Bahadur efficiency of rank tests for the change-point problem
type: Item
instance of: scholarly article
title: Bahadur efficiency of rank tests for the change-point problem
zbMATH Open document ID: 0668.62028
author: Jaap Praagman
published in: The Annals of Statistics
publication date: 1988-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,...,X_ N\) be independent random variables, then the sequence has a change-point at n, if \(X_ 1,X_ 2,...,X_ N\) have common distribution F and \(X_{n+1},...,X_ n\) have a common distribution G, \(G\neq F\). The change-point problem is to test the null- hypothesis of no change \((G=F)\) against the alternative of a change \(G<F\) at an unknown change point. The difference with the well-known two-sample problem is that in the change-point problem n is unknown. Let \(T_{n,k}\) denote a two-sample rank statistic based on samples \(X_ 1,...,X_ k\) and \(X_{k+1},...,X_ N\). Two classes of statistics, based on two-sample rank statistics, for the change-point problem are considered: statistics of sum-type \(S_ N=\sum^{N- 1}_{k=1}C_{N,k}T_{N,k}\) or statistics of max-type \(M_ N=\max_{1\leq k<N}C_{N,k}T_{N,k}\), where the \(C_{N,k}\) are weights. The rank statistics \(T_{N,k}\) considered are of the classical form \(T_{N,k}=\sum^{k}_{i=1}\) \(a_ N(R_ i)\), with \(a_ N(i)\) denoting the scores. In the paper the exact Bahadur slope of both \(S_ N\) and \(M_ N\) is derived, under suitable conditions imposed upon the weights \(C_{N,k}\) and the scores \(a_ N(i)\). It is shown that the Bahadur (exact) efficiency of tests with the same weights (the same scores) depends on those weights (scores), in contrast with the Pitman efficiency of these tests, which are known to be independent of the (common) weights (scores). As main result of the paper it is established that tests of type \(M_ N\) are at least as efficient in the sense of Bahadur as corresponding tests of type \(S_ N\), uniformly over all alternatives considered. The main tool in the proofs is a very general theorem on large deviations for empirical probability measures, due to \textit{P. Groeneboom}, \textit{J. Oosterhoff} and \textit{F. H. Ruymgaart} [Ann. Probab. 7, 553-586 (1979; Zbl 0425.60021)]. This latter result can be applied because the statistics of type \(S_ N\) and \(M_ N\) can be viewed as fu…
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G10
zbMATH DE Number: 4094590
zbMATH Keywords: Bahadur efficiency
zbMATH Keywords: linear rank test
zbMATH Keywords: change-point test
zbMATH Keywords: change-point problem
zbMATH Keywords: two-sample rank statistic
zbMATH Keywords: statistics of sum-type
zbMATH Keywords: exact Bahadur slope
zbMATH Keywords: Pitman efficiency
zbMATH Keywords: large deviations
zbMATH Keywords: functions of empirical distribution functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350700
OpenAlex ID: W2073970600
DOI: 10.1214/AOS/1176350700
description: scientific article; zbMATH DE number 4094590