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Thread: Hypothesis Testing

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    Hypothesis Testing

    Let $\displaystyle X$ have a pdf of the form $\displaystyle f(x; {\theta}) = {\theta}x^{\theta-1}$ where $\displaystyle {\theta}\in\{\theta: \theta=1, 2\}$.

    To test the simple hypothesis $\displaystyle H_0:\theta=1$ against the alternative simple hypothesis $\displaystyle H_1:\theta=2$, use a random sample $\displaystyle X_1, X_2$ of size $\displaystyle n=2$ and define the critical region to be $\displaystyle C = \{(x_1, x_2): \frac{3}{4} \leq x_1x_2 \}$.

    Find the power function of the test.

    I hate to ask for help when I feel like I don't have much of anything to show for my own efforts, but I feel stuck.

    I know I need $\displaystyle \gamma(\theta) = P_{\theta}[\frac{3}{4} \leq x_1x_2] = 1 - P_{\theta}[x_1x_2 \leq \frac{3}{4}]$

    I don't know how to set the limits of integration on the joint pdf to get the answer given in the back of the book:
    $\displaystyle 1-(\frac{3}{4})^\theta+\theta(\frac{3}{4})^{\theta}l og({\frac{3}{4})}
    $
    Last edited by Zithuan; Apr 7th 2009 at 05:58 PM.
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