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I am currently using (multivariate) mixture copulas to model a financial data set. The mixture has two components as follows:
$$C_{mixture}=wC_1+(1-w)C_2$$ where $C_1$ and $C_2$ are copulas. I have closed form solutions for the tail dependence coefficients of both copulas. Are there any general solutions for the tail dependence of the mixture?

InfiniteVariance
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  • What about example 3.2.3 [here](http://edoc.hu-berlin.de/master/grossmass-till-2007-09-28/PDF/grossmass.pdf)? – eric_kernfeld Apr 18 '16 at 20:07
  • Do you think this can be generalized to more than two dimensions? – InfiniteVariance Apr 18 '16 at 20:19
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    @eric Oh my. Grossmass is using $\lambda$ both as a mixing proportion *and* as the measure of tail dependence in the same formula (albeit with subscripts to distinguish upper and lower tail dependence). That's just *awful*. (How did Hardle allow it to get to the point of submission in that terrible state?) – Glen_b Apr 19 '16 at 00:52
  • So given that tail dependence is always a pairwise concept to me it comes down to the follwing question: Is the formula stated (tail dependence of mixture equals weighted tail dependence, where the weights are the mixture weights) applicable in higher dimension. The closed form solution I mentioned is already for pairs of variables in a multivariate copula context. – InfiniteVariance Apr 19 '16 at 11:07
  • @InfiniteVariance If you are dealing with $Y = [y_1... y_D]$, you could just define $X = [y_j, y_k]$ and apply the results to $X$. Does that give you what you need? – eric_kernfeld Apr 19 '16 at 19:17
  • So if I have, say, a multivariate $t$-copula for $Y = [y_1,y_2,y_3]$ with correlation coefficients $\rho_{12},\rho_{13},\rho_{23},$ and degree of freedom $\nu$ then $X=[y_1,y_2]$ will have tail dependence coefficient $t_{\nu+1}( - \sqrt{\nu+1} \sqrt{1-\rho_{12}} \sqrt{1+\rho_{12}})$ ,where I have this formula from Prop.1 in http://www.macs.hw.ac.uk/~mcneil/ftp/tCopula.pdf – InfiniteVariance Apr 19 '16 at 21:28
  • In a next step I would then compute the TDCs for all pairs of the second mixture component and apply formula 3.2.3 in Grossmass? – InfiniteVariance Apr 19 '16 at 21:35
  • @eric The reason why I am asking is that your answer implies a bivariate copula. If I remember correctly I can get marginal distributions from joint distributions (here that of $X$ and $Y$) through $F_X(x) = P(X \leq x) = \lim_{y \to \infty} P(X \leq x, Y \leq y) = \lim_{y \to \infty} F_{XY} (x, y)$. But is this equal to the specified $t$-copula in my example above? – InfiniteVariance Apr 20 '16 at 11:23
  • @InfiniteVariance No, you would need to get the marginal distribution by integrating over the other coordinates: $f_m(y_1, y_2) = \int_y f_j(y_1, y_2, y)dy$. You can probably use the references Demarta & McNeil mention in section 2.1. – eric_kernfeld Apr 20 '16 at 16:57
  • @eric Thanks! So I would first determine the copula of $X = [y_j, y_k]$ in the example you mentioned above before computing the TDC and applying the results from above? – InfiniteVariance Apr 20 '16 at 18:07
  • @ eric I guess this also related to this question here: http://stats.stackexchange.com/questions/206440/obtain-marginal-cdf-from-joint-cdf-by-simulation – InfiniteVariance Apr 20 '16 at 19:19
  • @InfiniteVariance It is similar, but I would suggest using analytic results rather than simulation, especially since your reference cites material on the multivariate t distribution. – eric_kernfeld Apr 21 '16 at 05:14

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