By Roger B. Nelsen

ISBN-10: 0387286594

ISBN-13: 9780387286594

ISBN-10: 0387286780

ISBN-13: 9780387286785

ISBN-10: 1441921095

ISBN-13: 9781441921093

The learn of copulas and their function in statistics is a brand new yet vigorously turning out to be box. during this ebook the scholar or practitioner of facts and likelihood will locate discussions of the basic houses of copulas and a few in their fundamental purposes. The purposes contain the research of dependence and measures of organization, and the development of households of bivariate distributions. This e-book 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)**

**Example text**

4. 12. If C ¢ is any n-subcopula, then for every u in Dom C ¢ , W n ( u) £ C ¢( u) £ M n ( u) . 13. For any n ≥ 3 and any u in In , there exists an ncopula C (which depends on u) such that C ( u) = W n ( u) . Proof (Sklar 1998). Let u = ( u1 , u 2 ,L , u n ) be a (fixed) point in In other than 0 = (0,0,…,0) or 1 = (1,1,…,1). There are two cases to consider. 1. Suppose 0 < u1 + u 2 + L + u n £ n – 1. Consider the set of points v where each v k is 0, 1, or tk = = ( v1 , v 2 ,L , v n ) min{( n - 1) u k ( u1 + u 2 + L + u n ) ,1} .

4 Copulas and Random Variables 29 (b) when q = 1, we have Gumbel’s bivariate logistic distribution; (c) when q = 0, X and Y are independent; and (d) the copula of X and Y is given by Cq ( u , v ) = uv . 3) This is the Ali-Mikhail-Haq family of copulas (Hutchinson and Lai 1990), which we will encounter again in Chapters 3 and 4. 15 Let X1 and Y1 be random variables with continuous distribution functions F1 and G1, respectively, and copula C. Let F2 and G2 be another pair of continuous distribution functions, and set X2 = F2( -1) ( F1 ( X1)) and Y2 = G2( -1) ( G1( Y1 )).

The following bivariate exponential distribution, first described by Marshall and Olkin (1967a,b), plays a similar role in a two-dimensional Poisson process. Consider a two-component system—such as a two engine aircraft, or a desktop computer with both a CPU (central processing unit) and a coprocessor. The components are subject to “shocks,” which are always “fatal” to one or both of the components. For example, one of the two aircraft engines may fail, or a massive explosion could destroy both engines simultaneously; or the CPU or the co-processor could fail, or a power surge could eliminate both simultaneously.

### An Introduction to Copulas (Springer Series in Statistics) by Roger B. Nelsen

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