BiCopPar2Beta {VineCopula}

Blomqvist's Beta Value of a Bivariate Copula

Description

This function computes the theoretical Blomqvist's beta value of a bivariate copula for given parameter values.

Usage

BiCopPar2Beta(family, par, par2 = 0, obj = NULL, check.pars = TRUE)

Arguments

family

integer; single number or vector of size n; defines the bivariate copula family:
0 = independence copula
2 = Student t copula (t-copula)
1 = Gaussian copula
3 = Clayton copula
4 = Gumbel copula
5 = Frank copula
6 = Joe copula
7 = BB1 copula
8 = BB6 copula
9 = BB7 copula
10 = BB8 copula
13 = rotated Clayton copula (180 degrees; ⁠survival Clayton'') \cr `14` = rotated Gumbel copula (180 degrees; ⁠survival Gumbel”)
16 = rotated Joe copula (180 degrees; ⁠survival Joe'') \cr `17` = rotated BB1 copula (180 degrees; ⁠survival BB1”)
18 = rotated BB6 copula (180 degrees; ⁠survival BB6'')\cr `19` = rotated BB7 copula (180 degrees; ⁠survival BB7”)
20 = rotated BB8 copula (180 degrees; “survival BB8”)
23 = rotated Clayton copula (90 degrees)
'24' = rotated Gumbel copula (90 degrees)
'26' = rotated Joe copula (90 degrees)
'27' = rotated BB1 copula (90 degrees)
'28' = rotated BB6 copula (90 degrees)
'29' = rotated BB7 copula (90 degrees)
'30' = rotated BB8 copula (90 degrees)
'33' = rotated Clayton copula (270 degrees)
'34' = rotated Gumbel copula (270 degrees)
'36' = rotated Joe copula (270 degrees)
'37' = rotated BB1 copula (270 degrees)
'38' = rotated BB6 copula (270 degrees)
'39' = rotated BB7 copula (270 degrees)
'40' = rotated BB8 copula (270 degrees)
'104' = Tawn type 1 copula
'114' = rotated Tawn type 1 copula (180 degrees)
'124' = rotated Tawn type 1 copula (90 degrees)
'134' = rotated Tawn type 1 copula (270 degrees)
'204' = Tawn type 2 copula
'214' = rotated Tawn type 2 copula (180 degrees)
'224' = rotated Tawn type 2 copula (90 degrees)
'234' = rotated Tawn type 2 copula (270 degrees)
Note that the Student's t-copula is not allowed since the CDF of the t-copula is not implemented (see BiCopCDF()).

par

numeric; single number or vector of size n; copula parameter.

par2

numeric; single number or vector of size n; second parameter for the two parameter BB1, BB6, BB7, BB8, Tawn type 1 and type 2 copulas (default: par2 = 0).

obj

BiCop object containing the family and parameter specification.

check.pars

logical; default is TRUE; if FALSE, checks for family/parameter-consistency are omitted (should only be used with care).

Details

If the family and parameter specification is stored in a BiCop() object obj, the alternative version

BiCopPar2Beta(obj)

can be used.

Value

Theoretical value of Blomqvist's beta corresponding to the bivariate copula family and parameter(s) par, par2.

Note

The number n can be chosen arbitrarily, but must agree across arguments.

Author(s)

Ulf Schepsmeier

References

Blomqvist, N. (1950). On a measure of dependence between two random variables. The Annals of Mathematical Statistics, 21(4), 593-600.

Nelsen, R. (2006). An introduction to copulas. Springer

Examples

## Example 1: Gaussian copula
BiCopPar2Beta(family = 1, par = 0.7)
BiCop(1, 0.7)$beta  # alternative

## Example 2: Clayton copula
BiCopPar2Beta(family = 3, par = 2)

## Example 3: different copula families
BiCopPar2Beta(family = c(3,4,6), par = 2:4)