Department of Mathematics ETH Zurich Switzerland ... › ... › Neslehova_presentation.pdf · Fallacies Dependence Concordance Measures Problems & References Appendix Motivation

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



Outline

The “continuous” vs. “non-continuous” case: typical fallacies


Concordance measures for non-continuous random variables

Problems & References

Some Theory Behind




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




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




Pathological Example

1/3 p 2/3

1/3

2/3

q




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




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




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




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




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.




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




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




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




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.




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 .




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




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



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).




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




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 ).



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Department of Mathematics ETH Zurich Switzerland ... › ... › Neslehova_presentation.pdf · Fallacies Dependence Concordance Measures Problems & References Appendix Motivation