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.
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References
Barakat H.M.: Measuring the asymptotic dependence between generalized order statistics. J. Stat. Theory Appl. 6(2), 106–117
Barakat H.M.: Comments on the rate of convergence to asymptotic independence between order statistics. Stat. Probab. Lett. 76(1), 35–38 (2006)
Burkschat, M.; Cramer, E.; Kamps, U.: Dual generalized order statistics Metron LXI(1), 13–26 (2003)
Chernyshov K.R.: Statistical linearization based on the maximal correlation. Control Commun. SIBCON 07, 29–36 (2007)
Cramer, E.: Contributions to Generalized Order Statistics. Habililationsschrift, University of Oldenburg (2003, reprint)
David H.A., Nagaraja H.N.: Order Statistics, 3rd edn. Wiley, New York (2003)
Dembo, A.; Kagan, A.; Shepp, L.A.: Remarks on the maximum correlation coefficient Bernoulli 7, 343–350 (2001)
Gebelein H.: Das statistische problem der korrelation als variations und eigenwertproblem und sein zusammenhang mit der ausgleichsrechnung. Z. Angew. Math. Mech. 21, 364–379 (1941)
Kamps U.: A Concept of Generalized Order Statistics. Teubner, Stuttgart (1995)
Lancaster H.O.: Some properties of the bivariate normal distribution considered in the form of a contingency table. Biometrika 44, 289–292 (1957)
Rényi A.: On measures of dependence. Acta Math. Acad. Sci. Hung. 10, 441–451 (1959)
Rohatgi V.K., Székely G.J.: On the background of some correlation inequalities. J. Stat. Comput. Simul. 40, 260–262 (1992)
Sarmanov, O.V.: Maximum correlation (Russian). Dokl. Akad. Nauk SSSR, 120, 52–55, 715–718 (1958)
Székely G.J., Móri T.F.: An extremal proerty of rectangular distributions. Statist. Probab. Lett. 3, 107–109 (1985)
Terrell G.R.: A characterization of rectangular distribution. Ann. Probab. II, 823–826 (1983)
Yaming Yu.: On the maximal correlation coefficient. Stat. Probab. Lett. 78(9), 1072–1075 (2008)
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The author is grateful to the referees for constructive suggestions and comments that improved the presentation substantially.
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Barakat, H.M. The maximal correlation for the generalized order statistics and dual generalized order statistics. Arab. J. Math. 1, 149–158 (2012). https://doi.org/10.1007/s40065-012-0002-9
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DOI: https://doi.org/10.1007/s40065-012-0002-9