Comparison between the locally most powerful unbiased and Rao's tests

Consider the problem of testing a simple hypothesis that [theta] = 0 against the alternative that [theta] [not equal to] 0. This paper makes an asymptotic comparison (up to o(n-1)) between Rao's test and the locally most powerful unbiased (LMPU) test under contiguous alternatives, [delta]n-1/2, both test having the same size [alpha] (up to o(n-1)). It is proved that for each [delta] and [alpha], the LMPU test is locally more powerful than the locally unbiased version of Rao's test. However, it also follows from the derivation that Rao's test, which is much simpler than the LMPU test, is almost as good as the latter in terms of power when the test size is small.