The maximal correlation for the generalized order statistics and dual ...

Arab J Math (2012) 1:149–158 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 ⎛ ⎞⎛ ⎞ n n−1     γ j ⎠ ⎝ (1 − u j )γ j −γ j+1 −1 ⎠ (1 − u n )γn −1 f U (1),...,U (n) (u 1 , . . . , u n ) = ⎝ j=1

j=1

on the cone {(u 1 , . . . , u n ) : 0 ≤ u 1 ≤ · · · ≤ u n < 1} ⊂ n , with parameters γ1 , . . . , γn > 0. The parameters  γ1 , . . . , γn are defined by γn = k > 0 and γr = k + n − r + Mr , r = 1, 2, . . . , n − 1, where Mr = n−1 j=r m j and m˜ = (m 1 m 2 . . . m n−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 E-mail: [email protected]

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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 ⎛ ⎞⎛ ⎞ n n−1   γ j −γ j+1 −1   ⎠ u nγn −1 γj⎠ ⎝ uj f Ud (1),...,Ud (n) (u 1 , . . . , u n ) = ⎝ j=1

j=1

on the cone {(u 1 , . . . , u n ) : 1 ≥ u 1 ≥ · · · ≥ u n > 0} ⊂ n . The quantile transformation X d (r ) = Fd−1 (Ud (r )), r = 1, 2, . . . , n, yields dgos based on arbitrary df Fd . Therefore, X d (1), . . . , X d (n) are arranged in descending order almost surely. Let B j , 1 ≤ j ≤ n, be independent rvs with respective Beta distribution Beta(γ j , 1), i.e., B j follows a power function distribution with exponent γ j . 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 B j , 1 ≤ j ≤ n (see Cramer [5] and Burkschat et al. [3]). Namely, ⎛ ⎞ r  X  (r ) ∼ F −1 ⎝1 − B j ⎠ , r = 1, 2, . . . , n, (1.1) j=1

and

⎛ X d (r ) ∼ Fd−1 ⎝

r 

⎞ B j ⎠ , r = 1, 2, . . . , n.

(1.2)

j=1

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 X 1:2 and X 2:2 (X 1:2 ≤ X 2:2 ) have finite variances, then their correlation coefficient satisfies the inequality corr(X 1:2 , X 2:2 ) ≤ 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  r (n − s + 1) corr(X r :n , X s:n ) ≤ , 1 ≤ r < s ≤ n, (1.3) s(n − r + 1) (here it is supposed that var(X r :n ) and var(X s: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 X d (r ), X d (s) for which m 1 = m 2 = · · · m s−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 m 1 = m 2 = · · · = m n−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 ), X d (r ) given X d (s) and X d (s) given X d (r ), respectively, as: (s − 1)! r −1 −s+1 (F(xr ))gm (F(xs )) (1 − F(xr ))m gm (r − 1)!(s − r − 1)! [h m (F(xs )) − h m (F(xr ))]s−r −1 f (xr ), F −1 (0) < xr < xs < F −1 (1), Cs−1   (1 − F(xr ))−γr +m+1 (1 − F(xs ))γs −1 f s|r (xs |xr ) = f X (s)|X (r ) (xs |xr ) = Cr −1 (s − r − 1)!

f r |s (xr |xs ) = f X

 (r )|X  (s)

(xr |xs ) =

[h m (F(xs )) − h m (F(xr ))]s−r −1 f (xs ), F −1 (0) < xr < xs < F −1 (1), (s − 1)!   r |s r −1 f d (xr |xs ) = f X d (r )|X d (s) (xr |xs ) = (1 − Fd (xr )) F m (xr )gm (r − 1)!(s − r − 1)! d

−s+1 gm (1− Fd (xs ))[h m (1− Fd (xs ))−h m (1 − Fd (xr ))]s−r −1 f d (xr ), Fd−1 (0) < xs < xr < Fd−1 (1),

123

(1.4)

(1.5)

(1.6)

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and 



f d (xs |xr ) = f X d (s)|X d (r ) (xs |xr ) = s|r

Cs−1 −γ +m+1 γ −1 (xr )Fd s (xs ) Fd r Cr −1 (s − r − 1)!

[h m (1 − Fd (xs )) − h m (1 − Fd (xr ))]s−r −1 f d (xs ), Fd−1 (0) < xs < xr < Fd−1 (1), where

gm (x) = h m (x) =

Cr −1 =

r

i=1 γr , r

(1.7)

1 m+1 [1 − (1 −

x)m+1 ], m = −1, − log(1 − x), m = −1, 1 − m+1 (1 − x)m+1 , m = −1, − log(1 − x), m = −1,

= 1, 2, . . . , n, with γn = k, f (x) =

∂ F(x) ∂x

and f d (x) =

∂ Fd (x) ∂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) s−r s−r γs s

(m + 1) Cs−1 j=1 ( m+1 + s − r − j), m = −1, γi = (2) Cr −1 = s−r , γ m = −1. s i=r +1 2 The main result Theorem 2.1 Let X  (1) ≤ X  (2) ≤ · · · ≤ X  (n) and X d (n) ≤ X d (n − 1) ≤ · · · ≤ X d (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 m 1 = m 2 = · · · = m s−1 = m. Furthermore, let X  (r ), X  (s), X d (r ) and X d (s) have finite variances. Then  Br (m) − Ar (m) As (m)   corr(X (r ), X (s)) ≤ ρn (r, s, m, γ˜s ) = × , (2.1) Ar (m) Bs (m) − As (m) and corr(X d (r ), X d (s)) ≤ ρn (r, s, m, γ˜s ), where

(2.2)



βj l j=1 1+β j ,

m = −1, 1, m = −1, ⎧

β j +1 l ⎪ m = −1, ⎨ j=1 β j +2 , Bl (m) =  ⎪ ⎩ 1 + lj=1 γ12 , m = −1,

Al (m) =

(2.3)

(2.4)

j

γ

j , with β1 = β j = m+1 with equality if

γ1 m+1

< −2, if m + 1 < 0, and γ˜s = (γ1 γ2 . . . γs ). Moreover (2.1) and (2.2) are satisfied

⎧ 1 ⎪ ⎨ F1 (x) = 1 − (1 − x) m+1 , 0 < x < 1, if m + 1 > 0, if m = −1, F(x) = F2 (x) = 1 − e−x , 0 < x < ∞, ⎪ 1 ⎩ m+1 F3 (x) = 1 − x , 1 < x < ∞, if m + 1 < 0,

and

⎧ 1 ⎪ ⎨ F1d (x) = x m+1 , 0 < x < 1, if m + 1 > 0, if m = −1, Fd (x) = F2d (x) = e x , −∞ < x < 0, ⎪ 1 ⎩ F (x) = (1 − x) m+1 , −∞ < x < 0, if m + 1 < 0. 3d

(2.5)

(2.6)

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Corollary 2.1 Let μi (xs ) = E(X i (r )|X i (s) = xs ) be the regression curve of X i (r ) given X i (s), where r |s  (r )|X  (s) = x ) X i (r ) and X i (s) are the gos based on the df Fi (x), i = 1, 2, 3. Similarly, let μid (xs ) = E(X id s id     be the regression curve of X id (r ) given X id (s), where X id (r ) and X id (s) are the dgos based on the df Fid (x), i = 1, 2, 3. Then, we have the following relations: r |s

r |s

r |s

r |s

(1) μ1d (xs ) − μ1 (xs ) = 1 − rs , for all 0 < xs < 1, m + 1 > 0. s|r s|r )(m+1) (2) μ1 (xr ) − μ1d (xr ) = γs (s−r +(s−r )(m+1) , for all 0 < xr < 1, m + 1 > 0. (3) (4) (5) (6)

μ2 (xs ) + μ2d (−xs ) = 0, for all 0 < xs < ∞, m = −1. s|r s|r μ2 (xr ) + μ2d (−xr ) = 0, for all 0 < xr < ∞, m = −1. r |s r |s μ3 (xs ) + μ3d (−xs ) = 1 − rs , for all 1 < xs < ∞, m + 1 < 0. s|r s|r )(m+1) μ3 (xr ) + μ3d (−xr ) = γs (s−r +(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 Theorem 2.1 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 corr(φ(X ), ψ(Y )) = |corr(X, Y )|.

(2.7)

φ,ψ

r |s

r |s

s|r

Therefore, the first step of our proof is to check the regression curves μi (xs ), μi (xr ), μid (xs ) and s|r μid (xr ), i = 1, 2, 3. On the other hand, we have Ft (X t (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 (X t (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 (X t (r )) 1

and X  (s) = φt (X t (s)), t = 1, 2, 3 (e.g., φ1 (x) = F −1 (F1 (x)) = F −1 (1 − (1 − x) 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 considered in Rohatgi and Székely [12], we have φ1 (x) = F −1 (x) and φ2 (x) = F −1 (1 − e−x ), with X 1 (i) = Ui:n is the ith uniform oos and X 2 (i) = Ri is the ith upper record value based on the exponential distribution). Thus (2.7) implies that corr(X  (r ), X  (s)) ≤ corr(X t (r ), X t (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(X d (r ), X d (s)) ≤ corr(X td (r ), X td (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 r |s μ1 (xs )

xs =

xr f r |s (xr |xs )d xr =

r xs , s

(2.8)

0

using (1.4) with F1 and the transformation xr = xs z, s|r μ1 (xr )

1 =1−

(1 − xs ) f s|r (xs |xr )d xs = xr

γs (s − r )(m + 1) xr + , γs + (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), r |s μ1d (xs )

1 = xs

123

r |s

xr f d (xr |xs )d xr =

 r r xs + 1 − , s s

(2.10)

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using (1.6) with F1d and the transformation xr = xs + (1 − xs )z, and s|r μ1d (xr )

xr =

s|r

xs f d (xs |xr )d xs = 0

γs xr , γs + (s − r )(m + 1)

(2.11)

using (1.7) with F1d , the transformation xs = xr z and Remark (1.1). Similarly, if m = −1, we get r |s μ2 (xs )

xs =

xr f r |s (xr |xs )d xr =

r xs , s

(2.12)

0

using (1.4) with F2 and the transformation xr = xs z, s|r μ2 (xr )

∞ = xr +

(xs − xr ) f s|r (xs |xr )d xs = xr + xr

s −r , γs

(2.13)

using (1.5) with F2 , the transformation xs = xr + z and Remark (1.1), r |s μ2d (xs )

0 =

r |s

xr f d (xr |xs )d xr =

r xs , s

(2.14)

xs

using (1.6) with F2d and the transformation xr = xs z, and s|r μ2d (xr )

xr =

s|r

xs f d (xs |xr )d xs = 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 r |s μ3 (xs )

xs =

xr f r |s (xr |xs )d xr =

 r r xs + 1 − , s s

(2.16)

1

using (1.4) with F3 and the transformation xr = xs + (1 − xs )z, s|r μ3 (xr )

∞ =

xs f s|r (xs |xr )d xs = xr

using (1.5) with F3 , the transformation xs = r |s μ3d (xs )

xr z

(2.17)

and Remark (1.1),

0 =

γs xr , γs + (s − r )(m + 1)

r |s

xr f d (xr |xs )d xr =

r xs , s

(2.18)

xs

using (1.6) with F3d and the transformation xr = xs z, and s|r μ3d (xr )

xr =1−

s|r

(1 − xs ) f d (xs |xr )d xs =

−∞

using (1.7) with F3d , the transformation 1 − xs =

γs (s − r )(m + 1) xr + , (2.19) γs + (s − r )(m + 1) γs + (s − r )(m + 1)

1−xr 1−z

and Remark (1.1). r |s

s|r

r |s

s|r

The relations (2.8)–(2.19) show that all the regression curves μi (xs ), μi (xr ), μid (xs ), and μid (xr ), i = 1, 2, 3, are linear. Finally, the following lemma completes the proof of Theorem 2.1.

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Lemma 2.1 For any 1 ≤ r < s ≤ n, we have



  (r ), X td (s)) = corr(X t (r ), X t (s)) = corr(X td

As (m) Br (m) − Ar (m) × = ρn (r, s, m, γ˜s ), (2.20) Ar (m) Bs (m) − As (m)

where Al (m) and Bl (m) are defined in (2.3) and (2.4), respectively. (1.1) and (1.2) take, respectively, the Proof We first consider

the case t = 1, i.e., Fr1 (x), F1d (x). In this case (r ) ∼ j=1 D j , where D j = B m+1 , 1 ≤ j ≤ n, be independent rvs with forms X 1 (r ) ∼ 1− rj=1 D j and X 1d j γ

β

j j respective power function distribution with exponent β j = m+1 . Therefore, E(D kj ) = β j +k , k = 1, 2, . . . ,   and E(X 1 (r )) = 1 − Ar (m), E(X 1d (r )) = Ar (m). Also, it can be easily shown that E(X 1 (r )X 1 (s)) =

r

β β β j s βj  (r )X  (s)) = 1 − Ar (m) − As (m) + rj=1 2+βj j sj=r +1 1+βj j and E(X 1d j=1 2+β j j=r +1 1+β j . Thus, 1d we get

  (r ), X 1d (s)) cov(X 1 (r ), X 1 (s)) = cov(X 1d r s  βj  βj = − Ar (m)As (m) 2 + βj 1 + βj j=1 j=r +1 ⎡ ⎤ 2 s r r     βj βj βj ⎣ ⎦ = − 1 + βj 2 + βj 1 + βj j=r +1 j=1 j=1 ⎡ ⎤ s r r    βj 1 + βj βj ⎣ ⎦ = As (m)(Br (m) − Ar (m)). − = 1 + βj 2 + βj 1 + βj j=1

j=1

(2.21)

j=1

On the other hand, using (2.21), we get  (r )) = Ar (m)(Br (m) − Ar (m)). var(X 1 (r )) = var(X 1d

(2.22)

By combining (2.21) with (2.22), we get the desired relation (2.20), in this case.  Second, case F2 (x), r F2d (x). In this case (1.1) and (1.2) take, respectively, the forms X 2 (r ) ∼ r we consider the  − j=1 log B j and X 2d (r ) ∼ j=1 log B j , where B j be independent rvs with respective Beta distribu tion Beta(γ j , 1). Therefore, E((− log B j )k ) = (k+1) , k = 1, 2, . . . , and thus E(X 2 (r )) = rj=1 γ1j and k γj  var(X 2 (r )) = rj=1 γ12 . Moreover, we have j

⎛ E(X 2 (r )X 2 (s)) = E ⎝

r   j=1

=

r  j=1

1 log Bj r 

2 + γ j2 i=1

i = j

2

⎞       s r r  r    1 1 1 1 ⎠ + log log + log log Bi Bj Bi Bj i=1 j=r +1

i=1i = j j=1

r  j=1

1 + γi γ j

r 

s 

i=1 j=r +1

1 . γi γ j

Therefore, we get cov(X 2 (r ), X 2 (s)) =

s s r r r r r  r        2 1 1 1 1 + + − = , 2 γγ γγ γγ γ γ2 j=1 j i=1 j=1 i j i=1 j=r +1 i j i=1 j=1 i j j=1 j i = j

and, thus   r  j=1    corr(X 2 (r ), X 2 (s)) =  s

1 γ j2

1 j=1 γ 2 j

123

 =

Br (−1) − Ar (−1) As (−1) × . Ar (−1) Bs (−1) − As (−1)

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 (r ) = −X  (r ), we get corr(X  (r ), X  (s)) = corr(−X  (r ), −X  (s)) = On the other hand, since X 2d 2 2 2 2d 2d   corr(X 2 (r ), X 2 (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 , 1 ≤ j ≤ n. Therefore, E(D kj ) = X 3 (r ) ∼ rj=1 D j and X 3d (r ) ∼ 1 − rj=1 D j , where D j = B m+1 j βj β j +k , k

 (l), l = r, s, exist = 1, 2, . . . , provided that β j < −k. Thus, the kth moments of X 3 (l) and X 3d provided that β j < −k, j = 1, 2, . . . , s, or equivalently γ j + k(m + 1) > 0, j = 1, 2, . . . , s. On the other hand, in view of Remark (1.1), we have γ j+1 + 2(m + 1) = γ j + (m + 1), j = 1, 3, . . . , s − 1. The last relation with the inequality γ j + (m + 1) > γ j + 2(m + 1), j = 1, 2, . . . , s, show that the condition β1 < −2 (or  (l), l = r, s. equivalently γ1 + 2(m + 1) > 0), implies there exist first and second moments of X 3 (l) and X 3d 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 Lemma 2.1, as well as the proof of Theorem 2.1. are completed. 

Proof of Corollary 2.1 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,  As (m) Br (m) − Ar (m) ρn (r, s, m, γ˜s ) = × 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 (X d (r ), X d (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 Theorem 2.1 the maximal correlation coefficient 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  r (n − s + 1) ρn (r, s, m, γ˜s ) = . s(n − r + 1) Moreover, F1 (x) = F1d (x) = x, 0 < x < 1. This case is considered in Rohatgi and Székely [12] (the relation (1.3), with corr(X r :n , X s:n ) = corr(X n−r +1:n , X n−s+1:n ), where X  (r ) = X r :n , X  (s) = X s:n , X d (r ) = X n−r +1:n , and X d (s) = X n−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) = rs . 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) = e x , −∞ < x < 0. This result reflects the fact that the correlation coefficient between the elements of the upper records is maximal 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 correlation coefficient between the elements of the upper records is the same as the maximal correlation coefficient between the elements of the lower records.  Br (0)−Ar (0) Remark 2.3 In Barakat [1] it is proved that ρn (r, s, 0, γ˜s ) = AArs (0) (0) × Bs (0)−As (0) is the correlation coefficient 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, m 1 , m 2 , . . . m n−1 . Moreover, in the same paper, it is proved  that the measure σr,s:n = 12ρn (r, s, 0, γ˜s ) provides a non-parametric criterion of asymptotically independence

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between the elements of gos and between the elements of dgos in general setting (where no any restriction is imposed on the parameters k, m 1 , m 2 , . . . m n−1 ). In the case of the upper and the lower record values  1−( 3 )r

 we have σr,s:n = 12 ( 43 )s−r × ( 43 s ) (see Barakat [1]). In other hand, in view of Theorem 2.1, we get 1−( 4 )   3 r 1−( )  σr,s:n = 12 ( 43 )s−r × ( 43 s ) ≤ 12 rs . The last relation reflects the fact that the asymptotic independence 1−( 4 )

between any two oos X r :n and X s: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 Rr and Rs , as well as the lower records L s and L r (the asymptotic independence between the upper records Rr and 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, γ˜s ) is the correlation coefficient between any two gos X t (r ) and X t (s) based on the df Ft , t =  (r ) and X  (s) based on the df F , t = 1, 2, 3, where no any restriction 1, 2, 3, or between any two dgos X td td td imposed on the parameters k, m 1 , m 2 , . . . m n−1 . Consequently, in this case the parameter m in ρn (r, s, m, γ˜s ) 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 Theorem 2.1. Corollary 3.1 For any r < s, we have



ρn (r, s, m, γ˜s ) ≤ min

As (m) , Ar (m)



Br (m) − Ar (m) . Bs (m) − As (m)

(3.1)

 occurs if, and only if, at least one Moreover, the asymptotic independence between the gos X r:n and X s:n Br (m)−Ar (m) of the relations AArs (m) (m) → 0 and Bs (m)−As (m) → 0 holds. 1+β j 2+β j

≥

βj 1+β j

> 0, for all j, we get 1 > Bk (m) ≥ Ak (m) > 0, for all k, and 1 > Bkl (m) ≥



β 1+β Akl (m) > 0, for all k < l, where Akl (m) = lj=k+1 1+βj j and Bkl (m) = lj=k+1 2+β jj . The combination of

Proof Since 1 >

r (m) the preceding relations yields 0 < BBrs (m)−A (m)−As (m) = (3.1), as well as the proof of Corollary 3.1.

Br (m)−Ar (m) Br s (m)Br (m)−Ar s (m)Ar (m)

< 1, which completes the proof of 

 are independent if, and only if, X  Corollary 3.2 For any r = s and any n, we have X r:n and X s:n d;r :n and  X d;s:n are independent.

Theorem 2.1 shows 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 uncorrelatedness 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, Theorem 2.1 enables 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. I

E

Corollary 3.3 Let X rE1 :n , X rI2 :n , X rC3 :n , X r4 :n and X r5 :n be the r1 th lower extreme (where r1 =constant, w.r.t. n), the r2 th lower intermediate (where r2 → ∞, rn2 → 0, as n → ∞), the r3 th central (where r3 → ∞, rn3 → λ ∈ (0, 1), as n → ∞), the r4 th upper intermediate (where r4 → ∞, rn4 → 1, as n → ∞) and the r5 th upper extreme (where n − s5 =constant, w.r.t. n) order statistic, respectively. Then E

(I) the convergence to the asymptotic independence of the couple (X rE1 :n , X r5 :n ) is faster than the couple I

(X rI2 :n , X r4 :n ).

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(II) the convergence to the asymptotic independence of the couple (X rE1 :n , X rC3 :n ) is faster than the couples (X rE1 :n , X rI2 :n ) and (X rI2 :n , X rC3 :n ). (III) the convergence to the asymptotic independence of the couple (X rI2 :n , X rC3 :n ) is faster than the couple √ (X rE1 :n , X rI2 :n ), if and only if r2 = ◦( n), as n → ∞. Proof The proof of (I), (II) and (III) follows immediately by noting, respectively, that  ρn (r1 , r5 , 0, γ˜r5 ) r1 (n − r5 + 1) (I) ρn (r1 , r5 , 0, γ˜r5 ), ρn (r2 , r4 , 0, γ˜r4 ) → 0 and ∼ → 0, as n → ∞, ρn (r2 , r4 , 0, γ˜r4 ) r2 (n − r4 + 1) since r1 , n − r5 + 1 = constant w.r.t. n and r2 , n − r4 + 1 → ∞, as n → ∞, (II) ρn (r1 , r2 , 0, γ˜r2 ), ρn (r1 , r3 , 0, γ˜r3 ), ρn (r2 , r3 , 0, γ˜r3 ) → 0, ρn (r1 , r3 , 0, γ˜r3 ) ∼ ρn (r1 , r2 , 0, γ˜r2 )



1 − λ r2 ρn (r1 , r3 , 0, γ˜r3 ) × → 0 and ∼ λ n ρn (r2 , r3 , 0, γ˜r3 )



r1 → 0, as n → ∞, r2

since, r1 = constant w.r.t. n and r2 → ∞, as n → ∞, and

 1−λ ρn (r2 , r3 , 0, γ˜r3 ) r2 (III) ρn (r1 , r2 , 0, γ˜r2 ), ρn (r2 , r3 , 0, γ˜r3 ) → 0 and × √ → 0, as n → ∞, ∼ ρn (r1 , r2 , 0, γ˜r2 ) λr1 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 X 10:100 and X 50:100 . For performing a goodness of fit test to identify the suitable limit distribution type of each of the statistics X 10:100 and X 50:100 , we first have to choose their types (extreme or intermediate or central type). The type of the order statistic X 50:100 can reasonably regarded as a central type, i.e., X 50:100 = X rC3 :n . However, on the one side, the type of the order statistic X 10:100 may be regarded as extreme type, i.e., X 10:100 = X rE1 :n , r1 = 10, n = 100, but on the other √ side, it may be regarded as lower intermediate type X 10:100 = X rI2 :n , r2 = [ 100], n = 100. In view of our requirement, Corollary 3.3, part (II), enables us to decide that the choice of extreme type for the order statistic X 10:100 is better than the choice of lower intermediate type. Example 3.2 (type II right censored samples) Let the censoring scheme be R1 = R2 = · · · = R M−1 = 0, R M = n − M. Therefore, we get β j = γ j = 2n + M − j + 1. Thus, if r < s, we get, after simple calculations, As (m) As (0) 2n − M − s + 1 = = Ar (m) Ar (0) 2n − M − r + 1 and Br (m) − Ar (m) Br (0) − Ar (0) r = = . Bs (m) − As (m) Bs (0) − As (0) s The last two relations, thus yield ρn (r, s, m, γ˜s ) =



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 X d (r ) and X d (s) are dependent for all r, s and n. On the other hand, by assuming that M = M(n) → ∞ and M n → 0, as n → ∞, we can easily deduce that Theorem 2.1 in [2], which is concerned with the asymptotic dependence between oos, will hold for this model.

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Acknowledgments The author is grateful to the referees for constructive suggestions and comments that improved the presentation 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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