The First-Median Test: A Two-Sided Version of the Control Median Test

Consider two independent samples (called x's and y's) of mutually independent observations from populations with c.d.f.'s F(x) and G(x) respectively. Let υ be the number of x's which are smaller than the median of the y-sample and let u be the number of y's which are less than the median of the x-sample. If the x-sample is regarded as the control group, then the control median test proposed by Kimball et al. 9, rejects the hypothesis that F(x) ≡ G(x) if u is small. The present paper discusses a symmetrized version of this test, the first median test, which is based on u if the median of the x-sample precedes the median of the y-sample and on υ, otherwise. The asymptotic distribution theory of both tests is developed. The tests are useful in analyzing life trial data because they permit the experimeter to reach a decision early. In the life trial situation it is important to minimize the expected number of observations required to reach a decision. It is shown that in large samples, when curtailed sampling is utilized, these procedures reach a decision before the standard median test 12, 13.