”statistics for near independence in multivariate extremes” a ...ericg/ledford.pdfstatistics for...
TRANSCRIPT
![Page 1: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/1.jpg)
”Statistics for Near Independence in MultivariateExtremes”
A.W. Ledford and J.A. Tawn
Discussion by Dan CooleyNCAR Extremes Reading Group
April 12, 2006
1
![Page 2: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/2.jpg)
Forward
These two Ledford and Tawn articles are a precursor to theHeffernan and Tawn paper we looked at several weeks ago.The first(?) alternative to traditional multivariate extremevalue theory.
� Review of classical multivariate extreme value theory
� Ledford and Tawn article 1
� A few notes on article 2
2
![Page 3: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/3.jpg)
Results from Univariate EVT
� Mt = maxi=1,...,t Xi.
� A dist. G is max stable if P(
Mt−btat
≤ x)
= P(X ≤ x) ⇔Gt(atx + bt) = G(x).
� F t(atx + bt)→ G(x),⇒ G is max stable
� All univariate max stable distributions are GEV.
3
![Page 4: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/4.jpg)
Results from Univariate EVT
� Mt = maxi=1,...,t Xi.
� A dist. G is max stable if P(
Mt−btat
≤ x)
= P(X ≤ x) ⇔Gt(atx + bt) = G(x).
� F t(atx + bt)→ G(x),⇒ G is max stable
� All univariate max stable distributions are GEV.
Max Stable Example: Unit Frechet
P{Xi ≤ x} = F (x) = exp{−x−1}
Normalizing constants: at = t, bt = 0
P{Mt/t ≤ x} = F t(tx) =[exp
{− 1
tx
}]t= exp
{− t
tx
}= exp{−x−1}
3
![Page 5: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/5.jpg)
Regular Variation
”Regularly varying functions are those functions which be-have asymptotically like power functions”.
limt→∞
U(tx)
U(t)= x−α
Heavy-tailed random variables have regularly varying tails.Example: Unit Frechet
limt→∞
1− F (tx)
1− F (t)= lim
t→∞
1− exp{−(tx)−1}1− exp{−(t)−1}
= limt→∞
(tx)−1 + O(t−2)
t−1 + O(t−2)
=1
x(e.g. 1− F (x) ∈ RV1)
If U ∈ RVα then U(x) = L(x)x−α.
4
![Page 6: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/6.jpg)
Multivariate EVT:
X1, X2, . . . iid F , X i = [Xi,1, . . . , Xi,d]T
� M t = [maxi=1,...,t Xi,1, . . . ,maxi=1,...,t Xi,d].
� A dist. G is max stable if P(
M t−btat
≤ x)
= P(X ≤ x) ⇔Gt(atx + bt) = G(x).
� F t(atx + bt)→ G(x)⇒ G is max stable [deHaan/Resnick, 1977].
� Q: Can the multivariate EVD be characterized?A: Yes, but not easily.
5
![Page 7: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/7.jpg)
Multivariate EVD’s: Characterization 1
WLOG, assume G is a d-dimensional EVD with unit Frechetmargins [Resnick, 1987]. Let E = [0,∞]d \ {0}. Then:
G(x) = exp(−
[∫ 1
0max
(w1
x1, . . . ,
wd
xd
)dH(w)
]),
whered∑
j=1
wj = 1,
and H is a finite measure on B = {x ∈ E : ||x|| = 1} suchthat ∫ 1
0wdH(w) = 1.
6
![Page 8: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/8.jpg)
Multivariate EVD’s: Characterization 2
Again, assume G is a d-dimensional EVD with unit Frechetmargins. Let E = [0,∞]d \ {0}. Then, there exists a non-homogeneous Poisson process on E with intensity measure
µ
{x ∈ E : ||x|| > r,
x
||x||∈ A
}=
H(A)
r,
such thatG(x) = exp (−µ{(0, x]c}) .
Idea: The value of G(x) depends only on the ”radius” (||x||)and the ”angular” ( x
||x||).
7
![Page 9: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/9.jpg)
Where does characterization 2 come from?
µ
{x ∈ E : ||x|| > r,
x
||x||∈ A
}=
H(A)
r,
Given a set B such that B ={x : x
||x|| ∈ A, ||x|| ∈ (a, b)}, mea-
sure is such that µ{B} = tµ{tB}.
8
![Page 10: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/10.jpg)
Where does characterization 2 come from?
Measure theory: Works for all ”shapes” B.
With such a measure, we get max stability:
Gt(tx) = [exp (−µ{(0, tx]c})]t = exp (−tµ{(0, tx]c}) = G(x)
9
![Page 11: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/11.jpg)
Multivariate regular variation
...n Kinda goes something like this n...
µx{tA} =P(x−1X ∈ tA)
P(||X|| > tx)
P(||X|| > tx)
P(||X|| > x)→ µ(A)t−α
� Still has angular idea
� Frechet case: α = 1
10
![Page 12: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/12.jpg)
Asymptotic Independence
Let (X1, X2) be a bivariate random variable with commonmarginals. Then X1 and X2 are asymptotically independentif
limu→∞
P{X1 > u|X2 > u} = 0.
Example: (X1, X2) bivariate normal with ρ < 1
Q: How best to handle the middle ground between asymp-totically dependent and complete independence?
11
![Page 13: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/13.jpg)
Example
12
![Page 14: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/14.jpg)
L&T: 1. Introduction
� Univariate review of threshold exceedance models (GPDand PP).
� Multivariate threshold models:
– point process model of de Haan (1985)
– model of Smith, Tawn, and Coles (1997)
� Asymptotic independence
� Problems with above approaches when components are(asymptotically) independent
13
![Page 15: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/15.jpg)
L&T: 2. Multivariate Threshold Model
� Equations 2.2 and 2.3 ??
� Equations 2.5 and 2.8
� Asymptotic independence case
14
![Page 16: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/16.jpg)
L&T: 3. Inference for the Model
� Develop censored likelihood (3.2)
� Assume a logistic dependence structure
� Develop score statistic under assumption of independence(3.4)
� Proposition 1
15
![Page 17: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/17.jpg)
L&T: 4. Variable Thresholds
Similar result holds
16
![Page 18: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/18.jpg)
L&T: 5. Inference for Asymptotic Independence
� Proposition 2
� η introduced
� Examples
� Estimation for η
17
![Page 19: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/19.jpg)
L&T: 6. Applications
� Simulations
� Two Examples (wave/surge and wind/rain)
18
![Page 20: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/20.jpg)
Dan’s simulated data
100 points of largest radius used, Xmin = min(X1, X2), GEVfit to Xmin, η = ξ
Normal (ρ = .8, Fr(1) Marg.) Logistic (α = .7)η .9 1η 0.684 0.889
SE 0.106 0.148
19
![Page 21: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/21.jpg)
L&T II
� From: P(Z1 > r, Z2 > r) ∼ L(r)r−1/η
To: P(Z1 > z1, Z2 > z2) ∼ L1(z1, z2)z−c11 z
−c22 + ...
where c1 + c2 = 1/η
� Make some assumptions, develop a likelihood
� Develop techniques for fitting a model (pretty cool results)
� Point process representation for asymptotic independence
20
![Page 22: ”Statistics for Near Independence in Multivariate Extremes” A ...ericg/ledford.pdfStatistics for Near Independence in Multivariate Extremes” A.W. Ledford and J.A. Tawn Discussion](https://reader033.vdocuments.net/reader033/viewer/2022052816/60a88f8681e4c42f671de324/html5/thumbnails/22.jpg)
References
de Haan, L. (1985). Extremes in higher dimensions: the model and some statistics.Proceedings 45th Session of the International Statistical Institute.
de Haan, L. and Resnick, S. (1977). Limit theory for multivariate sample extremes. Z.Wahrscheinlichkeitstheorie, 4:317–337.
Fougeres, A. (2004). Multivariate extremes. In B. Finkenstadt and H. Rootzen, editor,Extreme Values in Finance, Telecommunications and the Environment, pages 373–388. Chapman and Hall CRC Press, London.
Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions. Imperial College Press,London.
Mikosch, T. (2004). How to model multivariate extremes if one must?http://www.gloriamundi.org/picsresources/tm.pdf.
Resnick, S. (1987). Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.
Smith, R., Tawn, J., and Coles, S. (1997). Markov chain models for threshold ex-ceedances. Biometrika, 84:249–268.
21