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I was designing a hypothesis test, using the likelihood ratio test, and I found out that, under $H_{0}$, the test statistic $\Lambda$ equals the same constant $k$ no matter what the observations may be. Does that mean that the rejection region is of the following form? $$ C=\{ x, \Lambda(x)<k\}$$

Toney Shields
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    Please show full details. I ask because I expect it's the same potential error as your previous question. A pity you deleted instead of responding to clarify there. – Glen_b Oct 03 '16 at 23:08
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    @Glen_b What I did was the following: I wrote the likelihood ratio test, I pluged in the estimators of the the parameters (Which are function of the observations), I then noticed that under $H_{0}$ the test statistic always equal the same constant $k$ no matter what the observations may be, I did a simulation on Matlab and all the values of $\Lambda$ was close to 0 (values were of the form $x,xxx \times 10^{-16}$). realising this result was always true when $H_0$ is verified, I was wondering how can I extract the p-value, and since $\Lambda$ doesn't seem to have a distribution ... – Toney Shields Oct 04 '16 at 13:08
  • (at least to my knowledge), now that I look at my question, I can see that it's wrong – Toney Shields Oct 04 '16 at 13:08
  • Details should go in the question, but you're omitting the very details that would let us see why you have a constant value under $H_0$ (we're not mind-readers). In particular I suspect you're treating the $L_0$ term (ie. the one under H0) in the LRT statistic wrong (e.g. plugging in ML estimates derived under $H_1$). Please add complete details of what you did into your question, so we can see what you are doing. Alternatively if you know what went wrong, please add an answer. – Glen_b Oct 04 '16 at 21:56

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Your rejection region is either everything or it's nothing. The test has either a 100% false positive rate or a 100% false negative rate.

AdamO
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  • Somehow I suspect the problem occurred earlier... – eric_kernfeld Oct 04 '16 at 01:05
  • @eric_kernfeld what do you mean? – Toney Shields Oct 04 '16 at 17:15
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    @ToneyShields a likelihood ratio statistic should be a statistic. As in: it is a function of the data. If data can't change it, it's a dumb LR. You should check the assumptions of a likelihood ratio test. All kinds of problems surface when testing a hypothesis about, say, a uniform(0, $\theta$) distribution. Think about "why" and what the density function is, and so on. – AdamO Oct 04 '16 at 20:15