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I have the following question:

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What have I done so far:

I have showed that Y4 is sufficient and complete for theta. Now I need to apply Basu's theorem, and I am not exactly sure how to show that Y1 / Y4 or (Y1+Y2)/(Y3+Y4) are ancillary statistics. Help would be great.

Thanks.

1 Answers 1

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Use the fact that $(Y_1, Y_2, Y_3, Y_4)$ is equal in distribution to $\theta (Z_1, Z_2, Z_3, Z_4)$ where the $Z_i$'s represent the order statistics of four Uniform$(0, 1)$ random variables - this should be clear but prove it if it's not. Hence, for example, $\frac{Y_1}{Y_4}$ has the same distribution as $\frac{\theta Z_1}{\theta Z_4} = \frac{Z_1}{Z_4}$ which is free of $\theta$ and so ancillary. After applying Basu, we are victorious. The general theme here is that when you are looking at scale families you can easily get ancillary statistics by considering ratios.

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    (1): Let $U_1, ..., U_4$ be iid Uniform$(0, 1)$. Then $\theta U_1, ..., \theta U_4$ are iid Uniform$(0, \theta)$ is clear (just transform the pdf to get this). Then, the order statistics of $\theta U_1, ..., \theta U_4$ is equal in distribution to $Y_1, ..., Y_4$ since they are both order statistic of a Uniform$(0, \theta)$. But ordering $\theta U_1, ..., \theta U_4$ gives the same result as ordering $U_1, ..., U_4$, and multiplying by $\theta$, which is the same dist as $\theta Z_1, ..., \theta Z_4$ by definition of $Z_1, ..., Z_4$. (2): $\frac{ab}{ac} = \frac{b}{c}$ by cancellation :)2012-02-13