Summary
We shall disclose a relationship between the almost sure stability of weighted empirical distribution functions and sums of order statistics. First we obtain an extension of a theorem due to Csáki on the almost sure stability of the standardized uniform empirical distribution function. This result is then shown to be an essential tool to derive a characterization of the almost sure stability of the sum of k nupper order statistics from a sample of n independent observations from a distribution with positive support in the domain of attraction of a non-normal stable law, where 1≦k n≦n and k n→∞ as n→∞.
References
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Research performed while the author was at the Catholic University Nijmegen
Research supported by the Alexander von Humboldt Foundation while the author was visiting the University of Munich on leave from the University of Delaware
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Einmahl, J.H.J., Haeusler, E. & Mason, D.M. On the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics. Probab. Th. Rel. Fields 79, 59–74 (1988). https://doi.org/10.1007/BF00319104
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DOI: https://doi.org/10.1007/BF00319104
Keywords
- Distribution Function
- Stochastic Process
- Probability Theory
- Statistical Theory
- Order Statistic