Copulas: a Review and Recent Developments (2007)

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