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I am interesting into the problem of testing a specific hypothesis of the form $H_1:\{ \mu_1<\ldots <\mu_N\}$. In the literature I found, such a hypothesis is compared to whether $H_2:\{\mu_1=\ldots =\mu_N\}$ (all means equals) or the "full model" $H_2^{'}: \{(\mu_1,\ldots,\mu_N) \in R^N\}$ but not versus its complementary.

This puzzle me a lot because when testing $H_1$ Vs $H_2^{'}$ (full model) (at least as proposed in http://bayesfactor.blogspot.fr/2015/01/multiple-comparisons-with-bayesfactor-2.html), the associated bayes factor cannot exceed $N!$ whatever the evidence.

Moreover, using $H_1$ Vs $H_2$ (all mean equals e.g. http://www.sciencedirect.com/science/article/pii/S0167715214001862) seems to me to be a situation where one has to give very strong insight of why $\mu_i>\mu_{i+1}$ cannot happen (for similar consideration than How do you tell if you should use a one-tailed test or a two-tailed test in a t-test?)

So my question is :

Considering the situation that I have a theory according to which $\mu_1<\ldots <\mu_N$ that I want to test. Is there any reason to avoid comparing $H_1:\{ \mu_1<\ldots <\mu_N\}$ to its complementary ? or to prefer $H_2$ or $H_2^{'}$ ?

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