Minimal spacings of non-uniform densities
type: Item
instance of: scholarly article
title: Minimal spacings of non-uniform densities
zbMATH Open document ID: 0633.60019
DOI: 10.1016/0304-4149(87)90190-6
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: Typical of the results of this paper is the following: Suppose that \(X_ 1,X_ 2,..\). are i.i.d. random variables with a continuous probability density f(x), \(A<x<B<\infty\), such that \(\overline{\lim}_{x\to A+}f(x)<\infty\), \(\lim_{x\to B-}f(x)=\infty\) and f is non-decreasing in (B-\(\epsilon\),B) for some \(\epsilon >0\). If \(X_{1,n},...,X_{n,n}\) are the order statistics of \(X_ 1,...,X_ n\), the only possible non- trivial asymptotic distribution for the suitably scaled difference \(X_{n,n}-B\) is a Weibull distribution. Following ideas of \textit{P. Deheuvels} [Ann. Probab. 14, 194-208 (1986; Zbl 0594.60029)] the author proves that if the parameter \(\alpha\) of this Weibull distribution is in (0,\()\), then, as \(n\to \infty\), the k-th smallest of the spacings \(X_{i+1,n}-X_{i,n}\), \(1\leq i\leq n\), converges in distribution for each k to the k th smallest of the random variables \[ (\sum^{i+1}_{j=1}w_ j)^{1/\alpha}- (\sum^{i}_{j=1}w_ j)^{1/\alpha},\quad i\geq 1, \] where the \(w_ j's\) are i.i.d. exp(1). This and other similar results are applied to coverage problems of the circle and the line by random arcs or segments, and to the asymptotics of the scan statistics used in goodness-of-fit.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G99
zbMATH DE Number: 4030592
zbMATH Keywords: order statistics
zbMATH Keywords: Weibull distribution
zbMATH Keywords: spacings
zbMATH Keywords: coverage problems
zbMATH Keywords: circle
zbMATH Keywords: line
author: J. Hüsler
reviewed by: F. Papangelou
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90190-6
OpenAlex ID: W2053422489
cites work: The asymptotic distribution of the scan statistic under uniformity
cites work: Q3730807
cites work: On the influence of the extremes of an i.i.d. sequence on the maximal spacings
cites work: Q5633348
cites work: Q4162318
cites work: Random, nonuniform distribution of line segments on a circle
cites work: Random coverage of the circle and asymptotic distributions
cites work: Extremes and related properties of random sequences and processes
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description: scientific article; zbMATH DE number 4030592