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Limit results for linear combinations. (English) Zbl 0724.60020
Sums, Trimmed sums and extremes, Prog. Probab. 23, 377-391 (1991).
[For the entire collection see Zbl 0717.00017.]
Let \(U_{1n},...,U_{nn}\) be the increasing order statistics of a random sample from the uniform distribution on [0,1]. Define the L- statistic \(L_ n=n^{-1}\sum^{n}_{i=1}c_{ni}g(U_{ni})\) with \(c_{ni}\geq 0\) and g nondecreasing. By putting \(g=h(F^{-1})\) one obtains L-statistics for other distributions. The author studies asymptotic normality of \(L_ n\). A suitable centering is \(\mu_ n=n^{-1}(n+2)\int^{1-a_ n}_{a_ n}g(t)J_ n(t)dt,\) where \(a_ n=(n+2)^{-1}\) and \(J_ n\) is a step function determined by the \(c_{ni}\). It is assumed that \(J_ n\) as \(n\to \infty\) converges in a suitable way to a continuous function J. The function \(K(t)=\int_{[,t)}J(s)dg(s)\) on [0,1], with \(\int_{[a,b)}=- \int_{[b,a)}\) when \(b<a\), is the inverse of a distribution function H. It is shown that \(Z_ n=n^{1/2}(L_ n-\mu_ n)/\sigma_ n\) is asymptotically standard normal when H belongs to the domain of attraction of N(0,1). Then \(Z_ n-n^{-1/2} \sigma_ n^{- 1}\sum^{n}_{i=1}(Y_ i-E Y_{ni})\) and \(Z_ n-n^{-1/2} \sigma_ n^{-1}\sum^{n}_{i=1}(Y_ i-E Y_{ni})\) converge to zero in probability. Here \(Y_ i=H^{-1}(U_ i)\), \(Y_{ni}\) is a winsorized \(Y_ i\) and \(\sigma^ 2_ n=Var Y_{ni}\). Some extensions are given in remarks.
Reviewer: A.J.Stam (Winsum)

60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics