Bivariate Marshall-Olkin Copula

Computes CDF and simulations of the bivariate Marshall-Olkin copula.

cBivariateMO(u1, u2, dependencyParameter, ...)

crBivariateMO(numberSimulations = 10000, seed = 42, dependencyParameter)

Arguments

u1, u2	points at which to evaluate the copula.
dependencyParameter	correlation parameters, must be vector of length 2.
...	other parameters.
numberSimulations	Number of simulations.
seed	Simulation seed, 42 by default.

Value

Function :

• cBivariateMO returns the value of the copula.

• crBivariateMO returns simulated values of the copula.

Details

The bivariate Marshall-Olkin copula has CDF : $$C(u_{1}, u_{2}) = u_{1}u_{2}^{1 - \beta} \times% \textbf{1}_{\{u_{1}^{\alpha} \leq u_{2}^{\beta}\}} + % u_{1}^{1 - \alpha}u_{2} \times \textbf{1}_{\{u_{1}^{\alpha}% \geq u_{2}^{\beta}\}}$$ for \(u_{1}, u_{2}, \alpha, \beta \in [0, 1]\). It is the geometric mean of the independance and upper Fréchet bound copulas.

Examples

cBivariateMO(u1 = .76, u2 = 0.4, dependencyParameter = c(0.4, 0.3))
#> [1] 0.3392721

crBivariateMO(numberSimulations = 10, seed = 42, dependencyParameter = c(0.2, 0.5))
#>            [,1]       [,2]
#>  [1,] 0.9904496 0.26361815
#>  [2,] 0.9999990 0.47001156
#>  [3,] 0.6165533 0.31847468
#>  [4,] 0.9604149 0.13711462
#>  [5,] 0.9940986 0.26671480
#>  [6,] 0.9742787 0.75508067
#>  [7,] 0.9065275 0.61249833
#>  [8,] 0.5148039 0.06057856
#>  [9,] 0.9950895 0.27542915
#> [10,] 0.9706447 0.33692591