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The length of the shorth. (English) Zbl 0664.62040
Let \(\hat H_ n(\alpha)\) \((0<\alpha <1)\) denote the length of the shortest \(\alpha\)-fraction of the ordered sample \(X_{1:n},X_{2:n},...,X_{n:n}\), i.e., \[ \hat H_ n(\alpha)=\min \{X_{k+j:n}-X_{k:n}:\quad 1\leq k\leq k+j\leq n;\quad (j+1)/n\geq \alpha \}. \] Such quantities arise in the context of robust scale estimation. Using the concept of compact derivatives of statistical functionals, the asymptotic behaviour of \(\hat H_ n(\alpha)\) as n tends to infinity is investigated.
The length of the shortest \(\alpha\)-fraction may be regarded as a functional of the empirical distribution function. The author obtains his results by decomposing this functional into two factors which are sufficiently smooth near the respective limits to permit local replacement by linear operators. This is done by using the concept of compact differentiation of statistical functionals introduced by J. A. Reeds [On the definition of von Mises functionals. Ph. D. thesis, Harvard Univ. (1976)]. Indeed, in this paper the author masterly presents an example of the simplicity and usefulness of this method.
See also L. T. Fernholz [Von Mises calculus for statistical functionals. (1983; Zbl 0525.62031)], N. Reid [Ann. Stat. 9, 78-92 (1981; Zbl 0457.62031)] and W. Esty, R. Gillette, M. Hamilton and D. Taylor [Ann. Inst. Stat. Math. 37, 109-129 (1985; Zbl 0585.62058)].
Reviewer: J.Viollaz

MSC:
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
62F35 Robustness and adaptive procedures (parametric inference)
60F17 Functional limit theorems; invariance principles
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