Statistica Sinica

Yoshihide Kakizawa

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.