Copulas: a Review and Recent Developments (2007)

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We may also characterize other statistics which are relevant in reliability, lifemodeling or risk analysis. For example, one could be interested in the range X n:n ¡X 1:n or subranges X r1 :n ¡ X r2 :n for r 1 >r 2 . However, in order to derive explicitformulas, we need the joint distribution of X r1 :n and X r2 :n. In the case of independentand identically distributed random variables, Balakrishnan and Cohen (1991) givemore friendly formulas for the density. Nelsen (2003) found the copula C 1;n of X 1:nand X n:n :C 1;n (u; v) =v ¡ [maxf(1 ¡ u) 1 1n + v n ¡ 1; 0g] n ;see also Schmitz (2004).In the general case, the problem is open. One solution is then to use Monte Carlomethods, as suggested by Georges et al. (2001). A recent study on the degree ofassociation of pairs of ordered random variables is provided by Averous et al. (2005).In Anjos et al. (2005) we give a copula representation of the joint distributionfunction of r-th and s-th order statistics corresponding to X and Y giventhe associated copula C as follows. Consider a bivariate distribution function withcontinuous margins and n independent observations from the population (X; Y ).Let (X 1 ;Y 1 );::: ;(X n ;Y n ); n ¸ 2, be a sample from continuous distribution withcopula C and marginals F and G respectively. Let X r:n and Y s:n be the orderstatistics of the sample, 1 · r; s · n: Since F (x) andG(y) are continuous thepairs f(X 1 ;Y 1 );::: ;(X n ;Y n )g can be transformed into f(U 1 ;V 1 );::: ;(U n ;V n )g byU i = F (X i ) » U(0; 1) and V i = G(Y i ) » U(0; 1). Therefore, we get P (X r:n ·x; Y s:n · y) =P (U r:n · u; V s:n · v); where U r:n and V s:n are r-th and s-th orderstatistics corresponding to n independent observations from (U; V ).The marginal distributions of P (U r:n · u; V s:n · v) are Beta distributed randomvariables, i.e. U r:n » Beta(r; n ¡r +1) and V s:n » Beta(s; n¡s+1). Let ¯¡1¯¡1r;n¡r+1 ands;n¡s+1 be the inverses of these Beta distributions. The copula associated to orderstatistics of the pair (X r:n ;Y s:n ) is the same copula of the pair (U r:n ;V s:n ), i.e.C Xr:n ;Y s:n(w; t) =C Ur:n ;V s:n(w; t) =H Ur:n ;V s:n¡¯¡1r;n¡r+1(w);¯¡1s;n¡s+1(t) ¢ :Under the above notations the copula C Ur:n ;V s:nis given byC Ur:n;V s:n(w; t) =nXj=rnX X n!C ¡¯¡1r;n¡r+1(w);¯¡1s;n¡s+1(t) ¢ mm!(j ¡ m)!(k ¡ m)!(n ¡ j ¡ k + m)!k=sm(4)£ [¯¡1r;n¡r+1 (w) ¡ C(¯¡1r;n¡r+1 (w);¯¡1s;n¡s+1 (t))]j¡m£ [¯¡1s;n¡s+1£ [1 ¡ ¯¡1r;n¡r+1 (w) ¡ ¯¡1(t) ¡ C(¯¡1r;n¡r+1 (w);¯¡1s;n¡s+1 (t))]k¡ms;n¡s+1(t)+C¡¯¡1r;n¡r+1 (w);¯¡1where the third summation is over m 2 [max(0;j+ k ¡ n);min(j; k)]:s;n¡s+1 (t)¢ ] n¡j¡k+m ;20
Let R u = P nj=1 IfU j · ug and R v = P nj=1 IfV j · vg. In Barakat (2001) propertiesof joint distribution (R u ;R v ) are investigated, and as consequences, limitingdistribution results are obtained for the vector (X r:n ;Y s:n )where1· r; s · n. For¯xed r; s ¸ 1, as n !1,thepair(r; s) is called ¯xed rank (or the case of extremeorder statistics). When r; s !1as n !1,(r; s) is called increasing rank. Oneparticular rate of increase of special interest is when r n ! ¸1 and s n ! ¸2 as n !1such that 0 · ¸1;¸2 < 1or¸1 =0;¸2 = 1. Additionally, nine other cases coveringthe possible asymptotic distributions of bivariate order statistics are presented. Weconsider only the increasing rank case, but the method elaborated can be appliedsimilarly for the remaining cases.In Anjos et al. (2005) we extract the asymptotic copula, denoted by C a ,fromthelimiting distribution of (R u ;R v ) and use the corresponding approximation to evaluatethe joint distribution function of order statistics (X r:n ;Y s:n ); as follows.Theorem (weak convergence, Anjos et al. (2005)). Let min ¡ n ¡ r; r ¢ !1and min ¡ n ¡ s; s ¢ !1when n !1. Furthermore, let r n ! ¸1 and s n ! ¸2 asn !1such that 0 · ¸1;¸2 < 1 or ¸1 =0and ¸2 =1.Ifr ¡ nupnu(1 ¡ u)! ¿ 1ands ¡ nvpnv(1 ¡ v)! ¿ 2hold for given u; v 2 (0; 1) and ¯xed constants ¿ 1 and ¿ 2 , and additionally½ Ru ;R v= Corr(R u ;R v ) n !1 ¡¡¡¡¡¡¡¡! ½;thenÃPR u ¡ nupnu(1 ¡ u)·r ¡ nu pnu(1 ¡ u);R v ¡ nvpnv(1 ¡ v)·!p s ¡ nvnv(1 ¡ v)d ¡¡¡¡¡!n !1 © ½ (¿ 1 ;¿ 2 );where © ½ is the joint distribution function of bivariate normal distribution with zeromean vector and correlation coe±cient ½ = pC(u;v)¡uv¥uv(1¡u)(1¡v)Since C(w; t) =H(F ¡1 (w);G ¡1 (t)), the asymptotic copula C a of (R u ;R v )isgivenby C a (w; t) =© ½ (© ¡1 (w); © ¡1 (t)) and the corresponding survival copula has the formC a (w; t) =© ½ (© ¡1 (w); © ¡1 (t)):Under conditions in last theorem H Ur:n;Vs:n (u; v) can be approximated byµµ H Ur:n;Vs:n (u; v) ¼ © ½ ©µ¯r;n¡r+1 ¡1 (u); © ¡1 ¯s;n¡s+1 (v) ;since H Ur:n ;V s:n(u; v) =H Ru ;R v(r; s). Using the well know relation C(w; t) =w +t ¡ 1+C(1 ¡ w; 1 ¡ t) whereC(w; t) is the survival copula, we obtain the following21
Page 3 and 4: can be employed in probability theo
Page 5 and 6: y Genest et al. (1993) and Shi and
Page 7 and 8: Table 1: Distribution of (C 1 jC 2
Page 9 and 10: It should be mentioned here that gr
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: (1999)) even in the cases where the
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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