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

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

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, and In the classical oos, wherem = (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 corr(X r :n , X s: (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: and 1 For all 1 ≤ r < s ≤ n and all values of m, it is easy to prove the following relations (1) be gos and dgos based on arbitrary continuous df's F and F d , respectively, such that for any 1 ≤ r < s ≤ n we have

The main result
and 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 F id (x), i = 1, 2, 3. Then, we have the following relations: 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 Therefore, the first step of our proof is to check the regression curves μ r |s . Moreover, for the case of the oos and the upper record values which are considered in Rohatgi and Székely [12], we have φ 1 , with X 1 (i) = U i:n is the ith uniform oos and X 2 (i) = R i is the ith upper record value based on the exponential distribution). Thus (2.7) implies that using (1.4) with F 1 and the transformation x r = x s z, using (1.5) with F 2 , the transformation x s = x r + z and Remark (1.1), using (1.6) with F 2d and the transformation x r = x s z, and using (1.6) with F 3d and the transformation x r = x s z, and  Proof We first consider the case t = 1, i.e., F 1 (x), F 1d (x). In this case (1.1) and (1.2) take, respectively, the forms X 1 (r ) ∼ 1− r j=1 D j and X 1d (r ) ∼ r j=1 D j , where D j = B m+1 j , 1 ≤ j ≤ n, be independent rvs with respective power function distribution with exponent β j = γ j m+1 . Therefore, E(D k j ) = β j β j +k , k = 1, 2, . . . , and E(X 1 (r )) = 1 − A r (m), E(X 1d (r )) = A r (m). Also, it can be easily shown that E(X 1 (r )
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, 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 andF d , 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 Rényi, 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).
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 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 F 2d (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 F 2d (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.
Remark 2.3 In Barakat [1] it is proved that ρ n (r, s, 0,γ s ) = A s (0) is the correlation coefficient between any two uniform gos U (r ) and U (s) or any two uniform dgos U d (r ) and U d (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 σ r,s:n = 12ρ n (r, s, 0,γ s ) provides a non-parametric criterion of asymptotically independence 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 (r, s, m,γ s ) is the correlation coefficient between any two gos X t (r ) and X t (s) based on the df F t , t = 1, 2, 3, or between any two dgos X td (r ) and X td (s) based on the df F td , t = 1, 2, 3, where no any restriction imposed on the parameters k, m 1 , m 2 , . . . m n−1 . Consequently, in this case the parameter m in ρ n (r, s, m,γ s ) is related to the df's F t and F td , t = 1, 2, 3, and not to the gos or dgos themselves.

Discussion and applications
The following two results are direct consequences of Theorem 2.1.

Moreover, the asymptotic independence between the gos X r :n and X s:n occurs if, and only if, at least one of the relations A s (m)
A  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. (II) the convergence to the asymptotic independence of the couple (X E