On functionals of order statistics
Identifieur interne : 003497 ( Main/Exploration ); précédent : 003496; suivant : 003498On functionals of order statistics
Auteurs : W. SendlerSource :
- Metrika [ 0026-1335 ] ; 1982-12-01.
Abstract
Summary: Let gn be real functions,U ni, 1≤i≤n, the ordered sample ofn independentU(0,1) distributed random variables, andc ni(α), 1≤i≤n, 0≤α≤1 be (known) real numbers,n=1, 2, ... The random quantity $$T_n (\alpha ): = n^{ - 1} \sum\limits_{i = 1}^n {c_{ni} (\alpha )g_n (U_{ni} )} $$ , 0≤α≤1, is studied. Based on a method proposed byShorack [1972] the main result is the weak convergence of $$H_n : = n^{1/2} (T_n - \mu _n )$$ to Gaussian processes, where $$\mu _n (\alpha ): = \sum\limits_{i = 1}^n {c_{ni} (\alpha )} \int\limits_{(i - 1)/n}^{i/n} {g_n (t)dt} $$ , 0≤α≤1. The convergence is with respect to theSkorokhod [1956]-topologiesM 2,M 1 onD (I) and the ‖ ‖-topology onC(I), depending on the conditions imposed on thec ni(α).
Url:
DOI: 10.1007/BF01893363
Affiliations:
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