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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)

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