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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? – 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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