Estimating the distributions of scan statistics with high precision
type: Item
instance of: scholarly article
title: Estimating the distributions of scan statistics with high precision
zbMATH Open document ID: 1003.60075
DOI: 10.1023/A:1012205725339
author: George Haiman
published in: Extremes
publication date: 2002-11-21T00:00:00Z
review text: Let \(N(t)\) be a Poisson process with intensity \(\lambda\), let \(a,T>0\), and let \(L=T/a\) be integer. The author is interested in approximations of the probability \[ Q_L(n,\lambda)=Q_L=\Pr \Bigl\{\max_{0\leq t\leq T-a}(N(t+a)-N(t))<n \Bigr\} \] and analogous probabilities for min. The idea is to approximate \(Q_L\) for large \(L\) by \(Q_l\) with small \(l\) which can be easily computed by the Huntington-Naus exact formula [see \textit{R. J. Huntington} and \textit{J. I. Naus}, Ann. Probab. 3, 894-896 (1975; Zbl 0316.60014)]. New approximations are derived, e.g. \(Q_L\cong (2Q_2-Q_3)(1+Q_2-Q_3+2(Q_2-Q_3)^2)^{-(L-1)}\), and their relative errors are estimated. The obtained approximations are used to estimate the distribution and expectation of waiting time till a cluster. Probabilities connected with clusters on a circle and on a plain are also considered.
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 1833853
zbMATH Keywords: maximum cluster
zbMATH Keywords: multiple coverage
zbMATH Keywords: approximation
zbMATH Keywords: waiting time till cluster
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1833853