# zbMATH — the first resource for mathematics

The asymptotic distribution of trimmed sums. (English) Zbl 0647.62030
Let $$X_{1,n}\leq \cdot \cdot \cdot X_{n,n}$$ be the order statistics of n independent random variables with a common distribution function F, and let m(n) and k(n) be positive integers such that m(n)$$\to \infty$$, k(n)$$\to \infty$$, m(n)/n$$\to 0$$ and k(n)/n$$\to 0$$ as $$n\to \infty$$. The authors give a necessary and sufficient condition for the existence of normalizing constants $$A_ n>0$$ and $$B_ n$$ such that the sequence $T_ n=A_ n^{-1}\{\sum^{n-k(n)}_{i=m(n)+1}X_{i,n}-B_ n\}$ is stochastically compact. They show that if a subsequence $$\{T_{n'}\}$$ converges to V in distribution, then V is represented as follows: $V=\int_{0}^{-Z_ 1}(Z_ 1+x)d\psi_ 1(x)+Z+\int^{0}_{-Z_ 2}(Z_ 2+x)d\psi_ 2(x),$ where $$\psi_ 1$$ and $$\psi_ 2$$ are nondecreasing functions determined by m(n), k(n) and the tail of F, and $$(Z_ 1,Z,Z_ 2)$$ is a trivariate normal random vector.
They also give a necessary and sufficient condition for the existence of $$A_ n$$ and $$B_ n$$ such that $$T_ n$$ is asymptotically normal. A variant of Stigler’s theorem when m(n)/n$$\to \alpha$$, and k(n)/n$$\to 1- \beta$$, where $$0<\alpha <\beta <1$$, is also obtained.
Reviewer: T.Mori

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