CDVine (version 1.1-5)

CDVineAIC-BIC: AIC and BIC of C- and D-vine copula models

Description

These functions calculate the Akaike and Bayesian Information criteria of d-dimensional C- and D-vine copula models for a given copula data set.

Usage

CDVineAIC(data, family, par, 
          par2=rep(0,dim(data)[2]*(dim(data)[2]-1)/2), type)
CDVineBIC(data, family, par, 
          par2=rep(0,dim(data)[2]*(dim(data)[2]-1)/2), type)

Arguments

data

An N x d data matrix (with uniform margins).

family

A d*(d-1)/2 integer vector of C-/D-vine pair-copula families with values 0 = independence copula 1 = Gaussian copula 2 = Student t copula (t-copula) 3 = Clayton copula 4 = Gumbel copula

par

A d*(d-1)/2 vector of pair-copula parameters.

par2

A d*(d-1)/2 vector of second parameters for two parameter pair-copula families (default: par2 = rep(0,dim(data)[2]*(dim(data)[2]-1)/2)).

type

Type of the vine model: 1 or "CVine" = C-vine 2 or "DVine" = D-vine

Value

• AIC, BIC
  The computed AIC or BIC value, respectively.
• pair.AIC, pair.BIC
  An array of individual contributions to the AIC or BIC value for each pair-copula, respectively. Note: AIC = sum(pair.AIC) and similarly BIC = sum(pair.BIC).

Details

If $k$ denotes the number of parameters of a C-vine copula model with log-likelihood $l_{CVine}$ and parameter set $\boldsymbol{\theta}$, then the Akaike Information Criterion (AIC) by Akaike (1973) is defined as $$AIC := -2 l_{CVine}\left(\boldsymbol{\theta}|\boldsymbol{u}\right) + 2 k,$$ for observations $\boldsymbol{u}=(\boldsymbol{u}_1^\prime,...,\boldsymbol{u}_N^\prime)^\prime$. Similarly, the Bayesian Information Criterion (BIC) by Schwarz (1978) is given by $$BIC := -2 l_{CVine}\left(\boldsymbol{\theta}|\boldsymbol{u}\right) + \log(N) k.$$ The AIC and BIC expressions for D-vine copula models are defined accordingly.

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. Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics 6 (2), 461-464.

See Also

CDVineLogLik, CDVineVuongTest, CDVineClarkeTest

Examples

## Example 1: 3-dimensional D-vine model with Gaussian pair-copulas
data(worldindices)
Data = as.matrix(worldindices)[,1:3]
fam1 = c(1,1,1) 
par1 = c(0.2,0.3,0.4)

# calculate AIC and BIC
CDVineAIC(Data,fam1,par1,type=2)
CDVineBIC(Data,fam1,par1,type=2)


## Example 2: 6-dimensional C-vine model with Student t pair-copulas
## with 5 degrees of freedom
data(worldindices)
Data = as.matrix(worldindices)
dd = dim(Data)[2]*(dim(Data)[2]-1)/2
fam2 = rep(2,dd)
par2 = rep(0.5,dd)
nu2 = rep(5,dd)

# calculate AIC and BIC
CDVineAIC(Data,fam2,par2,nu2,type=1)
CDVineBIC(Data,fam2,par2,nu2,type=1)


## Example 3: 4-dimensional C-vine model with mixed pair-copulas
fam3 = c(5,1,3,14,3,2)
par3 = c(0.9,0.3,0.2,1.1,0.2,0.7)
nu3 = c(0,0,0,0,0,7)

# calculate AIC and BIC
CDVineAIC(Data[,1:4],fam3,par3,nu3,type=1)
CDVineBIC(Data[,1:4],fam3,par3,nu3,type=1)