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Distributions and expectations of order statistics for possibly dependent random variables. (English) Zbl 0790.62048
Summary: The class of all possible distribution functions of each order statistic for a sample of possibly dependent, identically distributed random variables is characterized. An effective method of determination of the extreme values for the expectation and variance of an arbitrary Borel function of order statistics is presented and applied for calculating the moments in the case of uniformly distributed random variables.

MSC:
62G30 Order statistics; empirical distribution functions
62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
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[1] Arnold, B.C.; Balakrishnan, N., Relations, bounds and approximations for order statistics, () · Zbl 0703.62064
[2] Dunford, N.; Schwartz, J.T., Linear operators I, (1958), Interscience New York
[3] Galambos, J., The asymptotic theory of extreme order statistics, (1978), Wiley New York · Zbl 0381.62039
[4] Ioffe, A.D.; Tihomirov, V.M., Theory of extremal problems, (1979), North-Holland New York · Zbl 0407.90051
[5] Lai, T.L.; Robbins, H., Maximally dependent random variables, (), 286-288 · Zbl 0352.60013
[6] Lai, L.T.; Robbins, H., A class of dependent random variables and their maxima, Z. wahrsch. verw. gebiete, 42, 89-111, (1978) · Zbl 0377.60022
[7] Leadbetter, M.R.; Lindgren, G.; Rootzen, H., Extremes and related properties of random sequences and processes, (1983), Springer-Verlag Amsterdam · Zbl 0518.60021
[8] Nikaid√≥, H., On von Neumann’s minimax theorem, Pacific J. math., 4, 65-72, (1954) · Zbl 0055.10004
[9] Rychlik, T., Stochastically extremal distributions of order statistics for dependent samples, Statist. probab. lett., 13, 337-341, (1992) · Zbl 0743.62018
[10] Rychlik, T., Weak limit theorems for stochastically largest order statistcs, (), 141-154
[11] Tchen, A., Inequalities for distributions with given marginals, Ann. probab., 8, 814-827, (1980) · Zbl 0459.62010
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