Abstract
Publisher Summary This chapter reviews the asymptotic comparison of tests. Asymptotic measures of relative efficiencies can be broadly classified in two categories including (1) local, and (2) nonlocal. A measure of performance that requires the alternative to approach the null is a local efficiency and a measure that lets the alternative stay fixed as n→∞ is a nonlocal efficiency. It seems fair to state that the most popular local efficiency is Pitman efficiency and the most popular nonlocal efficiency is Bahadur efficiency (ratio of exact Bahadur slopes). The concept of Pitman efficiency is based on Neyman–Pearsonian's viewpoint—that is, fixing α and making comparison in terms of powers. It turns out that for most of the reasonable tests the limiting power is 1 for any fixed alternative and α. The phenomenon that the nonlocal relative efficiencies converge to the local relative efficiencies as θ tends to a null value is well known.
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