A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

TLDR

Optimal or convenient tests of simple hypotheses can often be expressed in a form based on the sum of observations, as the likelihood ratio test for fixed sample size reduces to this structure. The test rejects H₀ when the sum of n independent observations falls below a threshold k, and its asymptotic properties are derived using the exponential decay of P(Sₙ ≤ na) governed by the minimum of the moment‑generating function of X − a. Each such test is characterized by an index ρ, and the ratio e = log ρ₁/log ρ₂ quantifies relative efficiency, implying that for large samples a size‑n sample under the first test yields error probabilities comparable to a size‑en sample under the second; when the two hypotheses are close, ρ can be approximated by assuming normality of X.

Abstract

In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.