MEASURES OF DEPENDENCE

1. MEASURES OF DEPENDENCE Application of copulas

2. 1. Introduction (cont.) • Linear Correlation: • -1≤≤1 • X and Yindependent  (X,Y) = 0. • (aX + b, cY + d) = sgn(ac)(X,Y)

3. 1. Introduction • Correlation one of the most widely used, and perhaps misunderstood, concept in statistics. • Advantages of linear correlation: • Straightforward to calculate • Easy to manipulate under linear operations • Natural measure of dependence in multivariate elliptical distribution

4. 1. Introduction (cont.) • Fallacies about linear correlation: • Marginal distributions and correlation coefficients determine the joint distribution. • Given marginals F1 and F2 all linear correlations between -1 and 1 are realizable.

5. 1. Introduction (cont.) • Shortcomings: • The variances of X and Y must be finite. Not ideal for heavy tailed distributions. • Zero correlation does not imply stochastic independence, except for the case of elliptic distributions. • Not invariant under nonlinear strictly monotonic transformations.

6. 1. Introduction (cont.) • Given two distributions F1 and F2, the attainable correlation is given by the following theorem: • Let(X,Y) a random variable with marginals F1 and F2 and unspecified dependence structure with finite variances. Then • The set of all possible correlations is a closed interval [min, max] and for the extremal correlations min<0< max. • minis attainable iff X and Y are perfectly negatively dependent (countermonotonic); maxis attainable iff X and Y are perfectly positively dependent (comonotonic). • min = -1if X and –Y are of the same type; max = 1, if X and Y are of the same type.

7. 1. Introduction (cont.) • Remark: two random variables are said to be of the same type if there exists  > 0 and  in R, such that Y =d X + . Conclusion: Need alternative measures of functional dependence.

8. 2. Measures of dependence. • Definition 2.1. (concordant, discordant)Observations (x1, y1) and (x2, y2) are concordant if (x1 – x2)(y1 – y2) > 0. They are discordant if (x1 – x2)(y1 – y2) < 0. • Definition 2.2. (Spearman’s rank correlation) Let X and Y be random variables with distributions F1 and F2 and joint distribution F. The population Spearman’s rank correlation is given by s = (F1(X), F2(Y)), where  is the usual linear correlation.

9. 2. Measures of dependence. • The sample estimator of the Spearman’s rank correlation is given by

10. 2. Measures of dependence. • Example 2.1.

11. 2. Measures of dependence. • Example 2.2.

12. 2. Measures of dependence. • Definition 2.3. Let {(Xi, Yi)} 1≤ i ≤ n be a random sample from a population (X, Y) with cdf F(x, y). There are nC2 distinct pairs of (Xi, Yi) of observations in the sample, and each pair is either concordant or discordant. Let c be the number of concordant pairs and d be the number of discordant pairs. Then an estimate of Kendall’s rank correlation for the sample is given by t = (c – d) / (c + d). The population version of Kendall’s rank correlation is given by

13. 2. Measures of dependence. • Note: the generalization of s and  to n > 2 is similar to the procedure for linear correlation. • Theorem 2.1. Let X and Y be random variables with cdf’s F1 and F2, and joint cdf F. Let  denote either s or . The following hold: •  is symmetric. • If X and Y are independent then  = 0. • -1 ≤  ≤ 1. • If T is strictly monotonic on the range of X, then

14. 2. Measures of dependence. • Advantages of these measures: • Invariant under monotonic transformations. • Perfect dependence corresponds to measures 1;  = 1 iff Y = T(X) for some monotone increasing T. • Robust against outliers. • Disadvantages: • Not easy to manipulate; not moment based correlations.

15. 3. The role of copulas. • Theorem 3.1. let X and Y be random variables with cdf’s F1(x) and F2(y) and joint cdf F(x, y). Let C(u, v) be the copula of X and Y. Then we have the following alternative formulas for Spearman’s and Kendall’s correlations respectively:

16. 3. The role of copulas. • Proof: we will use the following identity due to Hoeffding:

17. 3. The role of copulas. • Proof (cont.)the proof for Kendall’s identity will be presented in a more general setting. • Note: the proof of the last part of theorem 2.1 now follows easily from this theorem.

18. 3. The role of copulas. • Example3.1. Farlie-Gumbel-Morgenstern Family of copulas. note that   [-2/9, 2/9]. Limited range of dependence!

19. 3. The role of copulas. • Example 3.2. The Clayton-Cook-Johnson Family

20. 3. The role of copulas. • Theorem 3.2. Let (X1, Y1), (X2, Y2) be random vectors with joint cdf’s H1, and H2 and common marginals F (of X1 and X2) and G (of Y1 and Y2); let C1 and C2 denote the copulas of (X1, Y1) and (X2, Y2) respectively. Let K be defined by: K = P[(X1 – X2)(Y1 – Y2) > 0] - P [(X1 – X2)(Y1 – Y2) < 0]then:

21. 3. The role of copulas. Proof:

22. 3. The role of copulas. Proof (cont.) The result follows.

23. 3. The role of copulas. • Corollary 3.2. Under the hypothesis of theorem 3.1 the following hold: • K is symmetric in its arguments: K(C1, C2) = K(C2, C1) • K is non-decreasing in each argument: • Copulas can be replaced by survival copulas in K, i.e.

24. 3. The role of copulas. • Remark 3.2.1. If C has a singular component the formula for  can not be computed. For such copulas the following expression is used:This is the consequence of the following theorem (Li et al., 2002): • Theorem 3.3. Let C1 and C2 be copulas. Then

25. 3. The role of copulas. • Example 3.3. Recall M(u, v) = Min(u, v), W(u, v) = Max(u + v – 1, 0), (u, v) = uv. The support of M and W are the first and the second diagonals of I2. If g(u, v) is an integrable function with domain I2, then

26. 3. The role of copulas. • Example 3.3 cont.It follows that:

27. 3. The role of copulas. • If C is an Archimedean copula then the Kendall’stau can be evaluated directly using the generator. But first recall the theorem from last time:Let C be an Archimedean copula generated by . Then the distribution function of the random variable C(U, V) is given by

28. 3. The role of copulas. • Theorem 3.4. Let X and Y be random variables with an Archimedean copula C generated by . Then the Kendall’s tau of X and Y is given by:Proof: see Nelsen.

29. 3. The role of copulas. • Example 3.4. For the Gumbel family (t) = (-lnt). Therefore

30. 4. Tail dependence (Dr. Lin’s request). • Definition 4.1. Let X and Y be random variables with cdf’s F1 and F2. The coefficient of upper (lower) tail dependence of X and Y is

31. 4. Tail dependence • Can we use copulas to understand tail dependence? Recall the definition of survivor copula: C*(u, v) = u + v – 1 + C(1 – u, 1 – v). We now have:

32. 4. Tail dependence Therefore we get for the upper tail dependence: Likewise for the lower tail dependence we get:

33. 4. Tail dependence • Example 4.1. Gumbel copula:C(u, v) = exp[-{(-log(u)) + (-log(u)) }1/], u = 2 – 21/, l = 0.We see that for  > 1, C has tail dependence.

34. 4. Tail dependence • Example 4.2. Gaussian family of copulas.We need a different formula for u. The following can be derived from the definition of conditional probability and therefore the proof is not presented.

35. 4. Tail dependence • Example 4.2. (cont.)It can now be shown that Now if X and Y have bivariate normal distribution with correlation , it can be shown that u = 0. So the Gaussian copula does not have upper tail dependence.

36. 4. Tail dependence • Theorem 4.1.Let C be a strict Archimedean copula. If -1’(0) is finite, then C(u, v) does not have upper tail dependence. If C has upper tail dependence, then -1’(0) = - , and the coefficient of upper tail dependence is given by • Theorem 4.2. let  be strict generator. The coefficient of lower tail dependence for the copula C is given by

37. 4. Tail dependence • Example 4.3. Consider the Clayton family.C(u,v) = (u + v - 1)-1/. Then l = 2-1/;   0. • Example 4.4. It can be shown that the Frank family does not have any tail dependence.