Bicop.parameters_to_taildep

Bicop.parameters_to_taildep(self, parameters: numpy.ndarray) → numpy.ndarray

Converts the copula parameters to the tail dependence coefficients.

The result is a \(2 \times 2\)matrix \(M\)collecting the tail dependence coefficients in the four corners of the unit square: \(M(i, j)\)is the coefficient as \(U_1 \to i\)and \(U_2 \to j\), with \(i, j \in \{0, 1\}\)(0 = lower, 1 = upper). Thus \(M(0, 0)\)is the classical lower and \(M(1, 1)\)the classical upper tail dependence coefficient.

For the unrotated family, the lower \(\lambda_L = M(0, 0)\)and upper \(\lambda_U = M(1, 1)\)coefficients are:

Family	Lower \(\lambda_L\)	Upper \(\lambda_U\)
Independence, Gaussian, Frank	\(0\)	\(0\)
Student t	\(\lambda_t\) (see below)	\(\lambda_t\)
Clayton	\(2^{-1/\theta}\)	\(0\)
Gumbel	\(0\)	\(2 - 2^{1/\theta}\)
Joe	\(0\)	\(2 - 2^{1/\theta}\)
BB1	\(2^{-1/(\theta\delta)}\)	\(2 - 2^{1/\delta}\)
BB6	\(0\)	\(2 - 2^{1/(\theta\delta)}\)
BB7	\(2^{-1/\delta}\)	\(2 - 2^{1/\theta}\)
BB8	\(0\)	\(2 - 2^{1/\theta}\) if \(\delta=1\), else 0
Tawn (extreme-value)	\(0\)	\(2 (1 - A(1/2))\)
TLL (nonparametric)	NaN	NaN

Here \(\theta\)(and \(\delta\)for two-parameter families) denote the copula parameters and \(A\)the Pickands dependence function. For the Student t copula, with correlation \(\rho\), degrees of freedom \(\nu\), and \(t_{\nu+1}\)the Student t cdf, \(\lambda_t = 2\, t_{\nu+1}(-\sqrt{(\nu+1)(1-\rho)/(1+\rho)})\); it additionally has equal dependence in the two off-diagonal (discordant) corners, obtained by replacing \(\rho\)with \(-\rho\). All other parametric families have zero off-diagonal coefficients. Rotations permute the corners: 180 degrees swaps lower/upper, while 90/270 degrees move dependence to the off-diagonal corners.

Parameters:

parametersndarray, shape (n, m), dtype float

The parameters (must be a valid parametrization of the current family).