By Roger B. Nelsen
The research of copulas and their position in statistics is a brand new yet vigorously turning out to be box. during this ebook the scholar or practitioner of statistics and chance will locate discussions of the basic houses of copulas and a few in their basic functions. The purposes contain the examine of dependence and measures of organization, and the development of households of bivariate distributions. This publication is appropriate as a textual content or for self-study.
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Additional info for An Introduction to Copulas (Springer Series in Statistics)
1) H ( x , y ) = Cˆ ( F ( x ),G ( y )) . 4). We refer to Cˆ as the survival copula of X and Y. 6 Survival Copulas 33 joint survival function to its univariate margins in a manner completely analogous to the way in which a copula connects the joint distribution function to its margins. Care should be taken not to confuse the survival copula Cˆ with the joint survival function C for two uniform (0,1) random variables whose joint distribution function is the copula C. Note that C (u,v) = P[U > u ,V > v ] = 1 – u – v + C(u,v) = Cˆ (1 – u,1 – v).
15. The bivariate normal distribution with parameters m x , m y , s x2 , s y2 , and r is radially symmetric about the point ( m x , m y ) . The proof is straightforward (but tedious)—evaluate double integrals of the joint density over the shaded regions in Fig. 7(a). 7 Symmetry 37 H(a–x,b–y) (a–x,b–y) 1–v (a,b) (a+x,b+y) v H(a+x,b+y) 1–u (a) u (b) Fig. 7. 16. The bivariate normal is a member of the family of elliptically contoured distributions. The densities for such distributions have contours that are concentric ellipses with constant eccentricity.
Let Hq be the joint distribution function given by C ( u, v) = Ï1 - e - x - e - y + e - ( x + y +qxy ) , x ≥ 0, y ≥ 0, Hq ( x , y ) = Ì otherwise; Ó0, where q is a parameter in [0,1]. Then the marginal distribution functions are exponentials, with quasi-inverses F ( -1) (u) = - ln(1 - u ) and G ( -1) (v) = - ln(1 - v ) for u,v in I. Hence the corresponding copula is Cq ( u , v ) = u + v - 1 + (1 - u )(1 - v )e -q ln(1- u ) ln(1- v ) . 10. It is an exercise in many mathematical statistics texts to find an example of a bivariate distribution with standard normal margins that is not the standard bivariate normal with parameters m x = m y = 0, s x2 = s y2 = 1, and Pearson’s product-moment correlation coefficient r.