By Pravin K. Trivedi, David M. Zimmer, Visit Amazon's P. K. Trivedi Page, search results, Learn about Author Central, P. K. Trivedi,
Copula Modeling explores the copula process for econometrics modeling of joint parametric distributions. Copula Modeling demonstrates that useful implementation and estimation is comparatively user-friendly regardless of the complexity of its theoretical foundations. an enticing characteristic of parametrically particular copulas is that estimation and inference are in response to typical greatest chance approaches. hence, copulas could be envisioned utilizing machine econometric software program. this gives a considerable benefit of copulas over lately proposed simulation-based ways to joint modeling. Copulas are helpful in quite a few modeling events together with monetary markets, actuarial technology, and microeconometrics modeling. Copula Modeling offers practitioners and students with an invaluable consultant to copula modeling with a spotlight on estimation and misspecification. The authors conceal very important theoretical foundations. all through, the authors use Monte Carlo experiments and simulations to illustrate copula houses
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Additional info for Copula modeling: an introduction for practitioners
M). Let K be an m-variate distribution function with all marginals uniform on [0, 1]. If Fi (y) = exp[−ϕ −1 i Hi (y)], then H(y1 , . . , ym ) = ... K([F1 (y1 )]η1 , . . , [Fm (ym )]ηm )dΛ(η1 , . . , ηm ) is an m-dimensional distribution function with marginals H1 , . . , Hm . A special case of this theorem occurs when η η m η m 1 i K([F1 (y1 )] , . . , [Fm (ym )] ) = Πi=1 Fi (yi )] and the yi (i = 1, . . , m) are uniform [0, 1] variates. In this case the application of the theorem yields the m-dimensional copula −1 H(y1 , .
Then, 1 − F (y1 , y2 ) = e−y1 + e−y2 F (y1 , y2 ) 1 − F2 (y2 ) 1 − F1 (y1 ) + , = F1 (y1 ) F2 (y2 ) where F1 (y1 ) and F2 (y2 ) are univariate marginals. Observe that in this case there is no explicit dependence parameter. In the case of independence, since F (y1 , y2 ) = F1 (y1 )F2 (y2 ), 1 − F (y1 , y2 ) 1 − F1 (y1 )F2 (y2 ) = F (y1 , y2 ) F1 (y1 )F2 (y2 ) 1 − F2 (y2 ) 1 − F1 (y1 ) 1 − F2 (y2 ) 1 − F1 (y1 ) = + + . 3. Mixtures and Convex Sums 37 parameter θ: 1 − F (y1 , y2 ) 1 − F1 (y1 ) 1 − F2 (y2 ) = + F (y1 , y2 ) F1 (y1 ) F2 (y2 ) 1 − F1 (y1 ) 1 − F2 (y2 ) + (1 − θ) .
1 is Gumbel’s bivariate logistic distribution, denoted F (y1 , y2 ). Let (1 − F (y1 , y2 ))/F (y1 , y2 ) denote the bivariate survival odds ratio by analogy with the univariate survival function. Then, 1 − F (y1 , y2 ) = e−y1 + e−y2 F (y1 , y2 ) 1 − F2 (y2 ) 1 − F1 (y1 ) + , = F1 (y1 ) F2 (y2 ) where F1 (y1 ) and F2 (y2 ) are univariate marginals. Observe that in this case there is no explicit dependence parameter. In the case of independence, since F (y1 , y2 ) = F1 (y1 )F2 (y2 ), 1 − F (y1 , y2 ) 1 − F1 (y1 )F2 (y2 ) = F (y1 , y2 ) F1 (y1 )F2 (y2 ) 1 − F2 (y2 ) 1 − F1 (y1 ) 1 − F2 (y2 ) 1 − F1 (y1 ) = + + .