This paper examines a sequential testing procedure for choosing one of three simple hypotheses concerning the unknown mean $\mu$ of the normal distribution when the variance is known. The test is conducted by plotting $S_n$, the sum of the observations, versus $n$, the current sample size, until the point $(n, S_n)$ is contained within one of three triangular regions. When this occurs, sampling is terminated and the region containing $(n, S_n)$ determines which state of nature is accepted. Although we shall formally view the problem as one with only three states of nature $(\mu = \mu_1, \mu_2$ or $\mu_3)$, we shall proceed with the usual understanding that the performance of the test procedure should be evaluated for a wider class of states $(- \infty < \mu < \infty)$. The test is approximated by a corresponding exact test for the Wiener process. Formulas are derived which approximate the operating characteristics (OC) and the average sample size (ASN) for all values of $\mu$. The ASN function is compared with theoretical lower bounds. The testing procedure is compared with a modification of a three hypothesis testing procedure proposed by Sobel and Wald .
Gordon Simons. "A Sequential Three Hypothesis Test for Determining the Mean of a Normal Population with Known Variance." Ann. Math. Statist. 38 (5) 1365 - 1375, October, 1967. https://doi.org/10.1214/aoms/1177698692