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
\(\hat{\boldsymbol{\theta}}_1\) and
\(\hat{\boldsymbol{\theta}}_2\). The null hypothesis of
statistical indistinguishability of the two models is $$ $$$$H_0: P(m_i > 0) = 0.5\ \forall i=1,..,N, $$ where
\(m_i:=\log\left[\frac{c_1(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_1)}{c_2(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_2)}\right]\) for observations
\(\boldsymbol{u}_i,\ i=1,...,N\).
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 test statistic $$ $$$$\texttt{statistic} := B = \sum_{i=1}^N
\mathbf{1}_{(0,\infty)}(m_i), $$ where \(\mathbf{1}\) 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 \(Np = \frac{N}{2}\).
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.