By Roger B. Nelsen

The examine of copulas and their function in information is a brand new yet vigorously starting to be box. during this publication the coed or practitioner of data and chance will locate discussions of the elemental houses of copulas and a few in their fundamental functions. The purposes contain the examine of dependence and measures of organization, and the development of households of bivariate distributions. This ebook is appropriate as a textual content or for self-study.

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2). The proof in the n-dimensional case, however, is somewhat more involved [Moore and Spruill (1975), Deheuvels (1978), Sklar (1996)]. 10. Let H. C, Fj, Pi,"', F,. 9, and let FJ(-l), Fi(-l) , ... , respectively. (-l\u n )). 11. Let XI' X 2 ,'" , Xn be random variables with distribution functions FJ, F2, ... ,F,.. respectively, and joint distribution function H. 6) holds. If FJ, Fi,"', F,. are all continuous, C is unique. Otherwise, C is uniquely determined on Ran FJxRan Fix· .. xRan F,.. 8) nn(u) = u1u2 ..

Hence the survival copulas for the Marshall-Olkin bivariate exponential distribution yield a two parameter family of copulas given by . Ca f3(u, v) = mm(u , I-a v, uv 1-13 )= v, a> 13 u - v , uv l - f3 , u a ~ vf3. 3) This family is known both as the Marshall-Olkin family and the Generalized Cuadras-Auge family. , the case in which X and Yare exchangeable. 1) and indeed, Ca,o = CO,f3 = nand Cl,l = M. It is interesting to note that, although the copulas in this family have full support (for 0 < a,/3 < 1), they are neither absolutely continuous nor singular, I ut rather have both absolutely continuous and singular components Aa ,f3 and Sa,f3' respectively.

A singular bivariate distribution whose margins are Cauchy distributions. 5) with standard Cauchy margins: F(x) = I12 + (arctanx)/tr for all real x; and similarly for G(y). 4). However, the support of H is the image of the square lu -l/21 + -l/21 = 1/2 under the transformation u =F(x), v =G(y). This yields IXYI = I, Iv so that the support of this bivariate distribution consists of the four branches of the two rectangular hyperbolas xy = + I and X)' = -I. 2. A singular bivariate distribution whose margins are normal.