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A linear model has t-distributed noise $$ Y= bX + \epsilon $$ where $\epsilon \sim t(0, \sigma^2, df)$, with mean 0, variance $\sigma^2$ and dof $df$.

Given an independent sample $(x_i, y_i), i=1, \dots, n$, suppose we already have estimates of $b, \sigma^2, df$. How would you estimate a $1-\alpha$ prediction interval of $Y$ at $X=x$?

Is it still the same as when the noise $\epsilon$ is normal distributed: $$ \hat{y} \pm t_{\alpha/2, n-2} s_y \sqrt{\frac{1}{n} + \frac{(x-\bar{x})^2}{\sum (x_i-\bar{x})^2}} $$ where $$ s_y = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n-2}} $$

Thanks!

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  • $\begingroup$ The prediction interval should be of the form you stated except for the degrees of freedom. A little more though should be given to it. $\endgroup$ – Chamberlain Foncha Mar 7 '14 at 23:09

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