blomCOP: The Blomqvist Beta of a Copula

Description

Compute the Blomqvist Beta $\beta_\mathbf{C}$ of a copula (Nelsen, 2006, p. 182), which is defined at the middle or center of $\mathcal{I}^2$ as

$$\beta_\mathbf{C} = 4\times\mathbf{C}\biggl(\frac{1}{2},\frac{1}{2}\biggr) - 1\mbox{,}$$

where the $u = v = 1/2$ and thus shows that $\beta_\mathbf{C}$ is based on the median joint probability. The Blomqvist Beta is also called the medial correlation coefficient. Nelsen also reports that “although, the Blomqvist Beta depends only on the copula only through its value at the center of $\mathcal{I}^2$, but that [$\beta_\mathbf{C}$] nevertheless often provides an accurate approximation to both Spearman Rho rhoCOP and Kendall Tau tauCOP.” Kendall Tau $\tau_\mathbf{C}$, Gini Gamma $\gamma_\mathbf{C}$, and Spearman Rho $\rho_\mathbf{C}$ in relation to $\beta_\mathbf{C}$ satisfy the following inequalities (Nelsen, 2006, exer. 5.17, p. 185): $$\frac{1}{4}(1 + \beta_\mathbf{C})^2 - 1 \le \tau_\mathbf{C} \le 1 - \frac{1}{4}(1 - \beta_\mathbf{C})^2\mbox{,}$$ $$\frac{3}{16}(1 + \beta_\mathbf{C})^3 - 1 \le \rho_\mathbf{C} \le 1 - \frac{3}{16}(1 - \beta_\mathbf{C})^3\mbox{, and}$$ $$\frac{3}{8}(1 + \beta_\mathbf{C})^2 - 1 \le \tau_\mathbf{C} \le 1 - \frac{3}{8}(1 - \beta_\mathbf{C})^2\mbox{.}$$

A curious aside (Joe, 2014, p. 164) about the Gaussian copula is that Blomqvist Beta is equal to Kendall Tau (tauCOP): $\beta_\mathbf{C} = \tau_\mathbf{C}$ (see Note in med.regressCOP for a demonstration). Finally, a version of Blomqvist Beta defined outside the median is provided by blomCOPss.

Usage

blomCOP(cop=NULL, para=NULL, as.sample=FALSE,
                  ctype=c("joe", "weibull", "hazen", "1/n",
                                 "bernstein", "checkerboard"), ...)

Value

The value for $\beta_\mathbf{C}$ or sample $\hat\beta_n$ is returned.

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
as.sample: A logical controlling whether an optional R data.frame in para is used to compute the $\hat\beta_\mathbf{C}$ (see Note);
ctype: Argument of the same as EMPIRcop with the exception of the "joe" specific to the documentation here. The other choices trigger and are given over to the empirical copula; and
...: Additional arguments to pass to the copula or down to EMPIRcop if a sample version had been requested.

Author

W.H. Asquith

References

Genest, C., Carabarín-Aguirre, A., and Harvey, F., 2013, Copula parameter estimation using
Blomqvist's beta: Journal de la Soci$é$té Française de Statistique, v. 154, no. 1, pp. 5--24.

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.