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I am doing a two-tailed test, and the null distribution is a Gamma distribution $\Gamma(a,b)$. Let's say my test statistic is $R$. If I were to do a left tail test, then the P-value is computed as $P(R<R_{obs})$, where $R_{obs}$ is the observed value of $R$. Similarly, the P-value of a right tail test would be $P(R>R_{obs})$. For a two-tailed test, the P-value should be $1-P(|R|<R_{obs})=1-P(-R_{obs}<R<R_{obs})$. This where I'm having trouble with, since I have a Gamma distribution and $P(R<-R_{obs})$ is always 0, which implies $1-P(|R|<R_{obs})=1-P(R<R_{obs})$, and which is the same P-value of a right tail test. What am I missing?

Toney Shields
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    Two related points. First, you can't conduct a two-tailed test, or indeed any hypothesis test, until you specify a null hypothesis. Second, your definition of a two-tailed P-value appears to be copied from a different context where the null hypothesis is that the mean is zero and the null distribution is symmetric. Clearly neither of those things can be true here. – Gordon Smyth Feb 07 '21 at 08:38
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    The first question you should be asking is what parameter you are testing a hypothesis about. Is that $a$ or $b$ or $\mu=ab$ or something else? – Gordon Smyth Feb 07 '21 at 08:39
  • @GordonSmyth, Under the null hypothesis, $R=0$, but I have that as the sample size $N$ tends to $\infty$, my statistic converges to a certain value $c$. That is why my rejection region is $C=\{R:R\geq w_{1-\frac{\alpha}{2}}\}\cup\{R:R\leq w_{\frac{\alpha}{2}}\}$. $ w_{\alpha}$ is the quantile of the gamma distribution using significance level $\alpha$. – Toney Shields Feb 07 '21 at 09:06
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    The lower quantile is not equal to the negative of the upper quantile. That only works when the distribution is symmetric about 0. – John L Feb 07 '21 at 18:09
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    Toney, you still haven't got the idea of hypothesis testing. The null hypothesis has to be a statement about a population parameter. It is not a statement about an observed statistic like $R$. At this stage you haven't specified either a parameter of interest, or a null hypothesis, or a test statistic. Discussion will be fruitless unless you come to grips with those things. – Gordon Smyth Feb 07 '21 at 21:14
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    You're also getting sidetracked by considerations of what the test statistic might converge to asymptotically. Asymptotics are not relevant here. To construct a rejection region you need to consider the exact small sample distribution of the test statistic (whatever it might be) in relation to the null and alternative hypotheses (whatever they might be). – Gordon Smyth Feb 07 '21 at 23:20
  • @GordonSmyth, I appologize, I may have expressed my issue poorly. the statistic $L$ has a property, which under the null it has the value 0, otherwise, it is larger than $0$. So my null hypothesis is $L=0$ vs the alternative $L>0$. the estimator $R$ of $L$ has a Gamma distribution $\Gamma(a,b)$, where $a$ and $b$ are estimated from the sample. – Toney Shields Feb 08 '21 at 07:44
  • @GordonSmyrh, I am sorry, but I still am not sure what I'm not getting right. – Toney Shields Feb 08 '21 at 07:47

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