Download PDF by Roger B. Nelsen: An Introduction to Copulas

By Roger B. Nelsen

ISBN-10: 0387986235

ISBN-13: 9780387986234

ISBN-10: 1475730764

ISBN-13: 9781475730760

Copulas are services that subscribe to multivariate distribution capabilities to their one-dimensional margins. The research of copulas and their function in information is a brand new yet vigorously transforming into box. during this ebook the scholar or practitioner of records and chance will locate discussions of the basic homes of copulas and a few in their fundamental functions. The functions contain the research of dependence and measures of organization, and the development of households of bivariate distributions. With approximately 100 examples and over a hundred and fifty routines, this booklet is appropriate as a textual content or for self-study. the one prerequisite is an top point undergraduate direction in chance and mathematical records, even supposing a few familiarity with nonparametric information will be helpful. wisdom of measure-theoretic chance isn't really required. Roger B. Nelsen is Professor of arithmetic at Lewis & Clark collage in Portland, Oregon. he's additionally the writer of "Proofs with no phrases: routines in visible Thinking," released via the Mathematical organization of the US.

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Extra resources for An Introduction to Copulas

Example text

6; or equivalently, "P(-l)(t) = P-I)(l-t). We will now illustrate this procedure to find the copulas for the MarshallOlkin system of bivariate exponential distributions; and for the uniform distribution on a circle. 1 The Marshall-Olkin Bivariate Exponential Distribution The univariate exponential distribution plays a central role in mathematical statistics since it is the distribution of waiting time in a standard Poisson process. The following bivariate exponential distribution, first described by Marshall and Olkin (l967a,b), plays a similar role in a two-dimensional Poisson process.

Prove that (a) The distribution functions of X2 and Y2 are 1'2 and G2 , respectively; and (b) The copula of X2 and 12 is C. 23 Let X and Y be continuous random variables with copula C and a common univariate distribution function F. 16) are given by order statistic max(X,Y) min(X,y) where 0, 8, 1;, and C', respectively. , C ** = C, so *0* = i; C"'* = C, 2. 2), and let F(-I) and G(-I) be quasi-inverses of F and G, respectively. Then for any (u, v) in 12 C(u, v) = H(F(-l)(u), C(-l)(v)). 7 Symmetry If X is a random variable and a is a real number, we say that X is symmetric about a if the distribution functions of the random variables X - a and a - X are the same, that is, if for any x in R, P[X -a::; xl = P[a-X::; xl.

The random variables max(X, Y) and min(X, Y) are the order statistics for X and Y. Prove that the distribution functions of the order statistics are given by P[max(X, Y) :5: t] = C(F(t), G(t») and P[min(X, Y):5: t] = F(t) + G(t) - C(F(t), G(t)), so that when F = G, P[max(X, Y) :5: t] = bc(F(t)) and P[min(X, Y) :5: t] = 2F(t) - bc(F(t)). (b) Show that bounds on the distribution functions of the order statistics are given by max(F(t) + G(t) -1, 0):5: P[max(X, Y):5: t] :5: min(F(t), G(t» and max(F(t), G(t)) :5: P[min(X, Y) :5: t] :5: min(F(t) + G(t), 1).

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An Introduction to Copulas by Roger B. Nelsen

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