# Bahadur Efficiency of Tests for a Shift in Location of Normal Populations

## Summary

A sequence of independent random variables *X* _{1},...,*X* _{ N } is said to have a change point n, if *X* _{1},...,*X* _{ N } have a common distribution *F* and *X* _{n+1},..., *X* _{ N } have a common distribution *G*, *G* ≠ *F*. Consider the problem of testing the null hypothesis of no change, against the alternative of a one-sided change at an unknown change point n, when both *F* and *G* are normal with equal variance σ^{2}. Most of the test statistics for this problem can be interpreted as generalizations of two-sample statistics (*n* known). In this chapter we derive the Bahadur efficiencies for two classes of statistics that are generalizations of the two-sample likelihood ratio statistics. The asymptotic results are compared with some small-sample power estimates based on Monte Carlo experiments.

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