Definition of alternative hypothesis

Question

Which of these properties is required of the alternative hypothesis, by definition, in the general case:

a) If the alternative hypothesis is true, then the null hypothesis would necessarily tend to be rejected if a sufficient amount of data were taken.
b) If the null hypothesis is rejected, it follows that the alternative hypothesis must be true (perhaps given some assumptions)?

This seems like a simple question, but the definitions I've found online are all something like "the null and alternative hypotheses are two rival hypothesis that are tested."

If you can cite a source, it would be ideal, but you don't have to.

Answer 1

The Frequentist definition would be that the alternative hypothesis is the logical complement to the null hypothesis. The two hypotheses must be mutually exclusive, jointly cover the parameter space and be complementary. Bayesian methods don't require binary hypotheses.

EDIT To respond to comments. There is a tendency among some researchers to use $\mu=k$ as a null and an alternative of $\mu>k$. This might or might not be proper, particularly if the above definition is used.

This is usually used when it is implicitly known that $\mu<k$ is not part of the parameter space. For example, you cannot have negative calories. It is improper otherwise.

The use of an alternative such as $\mu>k$ is a problem for inference if, for example, in a z test one would find $z=-5$. Clearly, the null is rejected for most standard values of $\alpha$. However the inference and any decision which could follow from a null of $\mu=0$ since it is also clear that $\mu<0$.

The proper, one-sided, null hypothesis should have been $\mu\le{0}$, with an alternative of $\mu>0$. The role of formal hypothesis declarations in Frequentist inference and decision theory is two-fold.

First, it links the probability to a null hypothesis with well-defined frequencies. Second, it links the statements to a probabilistic version of modus tollens. Without a binary nature, that linkage is broken and the implied link between Aristotelean logic, frequencies, and set theory is also broken.

Answer 2

Answered in comments copied below:

This question (especially the particular framing of the comparison) reads like a question for a class, or a question from a textbook. Is something like that the case? "Perhaps given some assumptions" is impossibly vague -- what assumptions are included or excluded from consideration? Incidentally, neither is "true by definition" for any definition I've seen. (in particular a statement like "tend to be" isn't going to be part of a definition in any case; it's potentially a property of something once it's defined). Have you been given a definition? Did that definition mention assumptions? – Glen_b

Thanks, the note about power was helpful. I did some more searching. I think the technical term for my criterion (a) is that a test is "consistent" or "pointwise consistent in power," which means that as you take more and more data, the power gets closer and closer to 1. (I'm not 100% sure of this.) It's a desirable feature of an alternative hypothesis but maybe not part of the definition. By "given some assumptions" in (b), I guess I mean any assumptions you can justify based on your knowledge of the problem. (Like assuming particular drug might cure an illness but won't cause it.)