Page 138 - Special Topic Session (STS) - Volume 2

STS474 Hideatsu T.
has a skew t-distribution (Demarta and McNeil, 2005). The associated
copula is called a skew t-copula. = (1, . . . , )′ is a parameter
representing the degree of “distortion”.
(3) Nested Archimedean copula: The simplest extension of Archimedean
copula (2.1) is given by ( , . . . , ) = ( −1 ( ) + · · · + −1 ( )), and

1
1

its dependence structure is too symmetric. Thus the following nested
Archimedean copulas have been introduced in the literature.
 Fully nested Archimedean copula: Starting with ( , , ) ∶=
2
1
1
( −1 ( ) + −1 ( )), define recursively for d ≥ 3,
1
1
2
1
1
−1
( , . . . , ; , . . . , −1 ): = ( ( ) + 1 −1 (( , . . . , ; , . . . , −1 )))
1

1
2

2
1
1
1
 Partially nested Archimedean copula
() = (( , . . . , 1, 1 ; ), . . . , ( ,1 , . . . , , ; ) ; )
1,1
0
1

= ( −1 ∘ ( 1 −1 ( ) + ⋯ + 1 −1 ( 1, 1 )) + ⋯ + 0 −1 ∘
1,1
0
0
1
( −1 ( ,1 ) + ⋯ + 1 −1 ( , )))

See Joe (1997) or McNeil (2008) for exposition. Simple examples are
( , , ) = ( 1 −1 ( ) + −1 ∘ ( −1 ( ) + 2 −1 ( ))),
2
1
1
3
1
2
1
2
3
2
( , , , ) = ( −1 ∘ ( −1 ( ) + ( )) + −1
−1
1
2
2
1
2
2
1
3
1
2
1
3
∘ ( −1 ( ) + 3 −1 ( ))).
3
3
4
3
Because of its apparent rooted tree structure, this class of copulas is suited
for spatial data; prior knowledge on the spatial structure among individuals
can be utilized to construct the tree structure of nested Archimedean
copula. Conversely, one can in fact apply the method in Segers and
Uyttendaele (2014) to estimate the tree structure and thereby find which
kind of dependence are left over by the standard spatial autoregression.

3.1. Analyzing Procedure
We suggest the following procedure for the statistical analysis. Note that
all three copula families we discussed above are parametric. For simplicity
suppose that ’s are i.i.d. We tacitly assume that the model (3.1) ℎ ∼

2
(, ) has been fitted, but the residuals do not show i.i.d. behavior.

(i) By looking at 2-by-2 scatter plot or any other means, we need to check
whether significant tail dependence and/or asymmetric dependence is
found. If there is, then elliptical copula could be eliminated from the
candidate copulas.
(ii) For all the above model (1)–(3), we can compute either maximum
likelihood or pseudo-likelihood estimates (Genest et al., 1995),
depending on the assumption on the one-dimensional marginals. Use
AIC (or some other information criteria) to search for the best-fitted
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