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Weak convergence of the simple linear rank statistic under mixing conditions in the nonstationary case. (English) Zbl 0809.62010

Theory Probab. Appl. 38, No. 3, 405-422 (1993) and Teor. Veroyatn. Primen. 38, No. 3, 579-599 (1993).
Let \(\{X_{ni}\}\) be a triangular array of real-valued random variables with continuous distribution functions \(F_{ni} (x)\), \(\widehat {F}_ n (x)\) the corresponding empirical process, \(c_{ni} = g(i/n)\) an array of regression constants, \(\widehat {H}_ n\) the corresponding weighted empirical process, \(F_ n\), \(H_ n\) the expectations of \(\widehat {F}_ n\), \(\widehat {H}_ n\), respectively, and \(J\) a score function on \((0,1)\).
Considering \(\varphi\)-mixing as well as strong mixing, the asymptotic distribution theory for the linear rank statistic \[ S_ n(J) = n^{1/2} \int^ \infty_{-\infty} J((n/n + 1) \widehat {F}_ n (x)) d\widehat {H}_ n(x) \int^ \infty_{-\infty} J(F_ n(x)) dH_ n(x) \] is studied in the case where the underlying variables are nonstationary.

MSC:

62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
62G30 Order statistics; empirical distribution functions
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