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For the question in the link below, I am seeking the minimal sufficient statistic for $\theta$={$\beta_1$,$\beta_2$} in the linear regression model given.

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I have taken the ratio of likelihoods $L(x, \theta)$/$L(y,\theta)$ which has given me the result below:

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I know that I am looking for a condition that makes the likelihood ratio independent of $\theta$. From the first term in the product, when the sum of all $x_i$ is equal to the sum of all $y_i$, this term is made independent of $\theta$.

What about the second term in this product? Is there a second condition from this fraction that I need to synthesise into the solution, or have I already shown that the minimal sufficient statistic is the sum of all $x_i$? I can't find a way to algebraically break up that second fraction so that the product symbol goes away on each side.

Thanks.

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