Consistency of Certain Two-Sample Tests

Abstract

Let $X_1, \cdots, X_m; Y_1, \cdots, Y_n$ be independently distributed on the unit interval. Assume that the $X$'s are uniformly distributed and that the $Y$'s have an absolutely continuous distribution whose density $g(y)$ is bounded and has at most finitely many discontinuities. Let $Z_0 = 0, Z_{n + 1} = 1$, and let $Z_1 < \cdots < Z_n$ be the values of the $Y$'s arranged in increasing order. For each $i = 1, \cdots, n + 1$ let $S_i$ be the number of $X$'s which lie in the interval $\lbrack Z_{i - 1}, Z_i\rbrack$. For each nonnegative integer $r$, let $Q_n(r)$ be the proportion of values among $S_1, \cdots, S_{n + 1}$ which are equal to $r$. Suppose $m$ and $n$ approach infinity in the ratio $(m/n) = \alpha > 0$. In Section 2 it is shown that $$\operatornamewithlimits{\lim}{n \rightarrow \infty} \operatornamewithlimits{\sup}{r \geqq 0} |Q_n(r) - Q(r)| = 0$$ with probability one, where $$Q(r) = \alpha^r \int^1_0 \frac{g^2(y)}{\lbrack\alpha + g(y)\rbrack^{r + 1}}dy.$$ This result may be used to prove consistency of certain tests of the hypothesis that the two samples have the same continuous distribution. Several such examples are given in Section 3. A further property of one of these tests is briefly discussed in Section 4.