Power Bounds for a Smirnov Statistic in Testing the Hypothesis of Symmetry

Abstract

Lower and upper bounds on the power of a Smirnov test for symmetry $H_0: \bar{F} = F$ versus $H_1: \bar{F} \geqq F, \sup_x\lbrack\bar{F}(x) - F(x)\rbrack = \Delta > 0$ are obtained exactly or estimated for selected values of sample size $N$, level $\alpha$, and asymmetry $\Delta$. Furthermore the asymptotic power of the test as $N^{\frac{1}{2}}\Delta_N \rightarrow c$ is shown to be bounded by $\Phi(c - k_\alpha)$ and 1 if $c \geqq k_\alpha$ and by $\alpha$ and $2\Phi(c - k_\alpha)$ if $c < k_\alpha$, where $k_\alpha$ is the critical point. These bounds compare favorably in some respects with those of the Wilcoxon and other monotone rank tests studied in "Power bounds and asymptotic minimax results for one-sample rank tests," Ann. Math. Statist. 42 12-35.