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Say we have the following regression model: $$Y_i = \alpha + \beta(x_i - \mathrm{mean}(x)) + R_i$$

where $R_1,\ldots,R_{20} \sim G(0, \sigma)$

If we have $\mu(x) = \alpha + \beta(x - \mathrm{mean}(x))$, how do I go about finding the MLE of $\mu(5)$?

I have a given data set with some calculations done for me, but not sure how to approach this?

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    What distribution are you calling "$G$"?2012-06-28
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    Whatever might be the answer to the question I posted above, if you find the MLEs for $\alpha$ and $\beta$ and plug those in to $\alpha+\beta(x-\mathrm{mean}(x))$, that should be the MLE for $\mu(x)$. MLEs have that kind of invariance (or more precisely, maybe I should call it "equivariance").2012-06-28
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    How do you know this? Also G is the Gaussian distribution.2012-06-28
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    Usually $N$ or $\mathcal{N}$ is used for the normal or "Gaussian" distribution.2012-06-28

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