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
Fundação Calouste Gulbenkian
FCT/POCTI/FEDER ERAS
project
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A simple example

• Take r.v.s (R, ?), independent,
• and (X,Y) (R cos ?, R sin ?) .
• 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 ? 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 PA 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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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

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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) lt 1 / n .
• Hence C can not be fixed, we have
• C Cn
• and P(Cn) O (1/n) as n ? 8 .
• i.e. when n increases the set C moves, say,
  to the NE corner.

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Domain of attraction condition EVT

• There exist
• Functions a1, a2 gt0, b1, b2 real
• Parameters ?1 and ?2
• A measure ? on the positive quadrant
• 0, 8 2 \ (0,0) with
• ? (a A) a-1 ? (A) ?
• for each Borel set A, such that

for each Borel set A? with positive
distance to the origin.
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Remark
Relation ? is as in the example. But here we have
the marginal transformations
on top of that.
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Hence two steps

• Transformation of marginal distributions
• Use of homogeneity property of ?
• when pulling back the failure set.

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Conditions

• 1) Domain of attraction

2) We need estimators with
for i 1,2 with k ? k(n)?8 , k/n ? 0, n?8 .
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• 3) Cn is open and there exists (vn , wn) ? ? Cn
  such that (x , y) ? Cn ? x gt vn or y gt wn .
• 4) (stability condition on Cn ) The set

?
in does not depend on n where
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• Further S has positive distance from the
  origin.

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Before we go on, we simplify notation
Notation

• 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 gt 1 / 2 and
for i 1,2 where
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The Estimator Note that

Hence we propose the estimator
and we shall prove
Then
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More formally

• Write pn ? P Cn. Our estimator is

• Where

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Theorem

• Under our conditions

as n?8 provided ? (S) gt 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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Lemma

• For all real ? and x gt 0 , if ?n ? ? (n?8 )
  and cn cgt0,

provided
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Proposition

• Proof Recall

and
Combining the two we get
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The Lemma gives

• Similarly

Hence
?
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Finally we need to prove

• We do this in 3 steps.

Proposition 1 Define
We have
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Proof

• Just calculate the characteristic function and
  apply Condition 1.

Proposition 2 Define
we have
Proof
By the Lemma ? identity.
Next apply Lebesgues dominated convergence
Theorem.
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Proposition 3
Proof The left hand side is
By the Lemma ?
identity.

• The result follows by using statement and proof
  of Proposition 2

end of finite-dimensional case
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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?

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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
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where for 0 s 1 we define

• and ? is a homogeneous measure of degree 1.

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Conditions

• Cond. 1. Domain of attraction.
• Cond. 2. Need estimators
• such that

Cond. 3. Failure set Cn is open in C0,1 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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Cond. 5

Cond. 6

and
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Now the estimator for pn? PCn

• where

and
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Theorem

• Under our conditions

as n?8 provided ? (S) gt 0.