By Roger B. Nelsen
The learn of copulas and their function in information is a brand new yet vigorously becoming box. during this ebook the scholar or practitioner of records and chance will locate discussions of the elemental homes of copulas and a few in their basic purposes. The functions contain the examine of dependence and measures of organization, and the development of households of bivariate distributions. This ebook is acceptable as a textual content or for self-study.
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Extra resources for An Introduction to Copulas (Springer Series in Statistics)
Let X and Y be continuous random variables. Then X and Y are independent if and only if CXY = P. Much of the usefulness of copulas in the study of nonparametric statistics derives from the fact that for strictly monotone transformations of the random variables, copulas are either invariant or change in predictable ways. Recall that if the distribution function of a random variable X is continuous, and if a is a strictly monotone function whose domain contains RanX, then the distribution function of the random variable a (X) is also continuous.
0, with margins F and G given by x < -1, Ï0, y < 0, Ï0, Ô F ( x ) = Ì( x + 1) 2 , x Œ[-1,1], and G ( y ) = Ì -y Ó1 - e , y ≥ 0. ÔÓ1, x > 1, Quasi-inverses of F and G are given by F ( -1) (u) = 2 u - 1 and G ( -1) (v) = - ln(1 - v ) for u,v in I. 3 Sklar’s Theorem 23 uv . 9. Gumbel’s bivariate exponential distribution (Gumbel 1960a). 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].
3. Sklar’s theorem. Let H be a joint distribution function with margins F and G. Then there exists a copula C such that for all x,y in R, H(x,y) = C(F(x),G(y)). 1) If F and G are continuous, then C is unique; otherwise, C is uniquely determined on RanF¥ RanG. 1) is a joint distribution function with margins F and G. Proof. 5. 4 is a copula. The converse is a matter of straightforward verification. 1) gives an expression for joint distribution functions in terms of a copula and two univariate distribution functions.
An Introduction to Copulas (Springer Series in Statistics) by Roger B. Nelsen