The Annals of Mathematical Statistics

Asymptotically Efficient Tests Based on the Sums of Observations

John H. MacKay

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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.

Article information

Source
Ann. Math. Statist., Volume 30, Number 3 (1959), 806-813.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706209

Digital Object Identifier
doi:10.1214/aoms/1177706209

Mathematical Reviews number (MathSciNet)
MR107935

Zentralblatt MATH identifier
0090.11302

JSTOR
links.jstor.org

Citation

MacKay, John H. Asymptotically Efficient Tests Based on the Sums of Observations. Ann. Math. Statist. 30 (1959), no. 3, 806--813. doi:10.1214/aoms/1177706209. https://projecteuclid.org/euclid.aoms/1177706209


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