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The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
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In a second workflow supporting the same use case, you’ll see:
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https://alandix.com/academic/papers/synergy2024-epistemic/
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Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
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FIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdf
Slides astin
1. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures
XXXIIIrd International ASTIN Colloquium
Tail distribution and dependence measures
Arthur Charpentier, ACIA
2. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 2
• Clayton’s copulas
C(u,v,θ)=(u-θ+v-θ -1)-1/θ pour θ ≥ 1
A short introduction to copulas
3. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 3
Sklar’s Theorem (1959)
• Let (X,Y) be a pair of random variables, with joint c.d.f.
FXY(x,y) = P(X≤x,Y≤y) and let FX(x) = P(X≤x) , FY(y) = P(Y≤y).
• There is a copula C such that FXY(x,y) = C(FX(x),FY(y))
• Conversely, C satisfies C(u,v) = FXY(FX
-1(u),FY
-1(v))
where -1 denotes the generalized inverse, g-1(t)=inf{s,g(s)=t}.
A short introduction to copulas
4. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 4
A short introduction to copulas
Archimedian copulas
• Let φ:[0,1]→[0,+∞] be a convex strictly decreaasing function,
such that φ(1)=0.
C(u,v)= φ-¹(φ(u)+ φ(v)) for any 0≤u,v ≤1
is a copula, and f is called the generator of the copula.
• Ex φ(t)=t-θ-1 Clayton’s copulas
φ(t)=exp(-t1/θ) Gumbel’s copulas
φ(t)=-log[(e -θt-1)/(e -θ-1)] Frank’s copulas
5. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 5
A short introduction to copulas
Survival copulas
• Let (X,Y) be a random pair with copula C, and U=FX(x), V=FY(y)
• C is the c.d.f. of (U,V), then, the c.d.f. of (1-U,1-V) is C * defined
as C*(u,v)=u+v-1+C(1-u,1-v)
• C * satisfies
• C * is called survival copula, or dual copula
( ) ( ) ( ) ( )( )yF,xFCy,xFyY,xXP YX
*
XY ==>>
7. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 7
The lower tail conditional copula
Lower tail conditional copula
• Let (X,Y) be a pair of random variables with copula C, and let
U=FX(X) et V=FY(Y), such that C(x,y) = P(U ≤ x,V ≤ y)
• The lower tail conditional copula with threshold u,v is the
copula of (U,V) given U≤u and V≤v, and is denoted Φ(C,u,v)
• Given u,v in [0,1], Φ(C,u,v) is the copula of
(X,Y) given X ≤ VaR(X,u) and Y ≤ VaR(Y,v)
8. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 8
The lower tail conditional copula
9. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 9
The lower tail conditional copula
Lower tail conditional copula
• Clayton’s copula Φ(C,u,v) = C for any u,v in ]0,1]
Clayton’s copula is invariant by truncature
• Marshall-Olkin’s copula Cα,β(x,y)=min{x1-αy,xy1-β}
Φ(Cα,β, uβ,uα) = Cα,β
• Gumbel-Barnett’s copulas Cθ(x,y)=xyexp[-θlog(x)log(y)]
Φ(Cθ,u,v) = Cθ(u,v) where θ(u,v)=θ/[1+θlog(u)][1+θlog(v)]
10. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 10
The lower tail conditional copula
Lower tail conditional copula
• The joint c.d.f. of (U,V) given U≤u and V≤v is
• The marginal c.d.f.’s of (U,V) given U≤u and V≤v are
• The copula of (U,V) given U≤u and V≤v is (Sklar’s Theorem)
( ) ( ) ( )
( )v,uC
y,xC
vV,uUvV,xUPy,x)v,u,C(F =≤≤≤≤=
( ) ( ) ( )
( )v,uC
v,xC
vV,uUxUPx)v,u,C(FU =≤≤≤=
( )( ) ( ) ( )( )
( )v,uC
y)v,u,C(F,x)v,u,C(FC
y,xv,u,C
1
V
1
U
−−
=Φ
11. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 11
The upper tail conditional copula
Upper tail conditional copula
• Given u,v in [0,1], Φ(C*,1-u,1-v) is the copula of
(X,Y) given X > VaR(X,u) and Y > VaR(Y,v)
where C* is the survival copula of (X,Y)
• Using this duality property, studying upper tail dependence
could be done with the lower tail conditional copula Φ(C,u,v)
• These copulas could be extended to higher dimension
12. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 12
The lower tail conditional copula
Lower tail conditional copula
• Properties of the conditional copula : Dependence and tail
distribution (IME 2003), especially limiting results, when u,v
converge towards 0
• Archimedian copulas are stable : the conditional copula
obtained from an Archimedian copula, with an other generator
The generator of Φ(C,u,v) is φ(C(u,v)t) - φ(C(u,v))
13. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 13
The lower tail conditional copula
Application in Credit Risk models
• Let X and Y denote two times of defaults, and let C* denote
the survival copula at time 0
• Assume that no default occurred at time t>0, then, the copula
of the times of default is not C* : it is the copula of (X,Y) given
X>t and Y>t
• Ex : survival time are exponential (means µ and 2µ), and C* is a
Gumbel copula
14. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 14
The lower tail conditional copula
15. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 15
The lower tail conditional copula
16. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 16
Functional measures and conditional copulas
Dependence measures based on copula functions
• Kendall’s tau
• Spearman’s rho – rank correlation
could be extended to any other dependence measure
• Gini’s gamma or Blomqvist’s beta
• Lp distance of Schweizer and Wolf
17. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 17
Functional measures and conditional copulas
Lower / upper tail rank correlation
• Lower tail rank correlation defined as Sperman’s rho of (X,Y)
given X ≤ VaR(X,u) and Y ≤ VaR(Y,u), 0 ≤ u ≤ 1.
• Upper tail rank correlation defined as Sperman’s rho of (X,Y)
given X > VaR(X,u) and Y > VaR(Y,u), 0 ≤ u ≤1
( ) ( )( ) 3dxdyy,xu,u,C12u
]1,0[x]1,0[
−Φ=ρ
∫∫
( ) ( )( ) 3dxdyy,xu1,u1,C12u
]1,0[x]1,0[
*
−−−Φ=ρ
∫∫
18. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 18
Functional measures and conditional copulas
Other functional measures of dependence
• Upper tail Kendall’s tau defined as Kendall’s tau of (X,Y) given
X > VaR(X,u) and Y > VaR(Y,u), 0 ≤ u ≤1
( ) ( ) ( ) 1)y,x(u1,u1,Cd)y,x(u1,u1,C4u
]1,0[x]1,0[
**
−−−Φ−−Φ=τ
∫∫
19. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 19
Functional measures and conditional copulas
Other functional measures of dependence
• The rank correlation of (X,Y) given X ≤ VaR(X,u) is
• Or, more generally, (X,Y) given X < h and Y < h,
( ) ( )( ) 3dxdyy,x1,u,C12u
]1,0[x]1,0[
−Φ=ρ
∫∫
( ) ( ) ( )( )( ) 3dxdyy,xhF,hF,C12u
]1,0[x]1,0[
YX −Φ=ρ
∫∫
20. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 20
Functional measures and conditional copulas
21. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 21
Functional measures and conditional copulas
22. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 22
Functional measures and conditional copulas
23. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 23
Functional measures and conditional copulas
24. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 24
Functional measures and conditional copulas
25. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 25
Functional measures and conditional copulas
26. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 26
Functional measures and conditional copulas
27. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 27
Functional measures and conditional copulas
28. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 28
Functional measures and conditional copulas
• Clayton’s copula
29. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 29
Functional measures and conditional copulas
• Survival Clayton’s copula
30. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 30
Functional measures and conditional copulas
• Gaussian copula
31. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 31
Functional measures and conditional copulas
• Gumbel copula
32. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 32
Functional measures and conditional copulas
• Student copula
33. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 33
Functional measures and conditional copulas
0.010.600.000.100.510.17
0.030.600.000.120.510.20
0.110.600.000.170.510.25
0.270.600.120.260.510.33
0.600.600.600.600.600.60ρ(X,Y)
Frank
Survival
Clayton
Clayton
Survival
Gumbel
GumbelGaussian
( )%50ρ
( )%75ρ
( )%90ρ
( )%95ρ
34. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 34
Application : Loss-ALAE
• Dataset used in Frees and Valdez (1997)
• (Loss,ALAE) given LOSS>VaR(Loss,u), ALAE>VaR(ALAE,u)
• (Loss,ALAE) given LOSS>VaR(Loss,u)
35. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 35
Application : Loss-ALAE
36. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 36
Application : Loss-ALAE
37. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 37
Application : Loss-ALAE
38. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 38
Application : Loss-ALAE
39. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 39
Application : Loss-ALAE
40. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 40
Application : Motor-Household
• Dataset from Belguise,Levi (2003)
• « the best candidate is the HRT copula » which is the
survival/dual Clayton copula
41. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 41
Application : Motor-Household
42. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 42
Application : Motor-Household
43. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 43
Application : Motor-Household
44. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 44
• Dataset from the Society of Actuaries
• CEBRIAN, A., DENUIT, M. and P. LAMBERT, Analysis of
bivariate tail dependence using extreme value copulas: an
application to the SOA medical large claims database (2003)
Application : Group Medical Large Claims
45. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 45
Application : Group Medical Large Claims
46. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 46
Application : Group Medical Large Claims
47. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 47
Application : Group Medical Large Claims