Notation for one-sided hypothesis testing
I see the following notation for one-sided hypothesis testing:
- $H_0$: $K = 2$
- $H_1$: $K > 2$
I would find it more natural to write:
- $H_0$: $K \le 2$
- $H_1$: $K > 2$
Assume a situation where $K$ is not, by definition, limited to equal or be higher than two. $K$ is just a dummy variable and 2 a dummy number.
Where does the first notation come from and why is it in use?
1 answer
A partial answer may be better than none.
Not requiring that two hypotheses exhaust the parameter space is fairly clearly present in Neyman and Pearson (1933), so arguably no later than that. This paper is the origin of the Neyman-Pearson lemma.
They don't explicitly lay out hypotheses in that form you have though, this is still pretty early on. To find an explicitly laid out simple null and one-sided composite alternative, you might need to go a bit later, perhaps, to the generalizations of it.
Fisher was more focused on the null rather than laying out an explicit alternative or multiple alternatives.
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