Statistica Sinica 10(2000), 297-315

LOCAL COMPARISON OF RAO AND WALD STATISTICS

IN THE BAHADUR SENSE

Yoshihide Kakizawa

Hokkaido University

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 ($R_n$ and $W_n$) for testing $\theta=\theta_0$ are studied. It turns out that at alternative $\theta_0+\varepsilon$, the Bahadur slopes of these two statistics for the one-sided case are identical up to order $\varepsilon^4$, while for the two-sided case, they are identical only up to order $\varepsilon^2$, in general i.i.d. models and Gaussian stationary processes. We obtain the second- (first-) order Bahadur efficiency of $R_n$ and $W_n$ 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 $\varepsilon^5$-term of the Bahadur slope of $R_n$ for the one-sided case is different from that of $W_n$.

Key words and phrases: Bahadur slope, curved exponential family, Gaussian stationary process, large deviation theorem, Rao's statistic, spectral density, Wald's statistic.