Statistics and Probability Letters

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1 tatistics Probability Letters ) Contents lists available at ciencedirect tatistics Probability Letters journal homepage: On the relationships between copulas of order statistics marginal distributions Jorge Navarro a,, Fabio pizzichino b a Universidad de Murcia, pain b University of Rome La apienza, Italy a r t i c l e i n f o a b s t r a c t Article history: Received 5 May 2009 Received in revised form 30 November 2009 Accepted 30 November 2009 Available online 22 December 2009 In this paper we study the relationships between copulas of order statistics from heterogeneous samples the marginal distributions of the parent rom variables. pecifically, we study the copula of the order statistics obtained from a general rom vector X = X, X 2,..., X n ). We show that the copula of the order statistics from X only depends on the copula of X on the marginal distributions of X, X 2,..., X n through an exchangeable copula the average of the marginal distribution functions. We study in detail some relevant cases Elsevier B.V. All rights reserved.. Introduction Consider a vector F, F 2,..., F n ) of n arbitrarily chosen univariate distribution functions an arbitrary n-dimensional copula C. Then, the celebrated klar s theorem see, e.g., Nelsen, 2006, p. 8) ensures that there exists an n-dimensional distribution function F, that admits F, F 2,..., F n as its marginal distributions C as its connecting copula. In fact we can construct F by simply using the formula Fx, x 2,..., x n ) = C F x ), F 2 x 2 ),..., F n x n )). Thus no condition on the pair C, F, F 2,..., F n ) is requested for their compatibility. Consider now the connecting copula K the marginal distributions F :n, F 2:n,..., F n:n of the order statistics obtained from F. A well known result concerning order statistics see Avérous et al., 2005) is that the connecting copula of the order statistics of n independent identically distributed IID) rom variables X, X 2,..., X n, with common marginal distribution G, is a given copula that does not depend on G. That is, if C is the product copula F = F 2 = = F n = G, then K is a fixed copula i.e. it does not depend on G) called the order statistics copula. The picture radically changes if F i F j even if C is the product copula). In such a case we find that some relations exist both between K F, F 2,..., F n for K F :n, F 2:n,..., F n:n. Of course these relations depend on the choice of C. In this paper we are interested in discussing the relationships between the connecting copula of the order statistics the marginal distributions of the parent rom variables. In particular we show that for any vector of identically distributed rom variables, the connecting copula of the order statistics only depends on the connecting copula of the variables does not depend on their common marginal distribution. Furthermore, it will also emerge that this property cannot be generally true in the case of variables that have different marginal distributions. The problem considered here is equivalent to studying the copula for continuous ordered rom variables Y, Y 2,..., Y n, i.e., for variables such that PY Y 2 Y n ) =..) Corresponding address: Facultad de Matemáticas, Universidad de Murcia, 3000 Murcia, pain. Tel.: ; fax: address: jorgenav@um.es J. Navarro) /$ see front matter 2009 Elsevier B.V. All rights reserved. doi:0.06/j.spl

2 474 J. Navarro, F. pizzichino / tatistics Probability Letters ) Note that this problem reduces to the study of the vector of order statistics from a rom permutation of Y, Y 2,..., Y n. In fact, we will see that we can find a vector of exchangeable variables whose order statistics are jointly distributed as Y, Y 2,..., Y n ). Our paper is then strictly related with the wide literature concerning the distribution of order statistics; in particular we address the reader to basic references such as David Nagaraja 2003) Arnold et al. 2008). Trivially the condition.) implies that the marginal distributions are ordered in the usual stochastic sense. More details about the conditions to be satisfied by this vector of the marginal distributions have been given in the papers by Rychlik 994), Durante Jaworski 2008), Jaworski Rychlik 2008) Jaworski 2009) the references therein. ome copula representations for the distributions of order statistics were also obtained in Navarro pizzichino in press). However, the main purpose of this note is different since we want to underst what are the constraints to be satisfied by the connecting copula of a vector of order statistics. The rest of the paper is organized as follows. In ection 2 we develop a general analysis we give the main results. Examples enlightening details are presented in ection 3, where we study the general results for the bivariate case some particular cases of special interest. 2. Main results We consider a vector X = X, X 2,..., X n ) of rom variables with a continuous distribution function Fx, x 2,..., x n ). Let Cu, u 2,..., u n ) F, F 2,..., F n, respectively, denote the connecting copula the marginal distribution functions. Hence C is the joint distribution function of the variables F X ), F 2 X 2 ),..., F n X n ), from klar s theorem, Fx, x 2,..., x n ) = C F x ), F 2 x 2 ),..., F n x n )). It is well known that the rom variables X, X 2,..., X n are independent if only if the copula C is equal to the product copula C I u, u 2,..., u n ) = u u 2... u n for 0 < u i < i =, 2,..., n. Furthermore, we denote by P n the group of permutations of {, 2,..., n}, for π P n, set X π = X π), X π2),..., X πn) ). Concerning the joint distribution function F π the connecting copula C π of X π, we can write where F π x, x 2,..., x n ) = PX π) x, X π2) x 2,..., X πn) x n ) = C π Fπ) x ), F π2) x 2 ),..., F πn) x n ) ), C π u, u 2,..., u n ) = Cu π ), u π 2),..., u π n)). A rom vector X has an exchangeable distribution if X is equal in law to X π for all π P n or, equivalently, if its distribution function F is exchangeable, that is, if Fx, x 2,..., x n ) = Fx π), x π2),..., x πn) ) for all π P n. If F is exchangeable, then its copula C is exchangeable F = F 2 = = F n. Let us construct a rom vector X associated to X with an exchangeable distribution with order statistics which are equal in law to that of X. This rom vector X can be called the exchangeable rom vector associated to X. Denote by Π a rom variable uniformly distributed over P n. We now consider X = XΠ, that is, X = Xπ with probability /. The exchangeable distribution function F of the vector X is the uniform mixture of Fπ for π P n, that is, Fx, x 2,..., x n ) = F π x, x 2,..., x n ) = C π Fπ) x ), F π) x 2 ),..., F πn) x n ) ) the common marginal distribution G is the average of the marginal distributions of X, X 2,..., X n, that is, Gx) = n F i x). n i= Hence, using klar s theorem, F can also be written as Fx, x 2,..., x n ) = CGx ), Gx 2 ),..., Gx n )), where C is the copula of X. Thence, if G u) = inf{x [, + ] : Gx) = u} for 0 u, C can be written as C u, u 2,..., u n ) = C π Fπ) G u ) ), F π2) G u 2 ) ),..., F πn) G u n ) )) for 0 < u i < i =, 2,..., n. Note that C = CC, F, F 2,..., F n ), that is, C depends both on C on the marginal distributions F, F 2,..., F n. Also note that C is exchangeable but it is not necessarily equal to the exchangeable copula

3 J. Navarro, F. pizzichino / tatistics Probability Letters ) obtained from C, C Π u, u 2,..., u n ) = C π u, u 2,..., u n ). Note that C Π = C Π C), that is, C Π only depends on C. The exchangeable copulas C CΠ can be called the copula of mixtures the mixture of copulas, respectively. Now we can state the following properties. Lemma 2.. With the notation introduced above, the following properties hold. i) If F = F 2 = = F n, then C = CΠ. ii) If C is exchangeable, then C = C Π. iii) If C is exchangeable F = F 2 = = F n, then C = C Π = C. Proof. i) If F = F 2 = = F n, then G = F = F 2 = = F n. Thence, C u, u 2,..., u n ) = C π Fπ) G u ) ), F π2) G u 2 ) ),..., F πn) G u n ) )) = C π u, u 2,..., u n ) = C Π u, u 2,..., u n ). ii) If C is exchangeable, then C = C π for all π P n. Thence, C Π u, u 2,..., u n ) = C π u, u 2,..., u n ) = Cu, u 2,..., u n ) = Cu, u 2,..., u n ). The proof of iii) is immediate from i) ii). Note that if C = CΠ holds, then C only depends on C it does not depend on F, F 2,..., F n. We now pass to consider the joint distribution of the vector X O = X :n, X 2:n,..., X n:n ) of order statistics obtained from X = X, X 2,..., X n ). Analogously, we denote by XO = X:n, X2:n,..., Xn:n ) the vector of order statistics obtained from X. As it is easy to check see, e.g., pizzichino 200) Navarro et al. submitted for publication)), we have the following property. Lemma 2.2. The joint distribution of X O coincides with the one of XO. The proof is immediate since X O is equal in law to X π ) O for all π P n. As an immediate consequence we have that the copula of X O is equal to the copula of XO which only depends on C G. Moreover we have the following property. Proposition 2.3. If F = F 2 = = F n, then the copula of X O only depends on C. Proof. From Lemma 2. i) we have that if F = F 2 = = F n, then C = CΠ. Note that C is the joint distribution of U = G X ), U 2 = G X2 ),..., U n = G Xn ), that it only depends on C. Also note that the order statistics U :n, U 2:n,..., U n:n from U, U 2,..., U n satisfy U i:n = G Xi:n ) for i =, 2,..., n. Hence, the copula of X:n, X2:n,..., Xn:n ) is equal to the copula of U :n, U 2:n,..., U n:n ) which only depends on C. Now we can obtain the main result of the paper which can be stated as follow. Proposition 2.4. The copula of X O only depends on C = CC, F, F 2,..., F n ) the marginal distributions of X O only depend on C = CC, F, F 2,..., F n ) on G. Proof. From Lemma 2.2 we have that the joint distribution of X O coincides with the one of XO. Hence their copulas marginal distributions are equal. Moreover, as X, X2,..., Xn are identically distributed ID), from Proposition 2.3 we have that the copula of XO only depends on C. Hence the copula of XO only depends on C. Moreover, the marginal distributions of XO only depend on the distribution of X, that is, on C on G. Let us now denote by K F i:n i =, 2,..., n) the copula the marginal distributions of X :n, X 2:n,..., X n:n ), respectively. Notice that K only depends on C so we can write K = K C. However, note that C depends both on C on F, F 2,..., F n. Furthermore, F i:n x) = PX i:n x) = P Xi:n x) = PU i:n Gx)) for i =, 2,..., n, where U :n, U 2:n,..., U n:n are the order statistics from U i = G Xi:n ), i =, 2,..., n.

4 476 J. Navarro, F. pizzichino / tatistics Probability Letters ) From Lemma 2. Proposition 2.4, we can consider four special cases: i) If F = F 2 = = F n, then C = CΠ hence K only depends on the exchangeable copula C Π, that is, the mixture of copulas obtained from C. ii) If C is exchangeable, then C = C Π C u, u 2,..., u n ) = C F π) G u ) ), F π2) G u 2 ) ),..., F πn) G u n ) )). iii) If F is exchangeable, then K only depends on C = C Π = C. iv) If F = F 2 = = F n C is the product copula, then K is a fixed copula known as the order statistics copula. This last result was given by Avérous et al. 2005). 3. The bivariate case In this section we analyze in detail the special case of n = 2 rom variables X, X 2 with a bivariate distribution function F, connecting copula C with marginal distribution functions F F 2, respectively. Of course, from klar s theorem, we have Fx, y) = CF x), F 2 y)). Now consider the exchangeable rom vector X, X2 ) obtained from X, X 2 ) with distribution function Fx, y) = 2 Fx, y) + Fy, x) 2 = 2 CF x), F 2 y)) + 2 CF y), F 2 x)) = CGx), Gy)), where C is the connecting copula G = F + F 2 )/2 is the common marginal distribution. Hence C can be written as Cu, v) = 2 C F G u)), F 2 G v)) ) + 2 CF G v)), F 2 G u))). 3.2) The joint distribution function F O of the pair of order statistics X, X 2:2 ) can be written as F O x, y) = Fx, y) + Fy, x) Fx, x) 3.3) F O x, y) = Fy, y) for y < x the marginal distributions as F x) = F x) + F 2 x) Fx, x) = F x) + F 2 x) CF x), F 2 x)) F 2:2 y) = Fy, y) = CF y), F 2 y)). Notice that X, X 2:2 ) X, X2:2 ) are equal in law. Hence these equations can be respectively rewritten as F x) = 2Gx) CGx), Gx)) F 2:2 y) = CGy), Gy)). It is then self-evident that F F 2:2 only depend on the pair C G. Analogously, the joint distribution can be rewritten as F O x, y) = 2 CGx), Gy)) CGx), Gx)), 3.7) F O x, y) = CGy), Gy)) = F2:2 y) for y < x. 3.8) On the other h, if K is the connecting copula of the order statistics, then from klar s theorem, we have Hence, F O x, y) = K F x), F 2:2 y)). Ku, v) = F O F u), F v)). 2:2 3.) 3.4) 3.5) 3.6)

5 J. Navarro, F. pizzichino / tatistics Probability Letters ) Thence, for F u) F2:2 v), Ku, v) = 2 CGF u)), GF v))) 2:2 CGF u)), GF u))). 3.9) Analogously, for F u) > F2:2 v), we have Ku, v) = v. Recall that from 3.5) 3.6), F F 2:2 only depend on C G. Then Eqs. 3.9) 3.0) not only show that K depends on C, F, F 2 through C G, but also point out the relationships between K, on one side, F, F 2:2 on the other side. Also note that the support of K is included in the region {u, v) 0, ) 2 : F u) F2:2 v)}. Note that 3.6) can be rewritten as F 2:2 y) = δ CGy)), where δ Cu) = Cu, u) is the diagonal section of the copula C. Hence, if δ z) = inf{u [0, ] : δ Cu) = z}, then C z) = 2:2 G δ z)). F C Analogously, F can be related to the diagonal section of the survival copula associated to C as follows. The survival copula of X, X2 ) is a copula such that the survival function can be written as P X > x, X2 > y) = Gx), Gy)). It is well known that can be written as u, v) = u + v + C u, v). Also note that Thence F x) = P X > x, X2 > x) = Gx), Gx)). z) = G δ z)), F where δ u) = u, u) is the diagonal section of the survival copula δ z) = inf{u [0, ] : δ u) = z}. Note that δ only depend on C. Next we consider some special cases examples. Example 3. C = C I ). If X X 2 are independent, with marginal distribution functions F F 2, respectively, then Fx, y) = F x)f 2 y) Cu, v) = C I u, v) = uv is exchangeable. Hence, from 3.) 3.2), we have Fx, y) = where G = F + F 2 )/2 C u, v) = 2 F x)f 2 y) + 2 F y)f 2 x) = CGx), Gy)), 2 F G u))f 2 G v)) + 2 F 2G u))f G v)). Of course, C only depends on F F 2. Analogously, from 3.3) 3.4), the joint distribution of the order statistics is given by F O x, y) = F x)f 2 y) + F y)f 2 x) F x)f 2 x) F O x, y) = F y)f 2 y) for y < x. Alternatively, from 3.7) 3.8), it can be written as F O x, y) = 2 CGx), Gy)) CGx), Gx)) F O x, y) = CGy), Gy)) for y < x. The marginal distributions of the order statistics can be written as F x) = F x) + F 2 x) F x)f 2 x) = 2Gx) CGx), Gx)) = H Gx)) F 2:2 y) = F y)f 2 y) = CGy), Gy)) = H2:2 Gy)), where H u) = 2u Cu, u) H2:2 v) = Cv, v) depend on F F 2. Actually they only depend on the diagonal section of C = CF, F 2 ). Hence, F u) = i:2 G H i:2 u)) for i =, 2. Thence the connecting copula of the order statistics can be written as Ku, v) = F F u))f 2F v)) + 2:2 F F 2:2 v))f 2F u)) F F u))f 2G u)) = 2 CH u), H v)) 2:2 CH u), H u)) 3.0)

6 478 J. Navarro, F. pizzichino / tatistics Probability Letters ) for F u) F2:2 v) Ku, v) = v for F2:2 v) < F u). Note that in this case, F u) F2:2 v) resp. >) is equivalent to H u) H v) 2:2 >). Of course, C K are proper copulas that only depend on F F 2, so they can be denoted by CF,F 2 K F,F 2. Example 3.2 F = F 2 ). If X X 2 are identically distributed, then G = F = F 2 Fx, y) = CGx), Gy)). Hence, from 3.) 3.2), we have where Fx, y) = 2 CGx), Gy)) + 2 CGy), Gx)) = CGx), Gy)), 3.) Cu, v) = CΠ u, v) = 2 Cu, v) + Cv, u). 2 Thus C only depends on C does not depend on G = F = F 2. Analogously, from 3.7) 3.8), the joint distribution of the order statistics is given by F O x, y) = 2 CGx), Gy)) CGx), Gx)) 3.2) F O x, y) = CGy), Gy)) for y < x, from 3.5) 3.6), the marginal distributions by F x) = 2Gx) CGx), Gx)) = δ Gx)) 3.3) F 2:2 y) = CGy), Gy)) = δ C Gy)), where δ C x) = Cx, x) δ x) = x, x) are the diagonal sections of the connecting survival copulas of X, X 2 ). Thence F u) = G δ u)) v) = 2:2 G δ v)). C F Finally, the connecting copula of the order statistics can be written as Ku, v) = 2 C δ for v δ C δ 3.4) u), δ v)) δ C C δ u)) 3.5) u)) Ku, v) = v for v < δ C δ u)). Example 3.3 F exchangeable). This is a special case of Example 3.2 when C is exchangeable. Then C = CΠ = C. Hence, from 3.), we have Fx, y) = CGx), Gy)). Analogously, from 3.2), the joint distribution of the order statistics is given by F O x, y) = 2CGx), Gy)) CGx), Gx)) F O x, y) = CGy), Gy)) for y < x. Analogously, the marginal distributions are given by 3.3) 3.4). Thence, from 3.5), the connecting copula of the order statistics can be written as Ku, v) = 2C δ for v δ C δ u), δ v)) δ C C δ u)) 3.6) u)) Ku, v) = v for v < δ C δ u)). Example 3.4 F = F 2 C = C I ). This is a well known example Avérous et al., 2005) which is a special case of Examples Then we have Cu, v) = Cu, v) = uv. Hence Fx, y) = Gx)Gy), where G = F = F 2. Analogously, the joint distribution of the order statistics is given by F O x, y) = 2Gx)Gy) G 2 x) F O x, y) = G 2 y) for y < x the marginal distributions by F x) = 2Gx) G 2 x) F 2:2 y) = G 2 y). Thence, from 3.6), the connecting copula of the order statistics from IID rom variables can be written as Ku, v) = 2 u) /2 )v /2 ) u) /2 ) 2

7 J. Navarro, F. pizzichino / tatistics Probability Letters ) for 0 < u < u) /2 ) 2 v < Ku, v) = v for 0 < u < 0 < v < u) /2 ) 2. Of course, K is a fixed copula that does not depend on G = F = F 2. Acknowledgements The authors wish to thank the anonymous reviewers for their several helpful comments. We are specially grateful to a reviewer who suggested the names of the copulas C CΠ. JN is partially supported by Ministerio de Ciencia y Tecnología under grant MTM Fundación éneca under grant 08627/PI/08. F is partially supported by Progetto di Ricerca Università apienza 2008 Interazione e Dipendenza nei Modelli tocastici. References Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., A First Course in Order tatistics. In: Classics in Applied Mathematics, vol. 54. ociety for Industrial Applied Mathematics IAM), Philadelphia, PA, Unabridged republication of the 992 original. Avérous, J., Genest, C., Kochar,.C., On the dependence structure of order statistics. J. Multivariate Anal. 94, David, H.A., Nagaraja, H.N., Order tatistics, third ed. John Wiley & ons, Hoboken, New Jersey. Durante, F., Jaworski, P., Absolutely continuous copulas with given diagonal sections. Comm. tatist. Theory Methods 37, Jaworski, P., On copulas their diagonals. Inform. ci. 79, Jaworski, P., Rychlik, T., On distributions of order statistics for absolutely continuous copulas with applications to reliability. Kybernetika 44, Navarro, J., pizzichino, F., 200. Comparisons of series parallel systems with components sharing the same copula. Appl. toch. Models Buss. Industry in press). Navarro, J., pizzichino, F., Balakrishnan, N., Average systems their role in the study of coherent systems submitted for publication). Nelsen, R.B., An Introduction to Copulas, 2nd ed.. pringer, New York. Rychlik, T., 994. Distributions expectations of order statistics for possibly dependent rom variables. J. Multivariate Anal. 48, pizzichino, F., 200. ubjective probability models for lifetimes. In: Monographs on tatistics Applied Probability, vol. 9. Chapman & Hall/CRC, Boca Raton, FL.