Estimation of failure probability in higher-dimensional spaces

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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

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### 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

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
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.

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”
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.

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.

Remark

Relation ⑴ is as in the example.

But here we have the marginal transformations

on top of that.

Hence two steps:
• Transformation of marginal distributions
• Use of homogeneity property of υ

when pulling back the failure set.

Conditions

1) Domain of attraction:

2) We need estimators

with

for i = 1,2 with kk(n)→∞ ,k/n → 0, n→∞ .

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

in does not depend on n where

Then:

Condition 5 Sharpening of cond.1:

Condition 6 1 , 2 > 1 / 2 and

for i = 1,2 where

The EstimatorNote that

Hence we propose the estimator

and we shall prove

Then

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

Under our conditions

as n→∞ provided  (S) > 0.

For the proof note that by Cond. 5

and

Hence it is sufficient to prove

and

For both we need the following fundamental Lemma.

Lemma

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

provided

Proposition

ProofRecall

and

Combining the two we get

The Lemma gives

Similarly

Hence

Ɯ

Finally we need to prove

We do this in 3 steps.

Proposition 1Define

We have

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.

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

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?

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

where for 0 ≤ s ≤ 1 we define

and  is a homogeneous measure of degree –1.

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

Cond. 4

with

a fixed set (does not depend on n) and

Further:

Cond. 5

Cond. 6

and

Theorem

Under our conditions

as n→∞provided  (S) > 0.