CDVineVuongTest: Vuong test comparing two vine copula models
In CDVine: Statistical Inference of C- And D-Vine Copulas

Description Usage Arguments Details Value Author(s) References See Also Examples

This function performs a Vuong test between two d-dimensional C- or D-vine copula models, respectively.

CDVineVuongTest(data, Model1.order=1:dim(data)[2],
                Model2.order=1:dim(data)[2], Model1.family,
                Model2.family, Model1.par, Model2.par,
                Model1.par2=rep(0,dim(data)[2]*(dim(data)[2]-1)/2),
                Model2.par2=rep(0,dim(data)[2]*(dim(data)[2]-1)/2),
                Model1.type, Model2.type)

`data`	An N x d data matrix (with uniform margins).
`Model1.order, Model2.order`	Two numeric vectors giving the order of the variables in the first D-vine trees or of the C-vine root nodes in models 1 and 2 (default: `Model1.order` and `Model2.order = 1:dim(data)[2]`, i.e., standard order).
`Model1.family, Model2.family`	Two d*(d-1)/2 numeric vectors of the pair-copula families of models 1 and 2, respectively, 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)
`Model1.par, Model2.par`	Two d*(d-1)/2 numeric vectors of the (first) copula parameters of models 1 and 2, respectively.
`Model1.par2, Model2.par2`	Two d(d-1)/2 numeric vectors of the second copula parameters of models 1 and 2, respectively; necessary for t, BB1, BB6, BB7 and BB8 copulas. If no such families are included in `Model1.family`/`Model2.family`, these arguments do not need to be specified (default: `Model1.par2` and `Model2.par2 = rep(0,dim(data)[2](dim(data)[2]-1)/2)`).
`Model1.type, Model2.type`	Type of the respective vine model: `1` or `"CVine"` = C-vine `2` or `"DVine"` = D-vine

The likelihood-ratio based test proposed by Vuong (1989) can be used for comparing non-nested models. For this let c_1 and c_2 be two competing vine copulas in terms of their densities and with estimated parameter sets θ_1 and θ_2. We then compute the standardized sum, ν, of the log differences of their pointwise likelihoods m_i:=log[c_1(u_i|θ_1) / c_2(u_i|θ_2) ] for observations u_i in [0,1],i=1,...,N , i.e.,

statistic := ν = (1/n∑_{i=1}^N m_i) / ((∑_{i=1}^N (m_i - \bar{m} )^2)^0.5).

Vuong (1989) shows that ν is asymptotically standard normal. According to the null-hypothesis

H_0: E[m_i] = 0 forall i=1,...,N,

we hence prefer vine model 1 to vine model 2 at level α if

ν > Φ^{-1}(1-α/2),

where Φ^{-1} denotes the inverse of the standard normal distribution function. If ν<-Φ^{-1}(1-α/2) we choose model 2. If, however, |ν| <= Φ^{-1}(1-α/2), no decision among the models is possible.

Like AIC and BIC, the Vuong test statistic may be corrected for the number of parameters used in the models. There are two possible corrections; the Akaike and the Schwarz corrections, which correspond to the penalty terms in the AIC and the BIC, respectively.

`statistic, statistic.Akaike, statistic.Schwarz`	Test statistics without correction, with Akaike correction and with Schwarz correction.
`p.value, p.value.Akaike, p.value.Schwarz`	P-values of tests without correction, with Akaike correction and with Schwarz correction.

Jeffrey Dissmann, Ulf Schepsmeier

Vuong, Q. H. (1989). Ratio tests for model selection and non-nested hypotheses. Econometrica 57 (2), 307-333.

CDVineClarkeTest, CDVineAIC, CDVineBIC

## Not run: 
# load data set
data(worldindices)
d = dim(worldindices)[2]

# select the C-vine families and parameters
cvine = CDVineCopSelect(worldindices,c(1:6),type="CVine")

# select the D-vine families and parameters
dvine = CDVineCopSelect(worldindices,c(1:6),type="DVine")

# compare the two models based on the data
vuong = CDVineVuongTest(worldindices,1:d,1:d,cvine$family,dvine$family,
                        cvine$par,dvine$par,cvine$par2,dvine$par2,
                        Model1.type=1,Model2.type=2)
vuong$statistic
vuong$statistic.Schwarz
vuong$p.value
vuong$p.value.Schwarz

## End(Not run)