Asymptotically Efficient Tests Based on the Sums of Observations

Abstract

For tests, $\mathbf{\Phi} = \{\phi_k\}$, of composite hypotheses, $\omega_1$ and $\omega_2$, asymptotic efficiency is defined in terms of the behavior as $\alpha \rightarrow 0$ of the sample size $N_{\phi}$ required to reduce the maximum risk to $\alpha$. For problems where the $\omega_i$ contain elements $\theta_i$ whose relative densities satisfy $$\sup_{\omega_1} \inf_{t>0} E_{\theta}(f_2/f_1)^t = \inf_t E_i(f_2/f_1)^t = \sup_{\omega_2} \inf_{t<0} E_{\theta}(f_2/f_1)^t,$$ Chernoff's Theorem 1 [2] is applied to the non-randomized test $\mathbf{\Phi}^{\ast}$, with $\phi^{\ast}_k = 1$ or 0 according as $\Sigma \log (f_2/f_1) > 0$ or not, and proves $\mathbf{\Phi}^{\ast}$ asymptotically efficient (Theorem 2.1). The principal results of the paper are applications of Theorem 2.1 to tests of the difference $(\xi - \eta)$ of binomial probabilities with samples of relative size $m/n$. For $\omega_1 = \{\xi - \eta \leqq - \delta\}, \omega_2 = \{\xi - \eta \geqq \delta\}$, certain tests of the form $\phi^{\ast}_k = 1$ if and only if $\lambda(\hat \xi - \frac{1}{2}) > (\hat \eta - \frac{1}{2})$, with $\lambda$ increasing in $m/n$, turn out to be asymptotically efficient, while all tests of the form $\psi_k = 1, a_k, 0$ according as $(\hat \xi - \hat \eta)$ is greater than, equal to, or less than $c_k$ are asymptotically inefficient when $m \neq n$. For given relative sampling costs, the ratio $m/n$ may be chosen so that the asymptotic cost of observations is minimized.