Outline Copulae R-vines Model selection Limitations and challenges References

How to select a good vine

Ingrid Hobæk Haff

Universitetet i Oslo
ingrihaf@math.uio.no

International FocuStat Workshop on Focused Information

Criteria and Related Themes, May 9-11, 2016

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Copulae

Regular vines

Model selection and reduction

Limitations and challenges

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Copulae

• Remember that if the continuous variable X has the cdf FX ,

then U = FX (X ) is uniformly distributed [0, 1].
• In many cases, it is more convenient or natural to
study/model a transformation of the data, e.g. log(X ).
• In the copula world, one transforms the variables Xi with their
own cdfs Fi .
• For continuous variables, this is called the probability integral
transformation (PIT), and the resulting variables Ui = Fi (Xi )
follow a uniform distribution.

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Outline Copulae R-vines Model selection Limitations and challenges References

Copulae

• Copulae are tools for constructing multivariate distributions.

• The idea behind the PIT is to isolate the individual (marginal)
behaviour of the variables, to focus on their joint behaviour.
• Hence, a multivariate distribution can be split into
• the univariate margins
• a dependence structure.
• This dependence structure is called a copula.
• Definition: A copula C is a multivariate distribution with
uniform margins U[0, 1].

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Outline Copulae R-vines Model selection Limitations and challenges References

Sklar’s theorem [Sklar, 1959]

• Let X1 , . . . , Xd follow the joint distribution F1...d with margins

F1 , . . . , F d .
• Then, there exists a function C1...d such that

F1...d (x1 , . . . , xd ) = C1...d (F1 (x1 ), . . . , Fd (xd )),

where C1...d is a copula.

• This is true for any multivariate distribution, whether
continuous, discrete or a combination of the two.
• If F1...d is continuous, then the copula C1...d is unique.

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Outline Copulae R-vines Model selection Limitations and challenges References

Sklar’s theorem

• When the margins F1 , . . . , Fd in addition are absolutely

continuous and strictly increasing, one may express Sklar’s
theorem in terms of densities.
• Then
d
Y
f1...d (x1 , . . . , xd ) = c1...d (F1 (x1 ), . . . , Fd (xd )) fi (xi ), (1)
i=1

where c1...d be the density of C1...d , that is

∂ d C1...d
c1...d = ,
∂u1 . . . ∂ud
and f1...d the pdf corresponding to F1...d .

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Illustration

If we take this… …and divide it with the product of these…

0.4

0.4
0.25

0.3

0.3
0.2

density

density
0.2

0.2
0.15
Z
0.1

0.1

0.1
0.05

0.0

0.0
-4 -2 0 2 4 -4 -2 0 2 4
0

X X
4

2
4 Univariate standard normal densities
0 2
Y
0
-2 X
-2
-4
Bivariate standard normal density -4

…we get…
5

This is the density of a

4 3

bivariate Gaussian copula.

Z
21
0

0.8
1
0.6
0.8

Y 0.4 0.6

0.4 X
0.2
0.2

0 0

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Outline Copulae R-vines Model selection Limitations and challenges References

Illustration

If we take this… …and multiply it with the product of these…

Kreditt Operasjonell

0.0006

0.0014
5

0.0012
0.0005
4

0.0010
0.0004
3

0.0008
Z

Tetthet

Tetthet
0.0003
2

0.0006
0.0002
1

0.0004
0

0.0001

0.0002
1

0.8

0.0

0.0
1
0.6 0 5000 10000 15000 0 2000 4000 6000 8000
0.8
Verdi Verdi

Y 0.4

0.2
0.4 X
0.6
beta-density lognormal density
0.2

0 0

we get…
3
2.5

This is a bivariate density consisting of

1.5 2
Z

a Gaussian copula and beta- and

1

lognormal margins.
0.5
0

0.8

0.6 20

15
0.4
Y
10
X
0.2
5

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Outline Copulae R-vines Model selection Limitations and challenges References

Copulae – flexible enough?

• For bivariate models (d = 2), there exists a long and varied

list of copula families.
• As soon as d ≥ 3, the catalogue of available copulae is
significantly reduced [Genest et al., 2009].
• Several of the well-known copulae generalise to higher
dimensions.
• Unfortunately, their flexibility decreases with the dimension,
which restricts the range of dependence they are able to
reproduce.

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Copulae – flexible enough?

• Why not build a multivariate copula based merely on bivariate

ones?

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Copulae – flexible enough?

• Why not build a multivariate copula based merely on bivariate

ones?
• That is precisely the idea behind pair-copula constructions,
introduced by Joe [1997].

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Pair-copula constructions

Complete multivariate distribution Pair-copula construction

⋯
Gu
F
Ga

C t
M

⋯ Copula ⋯
t F

Gu

Gu
C
Ga

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Outline Copulae R-vines Model selection Limitations and challenges References

Building blocks
• The bivariate copulae constituting the construction need not
belong to the same family. The resulting multivariate
distribution will still be valid.
• One may for instance combine the following types of
pair-copulae Gumbel Clayton

1.0
1.0

0.8
0.8

0.6
0.6

V
v

0.4
0.4

0.2
0.2

0.0
0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

u U

Gaussian Student

1.0
1.0

0.8
0.8

0.6
0.6

V
V

0.4
0.4

0.2
0.2

0.0
0.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0
U
U

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Outline Copulae R-vines Model selection Limitations and challenges References

Pair-copula constructions (PCC)

• Let X1 , X2 , X3 be stochastic variables with cdf F123 and margins

F1 , F2 and F3 .
• Their pdf f123 may be factorised as

f123 (x1 , x2 , x3 ) =f3 (x3 )f2|3 (x2 |x3 )f1|23 (x1 |x2 , x3 ). (2)

• Expressing (2) in terms of the marginal pdfs and pair-copula

densities by the repeated use of (1), one obtains the
corresponding PCC.

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Outline Copulae R-vines Model selection Limitations and challenges References

Pair-copula constructions (PCC)

• More specifically,

f123 (x1 , x2 , x3 ) =f1 (x1 )f2 (x2 )f3 (x3 )

·c13 (F1 (x1 ), F3 (x3 ))c23 (F2 (x2 ), F3 (x3 ))
·c12|3 (F1|3 (x1 |x3 ), F2|3 (x2 |x3 )).

• Since f123 = f1 f2 f3 · c123 , then

c123 (F1 (x1 ), F2 (x2 ), F3 (x3 )) =c13 (F1 (x1 ), F3 (x3 ))c23 (F2 (x2 ), F3 (x3 ))
(3)
·c12|3 (F1|3 (x1 |x3 ), F2|3 (x2 |x3 )),

where c13 , c23 and c12|3 are the copula densities corresponding
to F13 , F23 and F12|3 , respectively.

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Pair-copula constructions (PCC)

• A five-dimensional copula may be decomposed as:

c12345 = c12 · c23 · c34 · c45 Level 1

·c13|2 · c24|3 · c35|4 Level 2
·c14|23 · c25|34 Level 3
·c15|234 Level 4.

• The copulae are organised in levels according to the number

of conditioning variables.
• Expression (3) is one of the three possible decompositions of
c123 , while in the five-dimensional case, there are as many as
480 different constructions.
• To help categorising and building them, Bedford and Cooke
[2001, 2002] and Kurowicka and Cooke [2006] introduced the
graphical models called vines.
Ingrid Hobæk Haff How to select a good vine
Outline Copulae R-vines Model selection Limitations and challenges References

Vines in 5 dimensions

2 3
12 13 4
1 2 3 4 5 T1 1 14 T1
12 23 34 45 5
15
23|1 13 24|1 14
12 23 34 45 T2
13|2 24|3 35|4 12 T2
25|1 15

13|2 24|3 35|4 T3 34|12 24|1

23|1 T3
14|23 25|34 25|1
35|12

14|23 25|34 T4 34|12 35|12 T4

15|234 45|123

D-vine C-vine

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Outline Copulae R-vines Model selection Limitations and challenges References

Regular vine
6 5 1 2
6,5 5,1 2,1

7,5 3,1 4,3

7 3 4 T1

6,1| 5 5,3| 1 3,2| 1

6,5 5,1 3,1 2,1

1,7|5 4,1| 3

7,5 4,3 T2

6,3| 15 5,2| 31
6,1| 5 5,3| 1 3,2| 1

7,3| 15 4,2| 13

7,1| 5 4,1| 3 T3

6,2| 315 5,4| 231

6,3| 15 5,2| 31 4,2| 13

7,2| 315

7,3| 15 T4

7,6| 2315 6,4| 2315

7,2| 315 6,2| 315 5,4| 231 T5

7,4| 62315
7,6| 2315 6,4| 2315 T6

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Outline Copulae R-vines Model selection Limitations and challenges References

Regular vines

• Many of the pair-copula arguments are conditional

distributions.
• These can be evaluated using a recursive formula [Joe, 1996]:

∂Cxvj |v −j (F (x|v −j ), F (vj |v −j ))

F (x|v ) = .
∂F (vj |v −j )

• In regular vines (R-vines), the copulae in question are, by

construction, always present in the preceding levels of the
structure.
• Inference on PCCs is in general demanding, whereas the
subclass of R-vines has many appealing computational
properties.

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Outline Copulae R-vines Model selection Limitations and challenges References

Vine matrix

• Dißmann et al. [2013] have proposed an efficient way of

storing the indices involved in the pair-copulae in a lower
triangular matrix:
 
1
c12345 = c12 · c23 · c34 · c45 5 2 
·c13|2 · c24|3 · c35|4  
←→ 4 5 3 
·c14|23 · c25|34 
3 4 5 4 

·c15|234
2 3 4 5 5

• The density of the R-vine may then be written in terms of the

indices of this matrix.

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Outline Copulae R-vines Model selection Limitations and challenges References

Vine inference

• Inference on these constructions requires

(i) the choice of structure
(ii) the choice of each pair-copula type
(iii) the estimation of the copula parameters.
• In principle, these three steps should be performed
simultaneously.
• Gruber and Czado [2016] have proposed a Bayesian method
for doing this, but computational complexity makes this
infeasible in medium to high dimensions.
• In practice, the three inference steps are therefore performed
sequentially.

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Outline Copulae R-vines Model selection Limitations and challenges References

Parameter estimation

• An R-vine is a special type of multivariate copula.

• When the its structure and copula types are given, one may in
principle use any estimator for multivariate copulae to
estimate its parameters.
• The model consists of an R-vine with parameters θ, combined
with univariate margins with parameters α.
• Even for rather low dimensions, the total number of
parameters is high.
• In higher dimensions, one therefore performs the estimation in
several steps.

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Outline Copulae R-vines Model selection Limitations and challenges References

Parameter estimation

• The log-likelihood function can be written as

l(α, θ; x 1 , . . . , x n ) = lM (α; x 1 , . . . , x n )+lC (θ; u 1 (α), . . . , u n (α)),

where u k (α) = (F1 (x1k ; α), . . . , Fd (xdk ; α)).

• The terms of lC can be grouped according to the level they
belong to, and the terms for level l depend on the copulae
from the levels 1, . . . , l, but not the ones after.
• The copula parameters may therefore be estimated level by
level, or even copula by copula if none of the copulae share
parameters.
• The state-of-the-art is to
1. estimate α in a separate step,
2. estimate θ levelPby level, using Fi (xik ; α̂) or
1 n
Fin (xik ) = n+1 j=1 I (xij ≤ xik ) as estimates of uki (α).

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Structure selection

• Two main types of structure selection strategies have been

proposed:
• building the vine top-down, with the aim of minimising the
dependence in the top levels
• building the vine bottom-up, with the aim of maximising the
dependence in the first levels.
• A procedure of the first type, based on partial correlations, is
suggested by Kurowicka [2011a].
• Dißmann et al. [2013] propose a procedure of the second type
based on Kendall’s τ coefficients.
• The latter has become the state-of-the-art.

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Outline Copulae R-vines Model selection Limitations and challenges References

Structure selection

• A key to the algorithm of Dißmann et al. [2013] is that each

level of an R-vine is a spanning tree.
• This is due to the proximity condition: two copulae from level
l can be combined into a copula on level l + 1 only if they
share all variables but one.
• The algorithm is:
1. Estimate τij for all pairs {i, j} ⊂ {1, . . . , d}. P
2. Select the spanning tree T1 that maximizes {i,j}∈T1 |τ̂ij |.
3. For levels l = 2, . . . , d − 2:
a. Estimate τij|v for all pairs {i, j} with conditioning set v ,
that fulfil the proximity condition.
b. P
Select the spanning tree Tl that maximises
{i,j}∈Tl |τ̂ij|v |.

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Outline Copulae R-vines Model selection Limitations and challenges References

Structure selection
We wish to construct an R-vine on five variables.
3 − 5 − 2 − 4
• Level 1 : For all 15 pairs {i, j}, we estimate τij .
|
There are 125 possible spanning trees. Assume
1
that this is the winner tree:
• Level 2 : There are now 3 possible spanning trees
15 − 25 − 24
and 4 conditional Kendall’s τ s to estimate:
τ12|5 , τ13|5 , τ23|5 , τ45|2 . Assume that this is the |

winner tree: 35

• Level 3 : There are now 3 possible spanning trees

and 3 conditional Kendall’s τ s to estimate: 12|5 − 45|2 − 23|5
τ13|25 , τ14|25 , τ34|25 . Assume that this is the winner
tree:
14|25 − 34|25
• Level 4 : This level is always given by the previous
ones.
Ingrid Hobæk Haff How to select a good vine
Outline Copulae R-vines Model selection Limitations and challenges References

Structure selection

• The unconditional Kendall’s τ s, needed to construct the first

level of the vine, can be estimated empirically.
• From the second level, the conditional Kendall’s τ s, τij|v , are
estimated semi-parametrically, as the empirical Kendall’s τ of
ûi|v and ûj|v , that are estimated parametrically based on
copulae from the previous level.
• This requires the simultaneous choice of copula types and
parameter estimation.
• Common practice is to select the type of each copula
separately by
1. computing the AIC for a list of candidate copulae
2. choosing the one with the best AIC.

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Model reduction

• A 20-dimensional (full) R-vine has at least 190 parameters.

For a 50-dimensional one, the number is at least 1225.
• In high-dimensional applications, it is therefore necessary to
reduce the number of parameters.
• One strategy is to identify independence copulae among the
pair-copulae.
• When c14|23 is an independence copula, it means that
X1 ⊥⊥ X4 |X2 , X3 and c14|23 (u, v ) = 1.
• There are two main methods for doing this: pruning and
truncation (Kurowicka [2011b], Brechmann et al. [2012],
Brechmann and Joe [2015]).

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Model reduction
• Pruning consists in testing each of the copulae in the
construction for independence.
• Typically, Cij|v is tested for independence by testing whether
τij|v is significantly different from 0.
• Truncation consists in finding a level after which all copulae
can be set to independence.
• Starting with a one-level vine, truncation is performed as
follows:
1. test whether one extra level of copulae makes the model
significantly better
2. if yes and the number of levels is < d − 1, return to 1
3. else return the truncation level.
• The log-likelihood ratio test of Vuong [1989] for non-nested
hypotheses is used as a criterion in step 1.
• The structure of each new level is selected using the algorithm
of Dißmann et al. [2013].
Ingrid Hobæk Haff How to select a good vine
Outline Copulae R-vines Model selection Limitations and challenges References

Model reduction
Pruning
c12 · c23 · c34 · c45
·c13|2 · c24|3 · c35|4
·c14|23 · c25|34
·c15|234

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Model reduction
Pruning Truncation
c12 · c23 · c34 · c45 c12 · c23 · c34 · c45
·c13|2 · c24|3 · c35|4 ·c13|2 · c24|3 · c35|4
·c14|23 · c25|34 − − − − −−
·c15|234 ·c14|23 · c25|34
·c15|234

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Model reduction
Pruning Truncation
c12 · c23 · c34 · c45 c12 · c23 · c34 · c45
·c13|2 · c24|3 · c35|4 ·c13|2 · c24|3 · c35|4
·c14|23 · c25|34 − − − − −−
·c15|234 ·c14|23 · c25|34
·c15|234

& .

c12 · c23 · c34 · c45

·c13|2 · c24|3 · c35|4
− − − − −−
·c14|23 · c25|34
·c15|234

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Outline Copulae R-vines Model selection Limitations and challenges References

Example 1: compound events

• In climatology, a compound event denotes an extreme event

that is caused by a combination of climate and weather
variables, that are not necessarily in an extreme state.
• In this setting, it is very important to model the dependence
between the various variables, and especially in the tails.
• Vines have been used to model the relationship between sea
surge and water levels in rivers running to the coast in
question.
• The vine was selected based on the standard vine selection
algorithm.
• A closer inspection showed that the AIC values for the top
five copulae were almost the same.
• The tail behaviour of these copulae was however widely
different.
Ingrid Hobæk Haff How to select a good vine
Outline Copulae R-vines Model selection Limitations and challenges References

Example 2: abalone data

• The data originate from a study by the Tasmanian

Aquaculture and Fisheries Institute.
• The harvest of abalones is subject to quotas.
• These quotas are based on the age distribution of the
abalones.
• Determining an abalone’s age is a highly time-consuming task.
• Hence, one would like to predict the age based on physical
measurements, such as weight and height.

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Outline Copulae R-vines Model selection Limitations and challenges References

Example 2: abalone data

• The Abalone data set was originally used for this purpose, and
consists of 4,177 samples of:
1. Sex 4. Height 7. Viscera weight
2. Length 5. Whole weight 8. Shell weight
3. Diameter 6. Shucked weight 9. Age.
• A vine model was used to estimate this conditional
distribution.
• The standard vine selection and truncation algorithms,
combined with pruning, resulted in a vine with 5 levels and no
independence copulae below this level.
• The estimated conditional distribution of age given the other
variables based on a one-level vine (with 7 parameters) is
almost the same as the one based on the selected, “best” vine
(with 25 parameters).
Ingrid Hobæk Haff How to select a good vine
Outline Copulae R-vines Model selection Limitations and challenges References

Limitations

• Most of the mentioned inference methods are heuristic.

• The selection and reduction methods
• do not take into account the intended use of the model
• are performed level by level
• are conditioned on the choices of copula types in preceding
levels.
• The truncation approach
• relies heavily on the model selection algorithm
• only considers whether all copulae after a certain level should
be independence.
• The method for choosing copula types
• does not take into account the intended use of the model
• is based on AIC, and usually combined with semi-parametric
estimation, which has been shown to be incorrect [Grønneberg
and Hjort, 2014].

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Challenges

• What should the ”benchmark” model be?

• The number of possible R-vine structures for a given data set
d−2
is huge even for medium dimensions (2( 2 )−1 d! for d
variables).
• When combined with all possible combinations of copula
types from even a moderately long list of candidates, the
number of possible vines becomes gargantuan.
• A smart (greedy) search algorithm for proposing candidate
models is therefore necessary.
• Perhaps one could do the selection in two steps:
• select the structure for an R-vine consisting of non-parametric
copulae
• select the parametric copula types when the structure is fixed.

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

Challenges

• Parameters/measures related to the vine rarely have closed

form expressions.
• The computation of potential ”focus parameters” will
generally require Monte Carlo methods.
• To make a focussed selection criterion computationally
efficient, one therefore needs to find good approximations to
the focus parameter.

Ingrid Hobæk Haff How to select a good vine

Outline Copulae R-vines Model selection Limitations and challenges References

A. Sklar. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Stat.

Univ. Paris, 8, 1959.
C. Genest, H. U. Gerber, M. J. Goovaerts, and R. J. A. Laeven. Editorial to the
special issue on modeling and measurement of multivariate risk in insurance and
finance. Insurance: Mathematics and Economics, 44(2), 2009.
H. Joe. Multivariate Models and Dependence Concepts. Chapman & Hall, London,
1997.
T. Bedford and R.M. Cooke. Probabilistic density decomposition for conditionally
dependent random variables modeled by vines. Annals of mathematics and
Artificial Intelligence, 32:245–268, 2001.
T. Bedford and R.M. Cooke. Vines – a new graphical model for dependent random
variables. Annals of Statistics, 30(4):1031–1068, 2002.
D. Kurowicka and R.M. Cooke. Uncertainty Analysis with High Dimensional
Dependence Modelling. Wiley, New York, 2006.
H. Joe. Distributions with Fixed Marginals and Related Topics, chapter Families of
m-variate distributions with given margins and m(m-1)/2 dependence parameters.
IMS, Hayward, CA, 1996.
Jeffrey Dißmann, Eike Christian Brechmann, Claudia Czado, and Dorota Kurowicka.
Selecting and estimating regular vine copulae and application to financial returns.
Computational Statistics and Data Analysis, 59:52–69, 2013.
L.F. Gruber and C. Czado. Bayesian model selection of regular vine copulas. Working
paper, 2016.
D. Kurowicka. Dependence Modeling: Vine Copula Handbook, chapter Optimal
truncation of vines, pages 233–248. World Scientific Publishing Co., 2011a.
Ingrid Hobæk Haff How to select a good vine
Outline Copulae R-vines Model selection Limitations and challenges References

D. Kurowicka. Optimal truncation of vines. In D. Kurowicka and H. Joe, editors,

Dependence Modeling: Vine Copula Handbook. World Scientific Publishing Co.,
2011b.
E.C. Brechmann, C. Czado, and K. Aas. Truncated regular vines in high dimensions
with application to financial data. Canadian Journal of Statistics, 40:68–85, 2012.
Eike C. Brechmann and Harry Joe. Truncation of vine copulas using fit indices.
Journal of Multivariate Analysis, 138:19–33, 2015.
Q. H. Vuong. Likelihood ratio tests for model selection and non-nested hypotheses.
Econometrica, 57:307–333, 1989.
S. Grønneberg and N.L. Hjort. The copula information criteria. Scandinavian Journal
of Statistics, 41:436–459, 2014.

Ingrid Hobæk Haff How to select a good vine