Abstract: Global optimality of likelihood ratio test statistics is well-known in the Bahadur sense. In this paper the behaviors of Rao and Wald statistics (and
) for testing
are studied. It turns out that at alternative
, the Bahadur slopes of these two statistics for the one-sided case are identical up to order
, while for the two-sided case, they are identical only up to order
, in general i.i.d. models and Gaussian stationary processes. We obtain the second- (first-) order Bahadur efficiency of
and
for the one- (two-) sided case. The third-order Bahadur efficiency depends on the statistical curvature. Two concrete examples are given. One is a curved exponential family, and the other is a Gaussian AR(1) process. The latter provides an example that the
-term of the Bahadur slope of
for the one-sided case is different from that of
.
Key words and phrases: Bahadur slope, curved exponential family, Gaussian stationary process, large deviation theorem, Rao's statistic, spectral density, Wald's statistic.