Compute the lower- and upper-tail dependency parameters (if they exist), respectively, of a copula according to Nelsen (2006, pp. 214--215). Graphical confirmation of the computations is important, and therefore, the function can also generate a plot. The dependency parameters are expressions of conditional probability that \(Y\) is greater than the \(100{\times}\)th percentile of its distribution \(G\) given that \(X\) is greater than the \(100{\times}t\)-th percentile of its distribution \(F\) as \(t\) approaches unity. Specifics in terms of quantile functions \(G^{(-1)}(t) = y(t)\) and \(F^{(-1)}(t) = x(t)\) follow.
The lower-tail dependence parameter \(\lambda^L_\mathbf{C}\) is defined as $$\lambda^L_\mathbf{C} = \lim_{t{\rightarrow 0^{+}}} \mathrm{Pr}[Y \le y(t)\mid X \le x(t)]\mbox{, and}$$ the upper-tail dependence parameter \(\lambda^U_\mathbf{C}\) with reversed inequalities is defined as $$\lambda^U_\mathbf{C} = \lim_{t{\rightarrow 1^{-}}} \mathrm{Pr}[Y > y(t)\mid X > x(t)]\mbox{.}$$
Nelsen (2006, p. 214) also notes that both \(\lambda^L_\mathbf{C}\) and \(\lambda^U_\mathbf{C}\) are nonparametric and depend only on the copula of \(X\) and \(Y\), and Nelsen shows that each can be computed if the above limits exist as follows: $$\lambda^L_\mathbf{C} = \lim_{t{\rightarrow 0^{+}}} \frac{\mathbf{C}(t,t)}{t} = \delta_\mathbf{C}'(0^{+})\mbox{\ and}$$ $$\lambda^U_\mathbf{C} = \lim_{t{\rightarrow 1^{-}}} \frac{1 - 2t - \mathbf{C}(t,t)}{1 - t} = 2 - \lim_{t{\rightarrow 1^{-}}} \frac{1 - \mathbf{C}(t,t)}{1-t} = 2 - \delta_\mathbf{C}'(1^{-})\mbox{,}$$ where \(\delta_\mathbf{C}'(t)\) is the derivative of the diagonal of the copula. Multiple presentations are shown because algebraic variants are shown across the literature.
If \(\lambda^L_\mathbf{C} \in (0,1]\), then \(\mathbf{C}\) has lower-tail dependence but if \(\lambda^L_\mathbf{C} = 0\), then \(\mathbf{C}\) has no lower-tail dependence. Likewise, if \(\lambda^U_\mathbf{C} \in (0,1]\), then \(\mathbf{C}\) has upper-tail dependence but if \(\lambda^U_\mathbf{C} = 0\), then \(\mathbf{C}\) has no upper-tail dependence.
taildepCOP(cop=NULL, para=NULL, tol=1e-6, divisor=2, plot=FALSE, ylim=NULL,
verbose=FALSE, ...)An R list is returned.
The rounded value of \(\lambda^L_\mathbf{C}\);
The rounded value of \(\lambda^U_\mathbf{C}\);
An attribute identifying the computational source: “taildepCOP”.
A copula function;
Vector of parameters or other data structure, if needed, to pass to the copula;
A tolerance on convergence;
The divisor on the incremental reductions towards \(0^+\) and \(0^-\) by the algorithm;
A logical plotting a diagnostic plot of the diagonal derivatives and label the limits;
Optional vertical limits if the plot is turned on. Although the dependence parameters are bounded as described above, numerical stability can be a problem. Stability is especially a problem if an empirical copula is being used; theefore, the bounds of the plot are left open unless the user locks them down with this argument;
Show incremental progress; and
Additional arguments to pass to the copula function.
W.H. Asquith
Charpentier, A., 2012, Copulas and tail dependence, part 1: R-bloggers, dated Sept. 17, 2012, accessed on February 2, 2019 at
https://www.r-bloggers.com/2012/09/copulas-and-tail-dependence-part-1/
Dobrić, J. and Schmid, F., 2005, Nonparametric estimation of the lower tail dependence \(\lambda^L\) in bivariate copulas: Journal of Applied Statistics, v. 32, no. 4, pp. 387--407.
Frahm, G., Junker, M., and Schmidt, R., 2005, Estimating the tail-dependence coefficient---
Properties and pitfalls: Insurance---Mathematics and Economics, v. 37, no. 1, pp. 80--100.
Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.
Schmidt, R., and Stadtmüller, U., 2006, Nonparametric estimation of tail dependence: The Scandinavian Journal of Statistics, v. 33, pp. 307--335.
COP, tailconCOP, tailordCOP, blomCOPss