Copulas: a Review and Recent Developments (2007)

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Therefore various attempts of \continuisation" of random variables were madeover the years, with grade transformation being one of them. This approach is verygeneral, applicable to all random variables for which their distribution functions canbe de¯ned. The so-called grade transformation, proposed by Szczesny (1991), resultsin a continuous, uniformly distributed variables and therefore can be considered anextension of probability integral transform.De¯nition (grade transformation, Szczesny (1991)). Let X be a random variablewith distribution function F (x). Then the cumulative distribution of a variableX transformed by its distribution function can be expressed asZP (F (X) · u) = I F (F (x);u)dF (x);whereI F (F (x);u)=½ 1; if F (x) · u;0; if F (x) >u:Let us now substitute I F by the randomizing transformation I ¤ Fde¯ned asI ¤ F (F (x);u)= 8
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)
Page 3 and 4: can be employed in probability theo
Page 5 and 6: y Genest et al. (1993) and Shi and
Page 7: Table 1: Distribution of (C 1 jC 2
Page 11 and 12: Nelsen et al. (2003) have used Bert
Page 13 and 14: 2.4 Time dependent copulasIn practi
Page 15 and 16: De¯nition (conditional pseudo-copu
Page 17 and 18: (i) Ã 1 (x 1 ;::: ;x n )=x 1 + :::
Page 19 and 20: (1999)) even in the cases where the
Page 21 and 22: Let R u = P nj=1 IfU j · ug and R
Page 23: usually wish to aggregate two-dimen
Page 26 and 27: 3.3 Copula representation via a loc
Page 28 and 29: such thatC(u; v) =© ru;v¡© ¡1 (
Page 30 and 31: Fisher-Tippett Theorem (version for
Page 32 and 33: et al. (2000). To measure contagion
Page 34 and 35: and the non-exchangeable copula (AL
Page 36 and 37: 0 2 4 6 8 10 120 1 2 3 4 5 60 2 4 6
Page 38 and 39: Bouye, E., Durrleman, V. Nikeghbali
Page 40 and 41: Embrechts, P., Lindskog, F., McNeil
Page 42 and 43: Hsing, T, KlÄuppelberg, C., Kuhn,
Page 44 and 45: Nelsen, R., Quesada-Molina, J., Rod

Copulas: a Review and Recent Developments (2007)

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