Copulas: a Review and Recent Developments (2007)

It should be mentioned here that grade transformation can be seen as analogousto a simple interpolation used by Schweitzer and Sklar (1974), and later by Nelsen(1999) as an example of a possible extension of any subcopula to a copula, whenmarginal distributions are not continuous. Due to its interesting properties, the gradetransformation is playing a special role in statistics enabling uni¯ed approach toanalysis of continuous, discrete or categorical data.2.2 Bernstein and Bertino's family of copulas2.2.1 Bernstein copulaSancetta and Satchell (2004) introduced a family of copulas de¯ned as Bernsteinpolynomials. The choice is motivated by the fact that Bernstein polynomials areclosed under di®erentiation and most of polynomial representations, e.g. Hermitepolynomials, Pad approximations, say, do not share the same properties in the contextof copula function.De¯nition (Bernstein copula, Sancetta and Satchell (2004)). Let ®( k 1m 1; ¢¢¢ ; k nbe real valued constant, k i 2f1; 2;:::g such that 1 · k i · m i ;i=1;::: ;n and substituteP ki ;m i(u i )=m i ()k i u k ii (1 ¡ u i) m i¡k i: If C Be :[0; 1] n ! [0; 1], whereC Be (u 1 ;::: ;u n )= X ¢¢¢ X µk1® ; ¢¢¢ ; k nP k1 ;mm 1 m1(u 1 ) :::P kn ;m n(u n )nk 1 k nsatis¯es the properties of the copula function from the Formal copula de¯nition, thenC Be is a Bernstein copula for any k i ¸ 1:The Bernstein copula generalizes the family of polynomial copula, which is aspecial case of copula with polynomial sections. An earlier reference for the Bernsteinpolinomial copula construction is Li et al. (1997). For more details on copulas withpolynomial sections see Nelsen (1999), Section 3.In Sancetta and Satchell (2004) are studied statistical properties in terms of bothdistributions and densities. In particular, it is shown that the coe±cients of theBernstein copula C Be have a direct interpretation as the points of some arbitraryapproximated copula C, i.e.C( k 1m 1; ¢¢¢ ; k nm n)=®( k 1m 1; ¢¢¢ ; k nm n).Sancetta and Satchell (2004) also develop a theory of approximation of multivariatedistributions in terms of Bernstein copula. They give the rate of consistency whenthe Bernstein copula density is estimated non-parametrically.The following important decomposition is achievednYC(u 1 ;::: ;u n )= u i + D(u 1 ;::: ;u n ); (u 1 ;::: ;u n ) 2 [0; 1] n ;i=1where D(u 1 ;::: ;u n ) is a perturbation term containing all information about the dependenceof (U 1 ;::: ;U n ), where U i » U(0; 1); i=1;::: ;n: This new representation9m n)