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Bounds on expectations of \(L\)-estimates for maximally and minimally stable samples. (English) Zbl 1355.62011

Summary: The random variables \(X_1,X_2,\dots,X_n\) are said to be maximally (resp. minimally) stable of order \(j\) \((j=1,2,\dots,n)\) if the distribution \(F_{(j)}\) of max\(\{X_{k_1},\dots,X_{k_j}\}\) (resp. \(G_{(j)}\) of min\(\{X_{k_1},\dots,X_{k_j}\})\) is the same for any \(j\) element subset \(\{k_1,\dots,k_j\}\) of \(\{1,2,\dots,n\}\). Under the assumption of maximal (resp. minimal) stability of order \(j\), we give lower and upper bounds on linear combinations of distribution functions of order statistics in terms of \(F_{(j)}\) (resp. \(G_{(j)}\)) and present the corresponding sharp two-sided bounds on expected \(L\)-estimates. Moreover, we evaluate the upper and lower deviations of the expectation of \(L\)-estimate from observations with a given dependence structure (copula) under maximally stable violations of the dependence assumption.

MSC:

62G30 Order statistics; empirical distribution functions
62E10 Characterization and structure theory of statistical distributions
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[1] DOI: 10.1016/S0167-7152(01)00014-1 · Zbl 1052.62057 · doi:10.1016/S0167-7152(01)00014-1
[2] DOI: 10.1016/0167-7152(92)90105-E · Zbl 0743.62018 · doi:10.1016/0167-7152(92)90105-E
[3] DOI: 10.1016/0167-7152(92)90071-C · Zbl 0761.62012 · doi:10.1016/0167-7152(92)90071-C
[4] Erdös P, Math Scand 13 pp 99– (1963) · doi:10.7146/math.scand.a-10692
[5] DOI: 10.1016/S0167-7152(01)00114-6 · Zbl 0994.62045 · doi:10.1016/S0167-7152(01)00114-6
[6] Galambos J, Bonferroni-type inequalities with applications (1996)
[7] DOI: 10.1080/02331888308802385 · Zbl 0808.62048 · doi:10.1080/02331888308802385
[8] DOI: 10.1080/01621459.1968.10480944 · doi:10.1080/01621459.1968.10480944
[9] DOI: 10.1080/03610928308831073 · Zbl 0786.62046 · doi:10.1080/03610928308831073
[10] DOI: 10.1016/S0378-3758(01)00321-4 · Zbl 1024.62022 · doi:10.1016/S0378-3758(01)00321-4
[11] DOI: 10.1002/0471722162 · Zbl 1053.62060 · doi:10.1002/0471722162
[12] DOI: 10.1214/aoms/1177729088 · Zbl 0050.35301 · doi:10.1214/aoms/1177729088
[13] DOI: 10.1023/A:1014681708984 · Zbl 1003.62046 · doi:10.1023/A:1014681708984
[14] Rychlik T, JIRSS 6 pp 141– (2007)
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