By Dorota Kurowicka; Roger Cooke
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Example text
Click a second time on ADD, and accept the default. Note that information on the uniform distribution is visible on-screen: the density function, the parameters, main quantiles and moments. • From the MODEL menu select Dependence, or simply click on the Dependence icon. This dependence input panel appears. From the dependence menu, select ADD NEW, and choose dependence tree. V 1 and V 2 are listed as available nodes. Double click on V 1, then on V 2. A tree is created with V 1 as root. The rank correlation between V 1 and V 2 is 0 by default.
N−1 . 5 (Conditional correlation) The conditional correlation of Y and Z given X ρYZ|X = ρ(Y |X, Z|X) = E(YZ | X) − E(Y | X)E(Z | X) σ (Y | X)σ (Z | X) is the product moment correlation computed with the conditional distribution of Y and Z given X. 29). 12 If a. X is distributed uniformly on the interval [0, 1], b. Y, Z are conditionally independent given X, c. Y |X and Z|X are distributed uniformly on [0, Xk ], k > 0, then |ρYZ|X − ρYZ;X | = which converges to 3 4 3k 2 (k − 1)2 ; 4(k 4 + 4k 2 + 3k + 1) as k → ∞.
B. The extremal correlation ρ = ρmin is attained if and only if X and Y are countermonotonic; similarly, the extremal correlation ρ = ρmax is attained if X and Y are comonotonic. From Hoeffding’s theorem, we get that ρmin < 0 < ρmax . There are examples for which this interval is very small (Embrechts et al. (2002)). 2 Let X = eZ , Y = eσ W , where Z, W are standard normal variables. Then X and Y are lognormally distributed, and: lim ρmin (X, Y ) = lim ρmax (X, Y ) = 0. σ →∞ σ →∞ Proof. From Hoeffding’s theorem ρmax , ρmin , are attained by taking W = Z, W = −Z respectively.