Estimation of failure probability in higher-dimensional spaces

Ana Ferreira, UTL, Lisbon, PortugalLaurens de Haan, UL, Lisbon Portugal

and EUR, Rotterdam, NLTao Lin, Xiamen University, China

Research partially supported by

Fundação Calouste Gulbenkian

FCT/POCTI/FEDER – ERAS project

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A simple exampleA 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

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• 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.

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• 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:The problem:

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A bit more formalA 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.

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Domain of attraction condition EVTDomain of attraction condition EVTThere 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.

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RemarkRemark

Relation is as in the example⑴ .But here we have the marginal transformations

on top of that.

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Hence two steps:Hence two steps:

1) Transformation of marginal distributions

2) Use of homogeneity property of υ

when pulling back the failure set.

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ConditionsConditions1) Domain of attraction:

2) We need estimators

with

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

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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

⑵

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Further : S has positive distance from the origin.

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Before we go on, we simplify notation:

NotationNotation

• Note that

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With this notation we can write

Cond. 1' :

Cond. 4' :

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Then:

Condition 5 Sharpening of cond.1:

Condition 6 1 , 2 > 1 / 2 and

for i = 1,2 where

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The EstimatorThe Estimator Note that

Hence we propose the estimator

and we shall prove

Then

•

•

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

• Where

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TheoremTheorem

Under our conditions

as n→∞ provided (S) > 0.

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For the proof note that by Cond. 5

and

Hence it is sufficient to prove

and

For both we need the following fundamental Lemma.

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LemmaLemma

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

provided

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PropositionProposition

ProofProof RecallRecall

and

Combining the two we get

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The Lemma gives

Similarly

Hence

Ɯ

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Finally we need to prove

We do this in 3 steps.

Proposition 1Proposition 1 Define

We have

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ProofProofJust calculate the characteristic function and

apply Condition 1.

Proposition 2Proposition 2 Define

we have

Next apply Lebesgue’s dominated convergence Theorem.

ProofProof

By the Lemma → identity.

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Proposition 3Proposition 3

The result follows by using statement and proof of Proposition 2

ProofProof The left hand side is

By the Lemma → identity.

end of finite-dimensional case

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Similar result in function spaceSimilar 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?

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EVT in EVT in C [0,1]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

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where for 0 ≤ s ≤ 1 we define

and is a homogeneous measure of degree –1.

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ConditionsConditionsCond. 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

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Cond. 4

with

a fixed set (does not depend on n) and

Further:

•

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and

•

•

Cond. 5

Cond. 6

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Now the estimator for pn: P{Cn} :

where

and

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TheoremTheoremUnder our conditions

as n→∞ provided (S) > 0.