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 \(\hat{\boldsymbol{\theta}}_1\) and
\(\hat{\boldsymbol{\theta}}_2\). We then compute the
standardized sum, \(\nu\), of the log differences of their pointwise
likelihoods
\(m_i:=\log\left[\frac{c_1(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_1)}{c_2(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_2)}\right]\) for observations \(\boldsymbol{u}_i\in[0,1],\
i=1,...,N\) , i.e.,
$$\texttt{statistic} := \nu = \frac{\frac{1}{n}\sum_{i=1}^N
m_i}{\sqrt{\sum_{i=1}^N\left(m_i - \bar{m} \right)^2}}. $$ Vuong
(1989) shows that \(\nu\) 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 \(\alpha\) if
$$\nu>\Phi^{-1}\left(1-\frac{\alpha}{2}\right), $$ where \(\Phi^{-1}\) denotes the inverse of the
standard normal distribution function. If
\(\nu<-\Phi^{-1}\left(1-\frac{\alpha}{2}\right)\)
we choose model 2. If, however,
\(|\nu|\leq\Phi^{-1}\left(1-\frac{\alpha}{2}\right)\), 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.