn), 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. The problem: • 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”

7. A bit more formal • Suppose we have n i.i.d. observations (X1,Y1), (X2,Y2), ... (Xn,Yn), 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 = Cn and P(Cn) = O (1/n) as n → ∞ . i.e. when n increases the set C moves, say, to the NE corner.

8. Domain of attraction condition EVT There exist • Functions a1, a2 >0, b1, b2 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.

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

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

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

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

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

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

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

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

22. The EstimatorNote that Hence we propose the estimator and we shall prove Then

23. More formally: • Write: pn:  P {Cn}. Our estimator is • Where

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

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.

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

27. Proposition ProofRecall and Combining the two we get

28. The Lemma gives Similarly Hence Ɯ

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

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

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

32. Similar result in function space 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?

33. EVT in C [0,1] 1. Definition of maximum: Let X1, X2, ... 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

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

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

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

37. Cond. 5 Cond. 6 and

38. Now the estimator for pn: P{Cn} : where and

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