Suppose I've done a linear regression, I've got $\hat{y}=Ax+b$ (where $x$ is probably a vector and $y$ may be a real or a vector). So our actual assumption is the model $y=Ax+b+e$, where $e$ is normal with mean zero and some fixed standard deviation $\sigma$.

But suppose $\sigma$ really depends on $x$. This makes sense; for example it might be larger when $x$ (and correspondingly, $y$) is larger, or something much more complicated.

In this context, can we will compute prediction intervals for an unknown value of $y$, given a new value of $x$?

What if we drop the normal noise entirely? Can we still use the central limit theorem to do something (say, get a prediction interval for an average of several estimates, or something)?

I should say: I would more appreciate an explanation, rather than a formula!


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