# Wilcoxon Rank-Sum Test $H_1$ notation

This notation is from Nonparametric Statistical Inference, 5th ed., by Gibbons and Chakraborti. On p. 290, it states:

Wilcoxon (1945) proposed a test where we accept the one-sided location alternative $H_L: \theta < 0$ ($X \overset{\text{ST}}{>}Y)$ if the sum of the ranks of the $X$'s is too large, or $H_L: \theta > 0$ ($X \overset{\text{ST}}{<}Y$) if the sum of the ranks of the $X$'s is too small.

What in the world does $\text{ST}$ mean? I'm also not quite sure what $\theta$ is supposed to mean either - on p. 289, it state:

The location alternative is that the populations have the same form but a different measure of central tendency. This can be expressed in symbols as follows:

$H_0: F_{Y}(x) = F_{X}(x)$ for all $x$

$H_L: F_{Y}(x) = F_{X}(x - \theta)$ for all $x$ and some $\theta \neq 0$

I think (from Pre-Calculus) that the shift $F_{X}(x) \to F_{X}(x - \theta)$ would be a right translation as long as $\theta > 0$ and a left translation as long as $\theta < 0$. So wouldn't $H_L$ be equivalent to saying $F_{X}(x) > F_{Y}(x)$ when $\theta > 0$ and $F_{X}(x) < F_{Y}(x)$ when $\theta < 0$ (in both cases, for at least one $x$?)?

• The notation is defined on page 229 (I have never seen the book aside from just now when I used Google books to locate "st" in it). – Glen_b Aug 10 '15 at 1:34
• Thank you @Glen_b, that is very helpful! :) Hopefully someday I can give this book a good read instead of just using it to look something up. – Clarinetist Aug 10 '15 at 1:36

The ST above the inequality refers to stochastic ordering. That is, we say that $X$ is stochastically larger than $Y$ if $P(X > t) > P(Y > t)$ for all $t$. $\theta$ is the location parameter of the family, so the alternative says that one distribution function is "shifted" with respect to the other. Your last observation is correct as this is the meaning of stochastic ordering.