VineCopula (version 1.6-1)

BiCopSelect: Selection and Maximum Likelihood Estimation of Bivariate Copula Families

Description

This function selects an appropriate bivariate copula family for given bivariate copula data using one of a range of methods. The corresponding parameter estimates are obtained by maximum likelihood estimation.

Usage

BiCopSelect(u1, u2, familyset = NA, selectioncrit = "AIC", indeptest = FALSE,
            level = 0.05, weights = NA, rotations = TRUE)

Arguments

u1, u2

Data vectors of equal length with values in [0,1].

familyset

Vector of bivariate copula families to select from (the independence copula MUST NOT be specified in this vector, otherwise it will be selected). The vector has to include at least one bivariate copula family that allows for positive and one tha

selectioncrit

Character indicating the criterion for bivariate copula selection. Possible choices: selectioncrit = "AIC" (default) or "BIC".

indeptest

Logical; whether a hypothesis test for the independence of u1 and u2 is performed before bivariate copula selection (default: indeptest = FALSE; see BiCopIndTest

level

Numeric; significance level of the independence test (default: level = 0.05).

weights

Numerical; weights for each observation (optional).

rotations

If TRUE, all rotations of the families in familyset are included.

Value

• An object of class BiCop, i.e., a list containing
• family
  The selected bivariate copula family.
• par, par2
  The estimated bivariate copula parameter(s).
• p.value.indeptest
  P-value of the independence test if performed.

Details

Copulas can be selected according to the Akaike and Bayesian Information Criteria (AIC and BIC, respectively). First all available copulas are fitted using maximum likelihood estimation. Then the criteria are computed for all available copula families (e.g., if u1 and u2 are negatively dependent, Clayton, Gumbel, Joe, BB1, BB6, BB7 and BB8 and their survival copulas are not considered) and the family with the minimum value is chosen. For observations $u_{i,j},\ i=1,...,N,\ j=1,2,$ the AIC of a bivariate copula family $c$ with parameter(s) $\boldsymbol{\theta}$ is defined as $$AIC := -2 \sum_{i=1}^N \ln[c(u_{i,1},u_{i,2}|\boldsymbol{\theta})] + 2k,$$ where $k=1$ for one parameter copulas and $k=2$ for the two parameter t-, BB1, BB6, BB7 and BB8 copulas. Similarly, the BIC is given by $$BIC := -2 \sum_{i=1}^N \ln[c(u_{i,1},u_{i,2}|\boldsymbol{\theta})] + \ln(N)k.$$ Evidently, if the BIC is chosen, the penalty for two parameter families is stronger than when using the AIC. Additionally a test for independence can be performed beforehand.

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. N. Petrov and F. Csaki (Eds.), Proceedings of the Second International Symposium on Information Theory Budapest, Akademiai Kiado, pp. 267-281. Brechmann, E. C. (2010). Truncated and simplified regular vines and their applications. Diploma thesis, Technische Universitaet Muenchen. http://mediatum.ub.tum.de/?id=1079285. Manner, H. (2007). Estimation and model selection of copulas with an application to exchange rates. METEOR research memorandum 07/056, Maastricht University. Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics 6 (2), 461-464.

See Also

RVineStructureSelect, RVineCopSelect, BiCopIndTest, BiCop

Examples

## Example 1: Gaussian copula with large dependence parameter
par1 <- 0.7
fam1 <- 1
dat1 <- BiCopSim(500, fam1, par1)

# select the bivariate copula family and estimate the parameter(s)
cop1 <- BiCopSelect(dat1[,1], dat1[,2], familyset = c(1:10),
                    indeptest = FALSE, level = 0.05)
cop1$family
cop1$par
cop1$par2


## Example 2: Gaussian copula with small dependence parameter
par2 <- 0.01
fam2 <- 1
dat2 <- BiCopSim(500, fam2, par2)

# select the bivariate copula family and estimate the parameter(s)
cop2 <- BiCopSelect(dat2[,1], dat2[,2], familyset = c(1:10),
                    indeptest = TRUE, level = 0.05)
cop2$family
cop2$par
cop2$par2


## Example 3: empirical data
data(daxreturns)
cop3 <- BiCopSelect(daxreturns[,1], daxreturns[,4],
                    familyset = c(1:10, 13, 14, 16,
                                  23, 24, 26, 33, 34, 36))
cop3$family
cop3$par
cop3$par2