Can we ever have $H_0: \theta\neq \theta_0$ (bilateral hypothesis)? Are there any theorems that show some sort of most powerful test for this case?
And what about $H_0: \theta<\theta_0$?
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I know that $H_0: \theta\leq \theta_0$ vs $H_1: \theta> \theta_0$ is the usual one-tailed hypothesis testing. – An old man in the sea. Feb 12 '17 at 18:24
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One can test anything. Not that the answer necessarily means much. From a Bayesian perspective, this null hypothesis and the opposite one are treated equivalently. – Xi'an Feb 12 '17 at 18:56
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As for the second case of testing $H_0: \theta < \theta_0$ against $H_1: \theta \ge \theta_0$ you may want to check out Karlin-Rubin Theorem. – Łukasz Grad Feb 12 '17 at 19:10
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@ŁukaszGrad But the version I know of the Karlin-Rubin theorem is for $H_0:\theta\leq \theta_0$. I'm not sure it also works for my question... – An old man in the sea. Feb 12 '17 at 19:13
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@Anoldmaninthesea. Karlin-Rubin ,as far as i know, is adequate for any test $H_0: \theta \in \Theta_0$, $H_1: \theta \in \Theta_1$ such that $$\forall_{\theta_0 \in \Theta_0, \theta_1 \in \Theta_1} \theta_0 < \theta_1$$. When all additional assumptions hold, that is – Łukasz Grad Feb 12 '17 at 19:26
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1Thanks for posting this as a question. While no proofs are discussed, you might be interested in: Why do statisticians say a non-significant result means “you can't reject the null” as opposed to accepting the null hypothesis?, which discusses this topic generally. – gung - Reinstate Monica Feb 12 '17 at 19:40
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@ŁukaszGrad Thanks for the input. Could you direct me to some references? I've never found a statement/discussion of the theorem that would comprehend my case. – An old man in the sea. Feb 12 '17 at 19:46
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@gung I had already read your answer, and found it informative... However, despite having also read the comments by silverfish, I wasn't able to adapt your reasoning to the usual unilateral case, let alone to the case I'm asking here. If you ever add that to your answer, please send me a signal... ;) – An old man in the sea. Feb 12 '17 at 19:56
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Yeah, it isn't the same, just related IMO. The one-sided case is also distinct. I suppose I could try to add something there, but it wouldn't quite fit I don't think. – gung - Reinstate Monica Feb 12 '17 at 20:00
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@gung Want me to ask a new question for it? ;) – An old man in the sea. Feb 12 '17 at 20:02
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1There's a difficulty with testing with the null being an open set -- the closure is in the alternative. This poses some problems – Glen_b Feb 13 '17 at 01:46
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@Xi'an Thanks for your comment. By the way, do you know of any reference that deals with these cases under the frequentist perspective? ;) – An old man in the sea. Feb 13 '17 at 12:21
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1$H_0: \theta \ne \theta_0$ can only make sense if $\theta$ can only assume some discrete values, i.e. it can't be arbitrarily close to $\theta_0$. If it's a usual continuous parameter, then this $H_0$ does not make sense. – amoeba Feb 18 '17 at 16:02
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@amoeba could you please elaborate a bit more? I'm still learning the ropes... – An old man in the sea. Feb 18 '17 at 19:25
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Yep, possible. Your "backwards" null hypothesis is the one-sample version of a test for equivalence. Take a look at the hypothesis $H_0: \: \theta_1 \neq \theta_2$ and how it is often handled with the two one-sided t test (TOST). Pay close attention to how p-values and type I errors are calculated.
Mike Anderson
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1Mike, thanks for your answer. Could you please add a bit more detail, and possibly some references? I've tried searching the sea of information that's the internet, but it's just immense... – An old man in the sea. Feb 12 '17 at 20:00
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3As far as I know, this is not what TOST does. The challenge in TOST is to re-interpret the practically useless "$H_0:\theta_1\ne\theta_2$" as $H_0:|\theta_1-\theta_2|\ge \Delta$ for some definite positive number $\Delta$. – whuber Feb 12 '17 at 20:55
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