# CDVineLogLik: Log-likelihood of C- and D-vine copula models

### Description

This function calculates the log-likelihood of d-dimensional C- and D-vine copula models for a given copula data set.

### Usage

 1 2 CDVineLogLik(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 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”) 14 = rotated Gumbel copula (180 degrees; “survival Gumbel”) 16 = rotated Joe copula (180 degrees; “survival Joe”) 17 = rotated BB1 copula (180 degrees; “survival BB1”) 18 = rotated BB6 copula (180 degrees; “survival BB6”) 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) 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

### Details

Let u=(u'_1,...,u'_N)' be d-dimensional observations with u_i=(u_{i,1},...,u_{i,d})' in [0,1]^d. Then the log-likelihood of a C-vine copula is given by

loglik:=l_{CVine}(θ|u)= ∑_{i=1}^N ∑_{j=1}^{d-1} ∑_{k=1}^{d-j} ln[c_{j,j+k|1,...,j-1}],

where

c_{j,j+k|1,...,j-1}:=c_{j,j+k|1:(j-1)}(F(u_{i,j}|u_{i,1},...,u_{i,j-1}),F(u_{i,j+k}|u_{i,1},...,u_{i,j-1})|θ_{j,j+k|1,...,j-1})

denote pair-copulas with parameter(s) θ_{j,j+k|1,...,j-1}.

Similarly, the log-likelihood of a d-dimensional D-vine copula is

loglik:=l_{DVine}(θ|u)= ∑_{i=1}^N ∑_{j=1}^{d-1} ∑_{k=1}^{d-j} ln[c_{k,k+j|k+1,...,k+j-1}],

again with pair-copula densities denoted by

c_{k,k+j|k+1,...,k+j-1}:=

c_{k,k+j|k+1,...,k+j-1}(F(u_{i,k}|u_{i,k+1},...,u_{i,k+j-1}),F(u_{i,k+j}|u_{i,k+1},...,u_{i,k+j-1})|θ_{k,k+j|k+1,...,k+j-1}).

Conditional distribution functions in both expressions are obtained recursively using the relationship

h(u|v,θ) := F(u|v) = \partial C_{uv_j|v_{-j}}(F(u|v_{-j}),F(v_j|v_{-j})) / \partial F(v_j|v_{-j}),

where C_{uv_j|v_{-j}} is a bivariate copula distribution function with parameter(s) θ and v_{-j} denotes a vector with the j-th component v_j removed. The notation of h-functions is introduced for convenience. For more details see Aas et al. (2009).

Note that both log-likelihoods can also be written as loglik=∑_{k=1}^{d(d-1)/2}ll_{k}, where ll_{k} are the individual contributions to the log-likelihood of each pair-copula.

### Value

 loglik The calculated log-likelihood value of the C- or D-vine copula model. ll An array of individual contributions to the log-likelihood for each pair-copula. Note: loglik = sum(ll). vv The stored transformations (h-functions) which may be used for posterior updates.

### Author(s)

Carlos Almeida, Ulf Schepsmeier

### References

Aas, K., C. Czado, A. Frigessi, and H. Bakken (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics 44 (2), 182-198.

BiCopHfunc, CDVineMLE, CDVineAIC, CDVineBIC

### Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ## 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 the log-likelihood logLik1 = CDVineLogLik(Data,fam1,par1,type=2) # check the above formula sum(logLik1$ll) logLik1$loglik ## 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 the log-likelihood logLik2 = CDVineLogLik(Data,fam2,par2,nu2,type=1) logLik2$loglik ## Example 3: 4-dimensional C-vine model with mixed pair-copulas fam3 = c(5,1,3,14,3,2) par3 = c(0.9,0.3,0.2,1.1,0.2,0.7) nu3 = c(0,0,0,0,0,7) # calculate the log-likelihood logLik3 = CDVineLogLik(Data[,1:4],fam3,par3,nu3,type=2) logLik3$loglik 

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