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I need to find an appropriate statistical test (likelihood ratio test, t-test, etc.) on the following: Let $\{X_i;Y_i\}^n_{i=1}$ be an i.i.d. sample of a random vector $(X;Y)$ and assume that $\bigl( \begin{smallmatrix} Y\\ X \end{smallmatrix} \bigr)$~$N$ $\left[\bigl( \begin{smallmatrix} \mu_1\\ \mu_2 \end{smallmatrix} \bigr), \bigl( \begin{smallmatrix} 1 & .5\\ .5 & 1 \end{smallmatrix} \bigr) \right]$. The hypotheses are: $H_0=\mu_1+\mu_2\le 1$; $H_1=\mu_1+\mu_2\gt 1$

By looking at this information, how do I know which test is the most appropriate? Is it because the data is i.i.d. I can simply take a likelihood ratio test? A good explanation on what test is more appropriate than another one would be greatly appreciated. This would definitely clear my mind.

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    $\begingroup$ Have you noticed that $X+Y\sim N(\mu_1+\mu_2, 3)$ and $X-Y\sim N(\mu_1-\mu_2, 1)$ are uncorrelated and jointly normal, whence they are independent? Thus you can digest your dataset into $\{(X_i+Y_i)\}$, view it as a set of iid realizations of a Normal distribution with known variance and unknown mean, and ask how to compare its mean to zero. This is an elementary textbook problem with a well-known answer (a Z test). $\endgroup$ – whuber♦ Oct 9 '12 at 4:54
  • $\begingroup$ @whuber thanks! I will look into this more carefully. Thanks for the insight. $\endgroup$ – CharlesM Oct 9 '12 at 4:58
  • $\begingroup$ @whuber what I find difficult though is that I face a composite hypothesis testing and I don't know how to set this is up. any suggestion would be welcomed $\endgroup$ – CharlesM Oct 9 '12 at 19:11
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    $\begingroup$ @whuber it is a previous year practice exam question - so yes not the test itself $\endgroup$ – CharlesM Oct 10 '12 at 17:10
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    $\begingroup$ @whuber Shouldn't the $X-Y$ distribution have $\mu_1-\mu_2$ as its mean? I realize it doesn't matter for this problem, but it just worries me to see the typo sitting there. $\endgroup$ – Glen_b May 16 '13 at 8:26
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Let's investigate the distribution of $Z=X+Y$.

$E[X+Y] = \mu_1 + \mu_2$

and

$var(Z) = var(X+Y) = var(X) + var(Y) + 2Cov(X,Y)$ which equals to 3 in your case.

What remains is testing $H_0: Z < 1$ which can be done with the usual t-test.

Hope this helps.

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