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Lemma B.2 Let C be a copula, and suppose C ( a, b ) = θ, where ( a, b ) is in (0 , 1) 2 and θ sat- isfies max ( a + b − 1 , 0) ≤ θ ≤ min ( a, b ). Then 

where C$_{U}$ and C$_{L}$ are the copulas given by 

where ( x ) + = max { x, 0 } . 

<img src='content_image/313798.jpg'>

<img src='content_image/313799.jpg'>

Lemma B.3 For fixed marginal distribution functions F$_{0}$ and F$_{1}$, sharp bounds on the joint distribution function F are given as follows: 

where 

<img src='content_image/313797.jpg'>

<img src='content_image/313796.jpg'>

