CDVineClarkeTest: Clarke 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 Clarke test between two d-dimensional C- or D-vine copula models, respectively.

CDVineClarkeTest(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 test proposed by Clarke (2007) allows to compare 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. The null hypothesis of statistical indistinguishability of the two models is

H_0: P(m_i > 0) = 0.5 forall i=1,..,N,

where m_i:=log[ c_1(u_i|θ_1) / c_2(u_i|θ_2) ] for observations u_i, i=1,...,N.

Since under statistical equivalence of the two models the log likelihood ratios of the single observations are uniformly distributed around zero and in expectation 50\% of the log likelihood ratios greater than zero, the tets statistic

statistic := B = ∑_{i=1}^N 1_{(0,∞)}(m_i),

where 1 is the indicator function, is distributed Binomial with parameters N and p=0.5, and critical values can easily be obtained. Model 1 is interpreted as statistically equivalent to model 2 if B is not significantly different from the expected value np=N/2.

Like AIC and BIC, the Clarke 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, Eike Brechmann

Clarke, K. A. (2007). A Simple Distribution-Free Test for Nonnested Model Selection. Political Analysis, 15, 347-363.

CDVineVuongTest, 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
clarke = CDVineClarkeTest(worldindices,1:d,1:d,cvine$family,dvine$family,
                          cvine$par,dvine$par,cvine$par2,dvine$par2,
                          Model1.type=1,Model2.type=2)
clarke$statistic
clarke$statistic.Schwarz
clarke$p.value
clarke$p.value.Schwarz

## End(Not run)