Lecture 13 Flashcards
What if a marginal distribution isn’t Gaussian ?
Impossible define joint distribution:
- Case when 2 variables = different marginal distribution
- Case for large number of marginal distribution → multivariate does not exist
→ use copula models
What are copulas ?
Relate two marginal distributions instead 2 series directly:
- Able to relate any kind of margin
- Possible to generate non-linear dependence
→ Pearson’s correlation not appropriate measure of dependence
What is the base of a copula ?
- 2 rv X and Y with marginal distributions Fx = Pr[X ≤ x] and Gy = Pr[Y ≤ y]
- Cdfs continuous
- Joint distribution H(x,y)= Pr[X ≤ x, Y ≤ y]
- All function’s range = [0,1]
→ H might not exist
What is the definition of a bivariate copula ?
Function C : [0,1] x [0,1] → [0,1]
What are the properties of a bivariate copula ?
• C(u,v) increases in u and v : if one marginal = cst, the other one increases
• C(u,0) = 0, C(u,1) = u , C(0,v) = 0, C(1,v) = v
o If one probability = 0 then joint also
o If one probability = 1 then joint determined by remaining one
• Pr[u1 ≤ U ≤ u2, v1 ≤ V ≤ v2] = C(u2,v2) – C(u2,v1) – C(u1,v2) + C(u1,v1) ≥ 0
What is copula’s theorem ?
H = joint distribution of C and Y with marginal distribution F, G then :
• Exist copula C s.t. H(x,y) = C [F(x), G(y)]
→ if F and G continuous, C = unique
What are the various measures of dependence and concordance ?
- Dependence = strength of relation between 2 variables
- Association = positive or negative relation
- Concordance = association where small values of one imply small values of the other and same for big values → X1 < X2 → Y1 < Y2 = (X1 -X2) (Y1 -Y2) > 0
- Discordant = inverse of concordant → (X1 -X2) (Y1 -Y2) < 0
What is the measure of concordance between rv X and Y ? What are its properties ?
κ(x,y)
- Defined for every pair of rv = completeness
- Normalized measure -1≤ κ ≤1 and κ(x,x) =1 and κ(x,-x) = -1
- Symmetric : κ(y,x) = κ(x,y)
- If X and Y are independent → κ(x,y) =0
- κ(x,-y) = κ(-x,y) = - κ(x,y)
→ measure of concordance = invariant w.r. to linear increasing transformations
Is Pearson’s correlation a measure of concordance ?
Only under normality, and is a measure of association
What are Kendall’s Tau and Spearman’s rho ?
Measures of concordance
What does it mean when two series are comonotonic ?
κ(x,y) = 1
What does it mean when two series are counter-monotonic ?
κ(x,y) = -1
What is Kendall’s tau for 2 rv ?
Probability of concordance - probability of discordance of two independent pairs
What is Spearman’s rho ?
multiple of probability of concordance - probability of discordance of 2 independent pairs
What happens if X2 and Y3 are independent in Spearman’s rho ?
ρs = distance between joint distribution of (X,Y) and independence
How can Spearman’s rho also be viewed ?
Pearson’s correlation between F and G or ranks of X and Y
What is the Pearson’s correlation ?
Natural scalar of linear dependence in elliptical distributions → misleading measure of dependence in more general situation
What are Pearson’s correlation’s properties ?
• ρ[X,Y] = invariant under linear transformations only
• ρ[X,Y] = bounded: -1 ≤ ρL ≤ ρ[X,Y] ≤ ρU ≤ 1
o ρU = comonotonic and ρL = counter-monotonic
- ρ[X,Y] for comonotonic (counter-monotonic) can be different from 1 (-1)
- ρ[X,Y] = 0 does not imply independence between X and Y
What is the first approach to modeling of non-linear dependence ?
estimating unrestricted joint density non-parametrically
→ deduce non-parametric estimate of associated unrestricted copula
What are the advantages and disadvantages of empirical copulas ?
• Advantage
o Not require any additional assumption on non-linear-dependence
• Drawback
o Complicated interpretations of patterns of non-linear dependence
o Likely to provide inaccurate and erratic results
What are the special cases of Elliptical copula
- Normal distribution
- T distribution
- Cauchy distribution
- Laplace distribution
- Uniform distribution
What is the condition for an Elliptical copula ?
- Random vector X ϵ R^n has multivariate elliptical distribution if density∶f(x)=|Σ|^(-1/2) g[(X-μ) Σ^(-1) (X-μ)] for some g∶ R→R+where Σ=PD
- Contours of equal density form ellipsoids in R^n
What are Archimedean Copulas ?
Copulas that are not derived from multivariate distribution functions
What is the Archimedean’s theorem ?
φ = continuous, strictly decreasing function from [0,1] to [0,∞) s.t. φ(1) =0 and φ^(-1) = inverse of φ. Function fro [0,1]^2 to [0,1] : C(u,v) = φ^(-1) [φ(u) + φ(v)] = copula only if φ = convex
