Suppose that $Y_1, Y_2,\ldots, Y_n$ are independent $N(\alpha,σ^2)$ Show that, if $\sigma$ is unknown, the likelihood ratio statistic for testing a value of $\alpha$ is given by $$D = n \log\left(1 + \frac{1}{n-1}T^2\right)\;,$$ where $$T = \frac{\hat{α} -\alpha}{\sqrt{s^2/n}}$$
Statistics (Likelihood ratio stat.)
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statistics
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0You haven't told us enough. Presumably the two parameters $\sigma$ and $\alpha$ should index a family of probability distributions. What family of distributions are you applying this to? – 2012-03-11
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0Please check to see that this is what you intended: there were some mismatched parentheses in the original, and the intent wasn’t entirely clear. – 2012-03-11
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0I think the recent edits make it clear. You're asking why the standard Student's t-test is a likelihood-ratio test. The way it's expressed here, with the test statistic depending only on $T^2$, you'd have to have a simple one-point null hypothesis and a two-sided alternative hypothesis. If I were presenting this to a class of students who know how to do only what they've been told how to do, I wouldn't give it as an exercise, but I might do it in class. For the kind who can be given the relevant definitions and then figure things out, it's not a bad exercise. – 2012-03-11
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0....and I've done it in front of classes a couple of times. – 2012-03-11
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0I up-voted this question after the necessary clarifications were done. But still at this time the vote total is $-1$. Is there something objectionable about the question? – 2012-03-11
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1Glad you got it, max. Could you "accept" my answer? (I.e. click on the green check-mark.) – 2012-03-13