Fallacies Dependence Concordance Measures Problems & References Appendix
Dependence of non-continuous random variables
Johanna Neslehova
Department of MathematicsETH ZurichSwitzerland
www.math.ethz.ch/~johanna
August 31, 2005
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Outline
The “continuous” vs. “non-continuous” case: typical fallacies
Dependence of non-continuous random variables
Concordance measures for non-continuous random variables
Problems & References
Some Theory Behind
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Motivation
Consider
• Two possibly dependent loss types and
Ni (T ) = number of losses of type i within the year T
• Dependence structures of the pair of non continuous randomvariables
(N1(T ),N2(T ))
Other examples
• Counting random variables like claim/loss frequencies ornumber of defaults in a portfolio
• Variables with jumps like losses censored by a certain threshold
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Distributions with continuous marginals
• Notation: (X ,Y ) ∼ F with marginals F1 and F2
F1 and F2 continuous
1. There exists a unique copula C such thatF (x , y) = C(F1(x),F2(y)) (Sklar’s theorem)
2. Modeling of the marginals and the copula can be doneseparately
3. C captures dependence properties which are invariat under a.s.strictly increasing transformations of the marginals
4. Scale and translation invariant measures of dependence arefunctions of C alone
➠ C is the (scale and location invariant) dependence structure
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Pathological Example
1/3 p 2/3
1/3
2/3
q
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Pathological Example
1/3 p 2/3
1/3
2/3
q
• (p, q) ∈ [0, 1/3] × [0, 1/3]:perfect positive dependence
• (p, q) = (1/√
3, 1/√
3):independence
• (p, q) ∈ [2/3, 1] × [2/3, 1]:perfect negative dependence
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Further Pitfalls
• The copula is not unique; i.e. several possible copulas exist
• The dependence structures of F are generally not the same asthe dependence structures of the possible copulas
F (x , y) ≤ F ∗(x , y) ∀x , y ∈ R 6⇒ C(u, v) ≤ C∗(u, v) ∀u, v ∈ [0, 1]
• Possible copulas remain invariant under strictly increasing andcontinuous transformations, but do not necessarily change inthe same way as the unique copula in the “continuous” case ifat least one of the transformations is decreasing
• Weak convergence of Fn does not imply the point-wiseconvergence of the corresponding possible copulas
• Any measure of association which depends only on the copulaof F is a constant
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Kendall’s Tau and Spearman’s rho in the
”Non-Continuous” Case?
”Naive” Idea: work with some of the non unique fitting copula
But: fitting copulas can differ considerably.
• For independent Bernoulli random variables X and YI τ(X ,Y ) ∈ [−3/4, 3/4]I ρ(X ,Y ) ∈ [−13/16, 13/16]
• for comonotonic Bernoulli random variables X and Y
I τ(X ,Y ) ∈ [0, 1]I ρ(X ,Y ) ∈ [1/2, 1]
Because: Difference between the probability of concordance anddiscordance 6= 4
∫C(u, v) dC∗(u, v) − 1
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Some Starting Points
• Investigation of the family of possible copulas correspondingto a fixed bivariate distribution
I In particular search for a suitable extension strategy whichwould produce a copula capturing the dependence structuresof the joint distribution function
• Investigation of possible bivariate distributions obtained froma fixed copula and marginals which follow some specifieddistribution (up to parameters), such as Poisson, binomial orBernoulli
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Extension Strategy: Examples
x
0.8
0.8
0.6
0.4
0.2
0.60.40.2
Subcopula for Bernoulli marginals:P[X = 0] = 0.4, P[Y = 0] = 0.3
P[X = 0,Y = 0] = 0.2.
Extension by a Gauss copula with
parameter % = 0.6.
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
The ”Standard” Extension Strategy
C(0.4, 0.4) = 0
0.40.2
0.6
0.60
x
0.8
0.8
0.2
0.4
y
1
10
C(0.4, 0.4) = 0.16
0.6
0.4
0.6
y
10.4
1
0.2
x
0.8
00.80.20
C(0.4, 0.4) = 0.3
0
0.8
10.8
y
x
1
0.4
0
0.6
0.6
0.2
0.40.2
The Standard Extension CS
• The standard extension copula of Schweizer and Sklar
• CS corresponds to the linear interpolation of the uniquesubcopula as well as the unique copula of the smoothed jdf
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Some Selected Properties of CS
Pros
• X and Y are independent if and only if CS is theindependence copula
• (X ∗,Y ∗) more concordant than (X ,Y ) if and only ifCS(u, v) ≤ C∗
S(u, v) for all u, v ∈ [0, 1]
• CS reacts on monotone transformations of the marginals asthe unique copula in the “continuous” case
Cons
• If X and Y are perfect monotonic dependent, CS does notcoincide with the Frechet-Hoeffding bounds
• Weak convergence does not imply the point-wise convergenceof the CS ’s
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Towards Kendall’s Tau and Spearman’s Rho
Difference between the probabilities of concordance anddiscordance equals 4
∫CS(u, v) dCS
∗(u, v) − 1
Consider
• Kendall’s tau: τ(CS ) = 4
∫ 1
0
∫ 1
0CS(u, v) dCS(u, v) − 1
• Spearman’s rho: ρ(CS ) = 12
∫ 1
0
∫ 1
0[CS(u, v) − uv ] du dv
However
• τ and ρ do not reach the bounds 1 and −1
• Exact bounds of τ and ρ are complicated and do not have thesame absolute value
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Exact Bounds for Kendall’s Tau and Spearman’s Rho
-0.5
0.6
-1
0.40.20
1.5
x
1
0.5
0.80
|τ (MS)/τ (WS )| for Binomial
distributions F1 = B(n, 0.4) and
F2 = B(n, x) with n =1, 4 and 10.
-0.5
0.6
-1
0.40.20
1.5
x
1
0.5
0.80
|ρ(MS)/ρ(WS )| for Binomial
distributions F1 = B(n, 0.4) and
F2 = B(n, x) with n =1, 4 and 10.
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Less Sharp Bounds for Kendall’s Tau and Spearman’s Rho
1. |τ(X ,Y )| ≤√
1 − E ∆F1(X )√
1 − E∆F2(Y ). The boundsare attained if Y = T (X ) a.s. where T is a strictly monotoneand continuous transformation on the range of X .
2. |ρ(X ,Y )| ≤√
1 − E ∆F1(X )2√
1 − E ∆F2(Y )2. The boundsare attained if Y = T (X ) a.s. where T is a strictly monotoneand continuous transformation on the range of X .
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Kendall’s Tau and Spearman’s Rho for Non-Continuous
Random Variables
τ(X ,Y ) =
41∫0
1∫0
CS(u, v) dCS(u, v) − 1
√[1 − E ∆F1(X )][1 − E∆F2(Y )]
ρ(X ,Y ) =
12
(1∫0
1∫0
(CS (u, v) − uv
)du dv
)
√[1 − E ∆F1(X )2][1 − E∆F2(Y )2]
• τ and ρ satisfy (modified) axioms of concordance measures
• Bounds are attained if Xa.s.= T (Y ) for T continuous and
strictly monotone
• τ and ρ are the sample versions for empirical distributions
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
τ , ρ and % for Binomial Distributions
0.2
0.60.4
0.4
x
0.20
0.6
1
0.8 1
0.8
F1 = B(2, 0.4), F2 = B(2, x).
0.4
0.6
0.2
0 0.40.2
x
10.8
0.8
0.6
1
F1 = B(4, 0.4), F2 = B(4, x).
0.6
0.6
0.4
0.2
0.40.20
x
0.8
0.8 1
1
F1 = B(10, 0.4), F2 = B(10, x).
-0.8
0.6 0.8
-1
0.4
-0.4
x
0.2
-0.6
0
-0.2
1
F1 = B(2, 0.4), F2 = B(2, x).
-0.6
0.6
-0.2
0.8
-0.4
-0.8
0 0.2
x
0.4 1
-1
F1 = B(4, 0.4), F2 = B(4, x).
-0.4
0.6
-0.6
-0.8
0.4
-1
0.20
-0.2
x
0.8 1
F1 = B(10, 0.4), F2 = B(10, x).J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Some Problems
• Modeling: dependence properties of a familyH = {C(F ,G ) : F ∈ F ,G ∈ G}
q
Ken
dall’
s ta
u
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
q
Ken
dall’
s ta
u
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
q
Ken
dall’
s ta
u
0 100 200 300
0.0
0.1
0.2
0.3
0.4
0.5
Kendall’s tau for binomial marginals and a Gauss copula (solidline), Frank copula (short-dashed line), Gumbel copula (dotted
line), and Frechet copula (long-dashed line).
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
References
M. Denuit and P. Lambert, 2005Constraints on concordance measures in bivariate discrete data
W. Hoeffding, 1940Maßstabinvariante Korrelationstheorie fur diskontinuierelicheVerteilungen
A. Marshall, 1996Copulas, marginals and joint distributions
J. Neslehova, 2004Dependence of non-continous random variables
J. Neslehova ETH Zurich
Dependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References Appendix
Some Theory Behind: Dependence Structure in the
Non-Continous Case
In the continuous case: C is the cdf. of the transformed vector(F1(X ),F2(Y )).Idea: In the non-continuous case, use a different transformation ofthe marginals:
ψ(x , u) := P[X < x ] + uP[X = x ] = F (x−) + u∆F (x)
Result: for a random vector (U,V ) with uniform marginals whichis independent of (X ,Y ), (ψ(X ,U), ψ(Y ,V )) has uniformmarginals and the corresponding unique copula is a possible copulaof (X ,Y ).Moreover: different dependence structure of (U,V ) leads todifferent possible copulas of (X ,Y ).
J. Neslehova ETH Zurich
Dependence of non-continuous random variables