DOI 10.1007/s40065-012-0002-9 R E S E A R C H A RT I C L E
H. M. Barakat
The maximal correlation for the generalized order statistics
and dual generalized order statistics
Received: 16 April 2011 / Accepted: 3 October 2011 / Published online: 11 April 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com
Abstract In each of three exhaustive and distinct cases, it is found a distribution for which the correlation coefficient between the elements of the generalized order statistics (gos) is maximal. The corresponding result for the dual generalized order statistics (dgos) is derived for other three different distributions. Moreover, some interesting relations for the regression curves between the elements of gos and dgos based on these distributions are obtained. As a consequence of this result, a non-parametric criterion of independence between gos and between dgos in a general setting is derived.
Mathematics Subject Classification (2010) 60F05· 62E20 · 62E15 · 62G30
1 Introduction
Kamps [9] introduced the concept of the generalized order statistics (gos) as a unification of several models of ascendingly ordered random variables (rvs). It is known that ordinary order statistics (oos), upper record values, sequential order statistics and progressive type II censored order statistics are special cases of gos. Uniform gos U(r) ≡ U(r, n, k, ˜m), r = 1, 2, . . . , n, are defined by their density function
fU(1),...,U(n)(u1, . . . , un) = ⎛ ⎝n j=1 γj ⎞ ⎠ ⎛ ⎝n−1 j=1 (1 − uj)γj−γj+1−1 ⎞ ⎠ (1 − un)γn−1
on the cone{(u1, . . . , un) : 0 ≤ u1≤ · · · ≤ un< 1} ⊂ n, with parameters γ1, . . . , γn> 0. The parameters γ1, . . . , γnare defined byγn = k > 0 and γr = k + n − r + Mr, r = 1, 2, . . . , n − 1, where Mr=nj−1=rmj
and ˜m = (m1m2. . . mn−1) ∈ n−1. gos based on some distribution function (df) F are defined via the quantile H. M. Barakat (
B
)Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
transformation X(r) = F−1(U(r)), r = 1, 2, . . . , n. Consequently, X(1) ≤ X(2) ≤ · · · ≤ X(n) holds almost surely. Burkschat et al. [3] have introduced the concept of dual generalized order statistics (dgos) to enable a common approach to descendingly ordered rvs like reversed order statistics and lower record values. Uniform dgos Ud(r) ≡ Ud(r, n, k, ˜m), r = 1, 2, . . . , n, are defined by their density function
fUd(1),...,Ud(n)(u1, . . . , un) = ⎛ ⎝n j=1 γj ⎞ ⎠ ⎛ ⎝n−1 j=1 uγjj−γj+1−1 ⎞ ⎠ uγn−1 n
on the cone {(u1, . . . , un) : 1 ≥ u1 ≥ · · · ≥ un > 0} ⊂ n. The quantile transformation Xd(r) = Fd−1(Ud(r)), r = 1, 2, . . . , n, yields dgos based on arbitrary df Fd. Therefore, Xd(1), . . . , Xd(n) are arranged
in descending order almost surely.
Let Bj, 1 ≤ j ≤ n, be independent rvs with respective Beta distribution Beta(γj, 1), i.e., Bj follows a
power function distribution with exponentγj. The central distribution theoretical result concerning gos and
dgos is that they can, respectively, be defined by the product of the independent power function distributed rvs Bj, 1 ≤ j ≤ n (see Cramer [5] and Burkschat et al. [3]). Namely,
X(r) ∼ F−1 ⎛ ⎝1 −r j=1 Bj ⎞ ⎠ , r = 1, 2, . . . , n, (1.1) and Xd(r) ∼ Fd−1 ⎛ ⎝r j=1 Bj ⎞ ⎠ , r = 1, 2, . . . , n. (1.2)
In the classical oos, where ˜m = (00 · · · 0) and k = 1, Terrell [15] proved that, when the sample size n = 2, and if the oos X1:2 and X2:2(X1:2 ≤ X2:2) have finite variances, then their correlation coefficient satisfies the inequality corr(X1:2, X2:2) ≤ 1/2 with equality if, and only if, F is uniformly distributed. Later, by considering the maximum correlation possible between any square-integrable functions of the uniform oos, and using the modified Jacobi polynomial, Székely and Móri [14] have shown that
corr(Xr:n, Xs:n) ≤
r(n − s + 1)
s(n − r + 1), 1 ≤ r < s ≤ n, (1.3)
(here it is supposed that var(Xr:n) and var(Xs:n) are finite and the sample size can be arbitrary). An interesting
alternative proof of (1.3) is given by Rohatgi and Székely [12] (see David and Nagaraja [6]). In this paper, using the method of Rohatgi and Székely [12], we extend the above result to a wide subclass of gos and dgos. Namely, for any 1 ≤ r < s ≤ n, we consider the gos X(r), X(s) and the dgos Xd(r), Xd(s) for which m1= m2= · · · ms−1= m. Moreover, we consider the three exhaustive and distinct cases m+1 > 0, m+1 = 0
and m+ 1 < 0. Clearly the m−gos and m−dgos, where m1 = m2 = · · · = mn−1 = m (see Kamps [9] and
Burkschat et al. [3]), are special cases of these subclasses. Using the results of Kamps [9], Cramer [5] and Burkschat et al. [3], we can write explicitly the conditional density functions of X(r) given X(s), X(s) given X(r), Xd(r) given Xd(s) and Xd(s) given Xd(r), respectively, as:
fr|s(xr|xs) = fX (r)|X(s) (xr|xs) = (s − 1)! (r − 1)!(s − r − 1)!(1 − F(xr))mgmr−1(F(xr))gm−s+1(F(xs)) [hm(F(xs)) − hm(F(xr))]s−r−1f(xr), F−1(0) < xr< xs< F−1(1), (1.4) fs|r(xs|xr) = fX (s)|X(r) (xs|xr) = Cs−1 Cr−1(s − r − 1)!(1 − F(x r))−γr+m+1(1 − F(xs))γs−1 [hm(F(xs)) − hm(F(xr))]s−r−1f(xs), F−1(0) < xr< xs< F−1(1), (1.5) fdr|s(xr|xs) = fX d(r)|Xd(s)(xr|xs) = (s − 1)! (r − 1)!(s − r − 1)!F m d (xr)g r−1 m (1 − Fd(xr)) gm−s+1(1−Fd(xs))[hm(1−Fd(xs))−hm(1 − Fd(xr))]s−r−1fd(xr), F−1(0)< xs< xr< F−1(1), (1.6)
and fds|r(xs|xr) = fX d(s)|Xd(r)(xs|xr) = Cs−1 Cr−1(s − r − 1)! F−γr+m+1 d (xr)F γs−1 d (xs) [hm(1 − Fd(xs)) − hm(1 − Fd(xr))]s−r−1fd(xs), Fd−1(0) < xs< xr< Fd−1(1), (1.7) where gm(x) = 1 m+1[1 − (1 − x) m+1], m = −1, − log(1 − x), m= −1, hm(x) = − 1 m+1(1 − x) m+1, m = −1, − log(1 − x), m= −1, Cr−1=ri=1γr, r = 1, 2, . . . , n, with γn= k, f (x) = ∂ F(x)∂x and fd(x) = ∂ F∂xd(x).
Remark 1.1 For all 1≤ r < s ≤ n and all values of m, it is easy to prove the following relations (1) γs− γr = −(s − r)(m + 1) (2) Cs−1 Cr−1 = s i=r+1γ i = (m + 1)s−rs−r j=1(mγ+1s + s − r − j), m = −1, γs−r s , m= −1.
2 The main result
Theorem 2.1 Let X(1) ≤ X(2) ≤ · · · ≤ X(n) and Xd(n) ≤ Xd(n − 1) ≤ · · · ≤ Xd(1) be gos and dgos based on arbitrary continuous df ’s F and Fd, respectively, such that for any 1 ≤ r < s ≤ n we have m1= m2= · · · = ms−1= m. Furthermore, let X(r), X(s), Xd(r) and Xd(s) have finite variances. Then
corr(X(r), X(s)) ≤ ρn(r, s, m, ˜γs) = As(m) Ar(m)× Br(m) − Ar(m) Bs(m) − As(m), (2.1) and corr(Xd(r), Xd(s)) ≤ ρn(r, s, m, ˜γs), (2.2) where Al(m) = l j=1 βj 1+βj, m = −1, 1, m= −1, (2.3) Bl(m) = ⎧ ⎪ ⎨ ⎪ ⎩ l j=1 βj+1 βj+2, m = −1, 1+lj=1 1 γ2 j, m = −1, (2.4)
βj = mγ+1j , with β1=mγ+11 < −2, if m +1 < 0, and ˜γs = (γ1γ2. . . γs). Moreover (2.1) and (2.2) are satisfied with equality if F(x) = ⎧ ⎪ ⎨ ⎪ ⎩ F1(x) = 1 − (1 − x) 1 m+1, 0 < x < 1, if m + 1 > 0, F2(x) = 1 − e−x, 0 < x < ∞, if m = −1, F3(x) = 1 − x 1 m+1, 1 < x < ∞, if m+ 1 < 0, (2.5) and Fd(x) = ⎧ ⎪ ⎨ ⎪ ⎩ F1d(x) = x 1 m+1, 0 < x < 1, if m+ 1 > 0, F2d(x) = ex, −∞ < x < 0, if m = −1, F3d(x) = (1 − x) 1 m+1, −∞ < x < 0, if m + 1 < 0. (2.6)
Corollary 2.1 Letμri|s(xs) = E(Xi(r)|Xi(s) = xs) be the regression curve of Xi(r) given Xi(s), where Xi(r) and Xi(s) are the gos based on the df Fi(x), i = 1, 2, 3. Similarly, let μi dr|s(xs) = E(Xi d (r)|Xi d (s) = xs) be the regression curve of Xi d(r) given Xi d(s), where Xi d(r) and Xi d(s) are the dgos based on the df Fi d(x), i = 1, 2, 3. Then, we have the following relations:
(1) μr1d|s(xs) − μ1r|s(xs) = 1 −rs, for all 0 < xs< 1, m + 1 > 0. (2) μs1|r(xr) − μs1d|r(xr) = γ(s−r)(m+1) s+(s−r)(m+1), for all 0 < xr < 1, m + 1 > 0. (3) μr2|s(xs) + μr2d|s(−xs) = 0, for all 0 < xs< ∞, m = −1. (4) μs2|r(xr) + μs2d|r(−xr) = 0, for all 0 < xr< ∞, m = −1. (5) μr3|s(xs) + μ3dr|s(−xs) = 1 −rs, for all 1 < xs< ∞, m + 1 < 0. (6) μs3|r(xr) + μ3ds|r(−xr) = γs(s−r)(m+1)+(s−r)(m+1), for all 1 < xr < ∞, m + 1 < 0.
Proof of Theorem 2.1 Following the method of Rohatgi and Székely [12], we see that the proof of Theorem2.1
will depend on the Sarmanov [13] result, which states that if X and Y are arbitrary rvs with finite variances and E(X|Y ) and E(Y |X) are both linear, then for any measurable functions φ and ψ, for which var(φ(X)) and var(φ(Y )) are finite
sup
φ,ψcorr(φ(X), ψ(Y )) = |corr(X, Y )|. (2.7)
Therefore, the first step of our proof is to check the regression curves μri|s(xs), μis|r(xr), μri d|s(xs) and μs|r
i d(xr), i = 1, 2, 3. On the other hand, we have Ft(Xt(r)) = U(r), t = 1, 2, 3, where U(1) ≤ · · · ≤ U(r) ≤ · · · ≤ U(n) are the uniform gos. Thus, for any m−gos X(r) based on an arbitrary continuous df F, we get X(r) = F−1(U(r)) = F−1(Ft(Xt(r))), where t = 1, 2, 3 if m > −1, m = −1, m < −1,
respectively. This means that, for every df F, there exists a function φt(x) such that X(r) = φt(Xt(r))
and X(s) = φt(Xt(s)), t = 1, 2, 3 (e.g., φ1(x) = F−1(F1(x)) = F−1(1 − (1 − x)
1
m+1) and φ2(x) =
F−1(F2(x)) = F−1(1 − e−x). Moreover, for the case of the oos and the upper record values which are
consid-ered in Rohatgi and Székely [12], we haveφ1(x) = F−1(x) and φ2(x) = F−1(1 − e−x), with X1(i) = Ui:nis
the i th uniform oos and X2(i) = Ri is the i th upper record value based on the exponential distribution). Thus
(2.7) implies that
corr(X(r), X(s)) ≤ corr(Xt(r), Xt(s)),
where t = 1 if m + 1 > 0, t = 2 if m + 1 = 0 and t = 3 if m + 1 < 0. A similar argument proves corr(Xd(r), Xd(s)) ≤ corr(Xtd(r), Xtd(s)),
where t = 1 if m + 1 > 0, t = 2 if m + 1 = 0 and t = 3 if m + 1 < 0. Now, if m+ 1 > 0, one can easily verify that
μr|s 1 (xs) = xs 0 xrfr|s(xr|xs)dxr = r sxs, (2.8)
using (1.4) with F1and the transformation xr = xsz,
μs|r 1 (xr) = 1 − 1 xr (1 − xs) fs|r(xs|xr)dxs= γ s γs+ (s − r)(m + 1) xr+ (s − r)(m + 1) γs+ (s − r)(m + 1), (2.9)
using (1.5) with F1, the transformation 1 − xs= (1 − xr)z and Remark (1.1),
μr|s 1d(xs) = 1 x xrfdr|s(xr|xs)dxr= r sxs+ 1−r s , (2.10)
using (1.6) with F1dand the transformation xr = xs+ (1 − xs)z, and μs|r 1d(xr) = xr 0 xsfds|r(xs|xr)dxs= γ s γs+ (s − r)(m + 1) xr, (2.11)
using (1.7) with F1d, the transformation xs = xrz and Remark (1.1). Similarly, if m = −1, we get
μr|s 2 (xs) = xs 0 xrfr|s(xr|xs)dxr = r sxs, (2.12)
using (1.4) with F2and the transformation xr = xsz,
μs|r 2 (xr) = xr+ ∞ xr (xs− xr) fs|r(xs|xr)dxs= xr+ s− r γs , (2.13)
using (1.5) with F2, the transformation xs= xr+ z and Remark (1.1),
μr|s 2d(xs) = 0 xs xrfdr|s(xr|xs)dxr = r sxs, (2.14)
using (1.6) with F2dand the transformation xr = xsz, and
μs|r 2d(xr) = xr −∞ xsfds|r(xs|xr)dxs= xr− s− r γs , (2.15)
using (1.7) with F2d, the transformation xs= xr− z and Remark (1.1). Finally, if m+ 1 < 0, one can easily
check that μr|s 3 (xs) = xs 1 xrfr|s(xr|xs)dxr= r sxs+ 1−r s , (2.16)
using (1.4) with F3and the transformation xr = xs+ (1 − xs)z,
μs|r 3 (xr) = ∞ xr xsfs|r(xs|xr)dxs= γs γs+ (s − r)(m + 1) xr, (2.17)
using (1.5) with F3, the transformation xs= xzr and Remark (1.1),
μr|s 3d(xs) = 0 xs xrfdr|s(xr|xs)dxr = r sxs, (2.18)
using (1.6) with F3dand the transformation xr = xsz, and
μs|r 3d(xr) = 1 − xr −∞ (1 − xs) fds|r(xs|xr)dxs= γs γs+ (s − r)(m + 1) xr+ (s − r)(m + 1) γs+ (s − r)(m + 1), (2.19)
using (1.7) with F3d, the transformation 1 − xs= 11−x−zr and Remark (1.1).
The relations (2.8)–(2.19) show that all the regression curvesμri|s(xs), μsi|r(xr), μri d|s(xs), and μi ds|r(xr), i =
Lemma 2.1 For any 1≤ r < s ≤ n, we have corr(Xt(r), Xt(s)) = corr(Xtd(r), Xtd (s)) = As(m) Ar(m)× Br(m) − Ar(m) Bs(m) − As(m) = ρ n(r, s, m, ˜γs), (2.20)
where Al(m) and Bl(m) are defined in (2.3) and (2.4), respectively.
Proof We first consider the case t = 1, i.e., F1(x), F1d(x). In this case (1.1) and (1.2) take, respectively, the
forms X1(r) ∼ 1−rj=1Djand X1d (r) ∼rj=1Dj, where Dj = Bmj+1, 1 ≤ j ≤ n, be independent rvs with
respective power function distribution with exponentβj = mγ+1j . Therefore, E(Dkj) = βj
βj+k, k = 1, 2, . . . , and E(X1(r)) = 1 − Ar(m), E(X1d(r)) = Ar(m). Also, it can be easily shown that E(X1(r)X1(s)) =
1− Ar(m) − As(m) +rj=1 βj 2+βj s j=r+1 βj 1+βj and E(X 1d(r)X1d(s)) = r j=1 βj 2+βj s j=r+1 βj 1+βj. Thus, we get cov(X1(r), X1(s)) = cov(X1d(r), X1d(s)) = r j=1 βj 2+ βj s j=r+1 βj 1+ βj − A r(m)As(m) = s j=r+1 βj 1+ βj ⎡ ⎣r j=1 βj 2+ βj − r j=1 β j 1+ βj 2⎤ ⎦ = s j=1 βj 1+ βj ⎡ ⎣r j=1 1+ βj 2+ βj − r j=1 βj 1+ βj ⎤ ⎦ = As(m)(Br(m) − Ar(m)). (2.21)
On the other hand, using (2.21), we get
var(X1(r)) = var(X1d (r)) = Ar(m)(Br(m) − Ar(m)). (2.22)
By combining (2.21) with (2.22), we get the desired relation (2.20), in this case.
Second, we consider the case F2(x), F2d(x). In this case (1.1) and (1.2) take, respectively, the forms X2(r) ∼
−r
j=1log Bj and X2d (r) ∼
r
j=1log Bj, where Bj be independent rvs with respective Beta
distribu-tion Beta(γj, 1). Therefore, E((− log Bj)k) = (k+1)γk
j , k = 1, 2, . . . , and thus E(X
2(r)) = r j=1γ1j and var(X2(r)) =rj=1γ12 j. Moreover, we have E(X2(r)X2(s)) = E ⎛ ⎝r j=1 log 1 Bj 2 + r i=1i = j r j=1 log 1 Bi log 1 Bj + r i=1 s j=r+1 log 1 Bi log 1 Bj ⎞ ⎠ = r j=1 2 γ2 j + r i=1i = j r j=1 1 γiγj + r i=1 s j=r+1 1 γiγj. Therefore, we get cov(X2(r), X2(s)) = r j=1 2 γ2 j + r i=1i = j r j=1 1 γiγj + r i=1 s j=r+1 1 γiγj − r i=1 s j=1 1 γiγj = r j=1 1 γ2 j , and, thus corr(X2(r), X2(s)) = r j=1γ12 j s j=1γ12 = As(−1) Ar(−1)× Br(−1) − Ar(−1) Bs(−1) − As(−1).
On the other hand, since X2d(r) = −X2(r), we get corr(X2d(r), X2d (s)) = corr(−X2(r), −X2(s)) = corr(X2(r), X2(s)). This completes the proof of (2.20), in the second case.
Finally, we consider the case F3(x), F3d(x). In this case (1.1) and (1.2) take, respectively, the forms
X3(r) ∼ rj=1Dj and X3d(r) ∼ 1 −rj=1Dj, where Dj = Bmj+1, 1 ≤ j ≤ n. Therefore, E(Dkj) = βj
βj+k, k = 1, 2, . . . , provided that βj < −k. Thus, the kth moments of X
3(l) and X3d (l), l = r, s, exist
provided thatβj < −k, j = 1, 2, . . . , s, or equivalently γj + k(m + 1) > 0, j = 1, 2, . . . , s. On the other
hand, in view of Remark (1.1), we haveγj+1+2(m +1) = γj+(m +1), j = 1, 3, . . . , s −1. The last relation
with the inequalityγj + (m + 1) > γj + 2(m + 1), j = 1, 2, . . . , s, show that the condition β1 < −2 (or
equivalentlyγ1+ 2(m + 1) > 0), implies there exist first and second moments of X3(l) and X3d (l), l = r, s.
Now, we can easily see that the proof of (2.20) in this case is similar as the proof of the first case. The proof of Lemma2.1, as well as the proof of Theorem2.1. are completed. Proof of Corollary2.1 The proof immediately follows using the relations (2.8)–(2.19).
Remark 2.1 The maximal coefficient of correlation between a pair of rvs(X, Y ), introduced by [8], is defined by the left hand side of equation (2.7). Therefore,
ρn(r, s, m, ˜γs) = As(m) Ar(m)× Br(m) − Ar(m) Bs(m) − As(m)
is the maximal coefficient of correlation between a pair of gos(X(r), X(s)) and a pair of dgos (Xd(r), Xd(s)), based on arbitrary continuous df’s F andFd, respectively. Rényi [11] gives a set of seven postulates which a
measure of dependence for a pair of rvs should satisfy. Of the dependence measures considered by R´enyi, only the maximal coefficient of correlation satisfies all seven postulates. Consequently, the maximal coefficient of correlation is conveniently applied to problems whose solution is considerably determined by characteristics of stochastic dependence such as the statistical linearization (e.g., Chernyshov [4]). The maximal coefficient of correlation, besides being a convenient measure of dependence, plays a critical role in various areas of statistics including correspondence analysis, optimal transformation for regression, and the theory of Markov processes, see Yaming [16]. Finally, It is worth remarking that in Theorem2.1the maximal correlation coeffi-cient between pairs of rvs is explicitly computed. Such exact computations are relatively rare. One well-known case, the Gaussian case, is due to Lancaster [10]; another example of such an explicit computation is the case of partial sums of i.i.d. rvs considered by Dembo et al. [7]. The third such case is order statistics given by (1.3). Remark 2.2 In the classical case of oos, where m= 0 and γi = n − i + 1, 1 ≤ i ≤ n, it is easy to show that
ρn(r, s, m, ˜γs) =
r(n − s + 1) s(n − r + 1).
Moreover, F1(x) = F1d(x) = x, 0 < x < 1. This case is considered in Rohatgi and Székely [12] (the
rela-tion (1.3), with corr(Xr:n, Xs:n) = corr(Xn−r+1:n, Xn−s+1:n), where X(r) = Xr:n, X(s) = Xs:n, Xd(r) = Xn−r+1:n, and Xd(s) = Xn−s+1:n). Finally, in the case of the upper record values, where m = −1 and γi = 1, 1 ≤ i ≤ n, it is easy to show that ρn(r, s, −1, 1) =
r
s. Moreover, F2(x) = 1 − e−x, 0 < x < ∞,
which again leads to the result of Rohatgi and Székely [12] for the upper record values case (see relation 2 of Rohatgi and Székely [12]). For lower record values the above result holds with F2d(x) = ex, −∞ < x < 0.
This result reflects the fact that the correlation coefficient between the elements of the upper records is max-imal for the exponentially distributed populations, while the correlation coefficient between the elements of the lower records is maximal for the reflected exponential distribution F2d(x). Moreover, the maximal
corre-lation coefficient between the elements of the upper records is the same as the maximal correcorre-lation coefficient between the elements of the lower records.
Remark 2.3 In Barakat [1] it is proved that ρn(r, s, 0, ˜γs) =
As(0)
Ar(0)×
Br(0)−Ar(0)
Bs(0)−As(0) is the correlation coeffi-cient between any two uniform gos U(r) and U(s) or any two uniform dgos Ud(r) and Ud(s), where no any restriction is imposed on the parameters k, m1, m2, . . . mn−1. Moreover, in the same paper, it is proved
between the elements of gos and between the elements of dgos in general setting (where no any restriction is imposed on the parameters k, m1, m2, . . . mn−1). In the case of the upper and the lower record values
we have σr,s:n = 12 (3 4)s−r × ( 1−(3 4)r 1−(3 4)s) (see Barakat [
1]). In other hand, in view of Theorem2.1, we get σ r,s:n = 12 (3 4)s−r × ( 1−(3 4)r 1−(3 4)s) ≤ 12 r
s. The last relation reflects the fact that the asymptotic independence
between any two oos Xr:nand Xs:n(which occurs, in view of the result of Barakat [1], if, and only if,rs → 0,
as n → ∞) implies the asymptotic independence between the upper records Rrand Rs, as well as the lower records Lsand Lr (the asymptotic independence between the upper records Rrand Rs, as well as the lower
records occurs, in view of the result of Barakat [1], if, and only if, s− r → ∞, as n → ∞).
Remark 2.4 Since (2.20), with (2.3) and (2.4), is proved using the relations (1.1) and (1.2), we deduce that ρn(r, s, m, ˜γs) is the correlation coefficient between any two gos Xt(r) and Xt(s) based on the df Ft, t =
1, 2, 3, or between any two dgos Xtd (r) and Xtd (s) based on the df Ftd, t = 1, 2, 3, where no any restriction
imposed on the parameters k, m1, m2, . . . mn−1. Consequently, in this case the parameter m in ρn(r, s, m, ˜γs)
is related to the df’s Ft and Ftd, t = 1, 2, 3, and not to the gos or dgos themselves.
3 Discussion and applications
The following two results are direct consequences of Theorem2.1. Corollary 3.1 For any r < s, we have
ρn(r, s, m, ˜γs) ≤ min As(m) Ar(m), Br(m) − Ar(m) Bs(m) − As(m) . (3.1) Moreover, the asymptotic independence between the gos Xr:n and Xs:noccurs if, and only if, at least one of the relations As(m) Ar(m) → 0 and Br(m)−Ar(m) Bs(m)−As(m) → 0 holds. Proof Since 1> 1+β2+βj j ≥ βj
1+βj > 0, for all j, we get 1 > Bk(m) ≥ Ak(m) > 0, for all k, and 1 > Bkl(m) ≥
Akl(m) > 0, for all k < l, where Akl(m) =lj=k+1 β
j 1+βj and Bkl(m) = l j=k+1 1+βj 2+βj. The combination of the preceding relations yields 0< Br(m)−Ar(m)
Bs(m)−As(m) =
Br(m)−Ar(m)
Br s(m)Br(m)−Ar s(m)Ar(m) < 1, which completes the proof of
(3.1), as well as the proof of Corollary3.1.
Corollary 3.2 For any r = s and any n, we have Xr:nand Xs:n are independent if, and only if, Xd;r:nand Xd;s:nare independent.
Theorem2.1shows that for the dfs given in (2.5) (as well as (2.6)) are the maximal correlation between the elements of the gos (as well as the elements of dgos) equals the (Pearson) correlation. Thus, the uncorrelated-ness of these elements implies their independence. This fact implies that in any real-world problems the linear relationship is the only possible relation between these elements. For example, the linear relationship is the only possible relation between any two upper records and between any two lower records for the exponentially and reflected exponentially distributed populations, respectively. Moreover, Theorem2.1enables us to derive some interesting results concerning the rates of the convergence to the asymptotic independence between different types of gos as well as dgos. The following consequence gives some of these results for the oos. Although, the proof of this consequence is simple, but to the best of the author knowledge this result is new. Corollary 3.3 Let XrE1:n, X I r2:n, X C r3:n, X I r4:n and X E
r5:n be the r1th lower extreme (where r1=constant, w.r.t.
n), the r2th lower intermediate (where r2→ ∞,rn2 → 0, as n → ∞), the r3th central (where r3→ ∞, r3
n → λ ∈ (0, 1), as n → ∞), the r4th upper intermediate (where r4→ ∞,rn4 → 1, as n → ∞) and the r5th upper extreme (where n− s5=constant, w.r.t. n) order statistic, respectively. Then
(I) the convergence to the asymptotic independence of the couple(XrE1:n, X
E
r5:n) is faster than the couple
(XI r:n, X
I r:n).
(II) the convergence to the asymptotic independence of the couple(XrE1:n, XrC3:n) is faster than the couples (XE r1:n, X I r2:n) and (X I r2:n, X C r3:n).
(III) the convergence to the asymptotic independence of the couple(XrI2:n, XCr3:n) is faster than the couple (XE r1:n, X I r2:n), if and only if r2= ◦( √ n), as n → ∞.
Proof The proof of (I), (II) and (III) follows immediately by noting, respectively, that (I) ρn(r1, r5, 0, ˜γr5), ρn(r2, r4, 0, ˜γr4) → 0 and ρn(r1, r5, 0, ˜γr5) ρn(r2, r4, 0, ˜γr4) ∼ r1(n − r5+ 1) r2(n − r4+ 1)→ 0, as n → ∞,
since r1, n − r5+ 1 = constant w.r.t. n and r2, n − r4+ 1 → ∞, as n → ∞,
(II) ρn(r1, r2, 0, ˜γr2), ρn(r1, r3, 0, ˜γr3), ρn(r2, r3, 0, ˜γr3) → 0, ρn(r1, r3, 0, ˜γr3) ρn(r1, r2, 0, ˜γr2) ∼ 1− λ λ × r2 n → 0 and ρn(r1, r3, 0, ˜γr3) ρn(r2, r3, 0, ˜γr3) ∼ r1 r2 → 0, as n → ∞,
since, r1= constant w.r.t. n and r2→ ∞, as n → ∞,
and (III) ρn(r1, r2, 0, ˜γr2), ρn(r2, r3, 0, ˜γr3) → 0 and ρn(r2, r3, 0, ˜γr3) ρn(r1, r2, 0, ˜γr2) ∼ 1− λ λr1 × r2 √ n → 0, as n → ∞, if and only if r2= ◦(√n), as n → ∞, since r1= constant w.r.t. n, as n → ∞.
Example 3.1 (the determination of a suitable type of a given order statistic). As an interesting application of the above consequence, we consider a requirement of a certain statistical problem which stipulates the asymptotic independence between the two order statistics X10:100 and X50:100. For performing a goodness
of fit test to identify the suitable limit distribution type of each of the statistics X10:100and X50:100, we first
have to choose their types (extreme or intermediate or central type). The type of the order statistic X50:100can reasonably regarded as a central type, i.e., X50:100 = XrC3:n. However, on the one side, the type of the order statistic X10:100may be regarded as extreme type, i.e., X10:100 = XrE
1:n, r1 = 10, n = 100, but on the other side, it may be regarded as lower intermediate type X10:100 = XrI2:n, r2= [
√
100], n = 100. In view of our requirement, Corollary3.3, part (II), enables us to decide that the choice of extreme type for the order statistic
X10:100is better than the choice of lower intermediate type.
Example 3.2 (type II right censored samples) Let the censoring scheme be R1 = R2 = · · · = RM−1 =
0, RM = n − M. Therefore, we get βj = γj = 2n + M − j + 1. Thus, if r < s, we get, after simple
calculations, As(m) Ar(m) = As(0) Ar(0) = 2n− M − s + 1 2n− M − r + 1 and Br(m) − Ar(m) Bs(m) − As(m) = Br(0) − Ar(0) Bs(0) − As(0) = r s. The last two relations, thus yield
ρn(r, s, m, ˜γs) =
r(2n − M − s + 1) s(2n − M − r + 1).
Therefore, if M is constant with respect to n then X(r) and X(s) as well as Xd(r) and Xd(s) are depen-dent for all r, s and n. On the other hand, by assuming that M = M(n) → ∞ and Mn → 0, as n → ∞, we can easily deduce that Theorem2.1in [2], which is concerned with the asymptotic dependence between oos, will hold for this model.
Acknowledgments The author is grateful to the referees for constructive suggestions and comments that improved the
presen-tation substantially.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use,
distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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