Bahadur efficiency of rank tests for the change-point problem
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 functions of empirical distribution functions.
functions of empirical distribution functions
scientific article; zbMATH DE number 4094590