Transcript Estimation of failure probability in higher

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
2
• 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.
3
4
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”
5
6
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.
7
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)
for each Borel set A, such that
for each Borel set A⊂
distance to the origin.
⑴
with positive
8
Remark
Relation ⑴ is as in the example.
But here we have the marginal transformations
on top of that.
9
10
11
12
Hence two steps:
1) Transformation of marginal distributions
2) Use of homogeneity property of υ
when pulling back the failure set.
13
Conditions
1) Domain of attraction:
2) We need estimators
with
for i = 1,2 with k  k(n)→∞ , k/n → 0, n→∞
14
.
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
15
Further : S has positive distance from the
origin.
16
17
18
Before we go on, we simplify notation:
Notation
• Note that
19
With this notation we can write
Cond. 1' :
Cond. 4' :
20
Condition 5 Sharpening of cond.1:
Then:
Condition 6
1 , 2 > 1 / 2
and
for i = 1,2 where
21
The Estimator
Note that
•
•
Hence we propose the estimator
and we shall prove
Then
22
More formally:
• Write: pn:  P {Cn}. Our estimator is
• Where
23
Theorem
Under our conditions
as n→∞ provided  (S) > 0.
24
For the proof note that by Cond. 5
and
Hence it is sufficient to prove
and
For both we need the following fundamental Lemma.
25
Lemma
For all real γ and x > 0 , if γn → γ (n→∞ )
and cn ≥ c>0,
provided
26
Proposition
Proof
Recall
and
Combining the two we get
27
The Lemma gives
Similarly
Hence
Ɯ
28
Finally we need to prove
We do this in 3 steps.
Proposition 1
Define
We have
29
Proof
Just calculate the characteristic function and
apply Condition 1.
Proposition 2
Define
we have
Proof
By the Lemma
→ identity.
Next apply Lebesgue’s dominated convergence Theorem.
30
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
31
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?
32
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
33
where for 0 ≤ s ≤ 1 we define
and  is a homogeneous measure of degree –1.
34
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
35
Cond. 4
•
with
a fixed set (does not depend on n) and
Further:
36
Cond. 5
•
Cond. 6
•
and
37
Now the estimator for pn: P{Cn} :
where
and
38
Theorem
Under our conditions
as n→∞ provided  (S) > 0.
39