I'm struggling a bit with the notation to get the prediction interval from a maximum likelihood regression.

Having 2 random variables $X, Y$, where they relate as $y = f(x; \beta) + \epsilon$, where $\epsilon \sim N(0, \sigma^2)$ and $\beta$ are coefficients that relate $X$ to $Y$, we do not know how the function $f$ uses the coefficients.

So i want to predict a new value $y_{new}$ given $x_{new}$, and give an interval around $y_{new}$.

The prediction should then be: $$\hat y_{new} = f(x_{new}; \hat \beta) + \epsilon $$

Then, the interval should be $$[\mu_{new} - z\sigma_{pred}, \mu_{new} + z\sigma_{pred}]$$

Here is where my doubts start, since we do not know $\mu_{new}$ nor $\sigma_{pred}$ we have to estimate them. We estimate $\mu_{new}$ with $\hat y_{new}$, so far so good. For $\sigma_{pred}$ i think it should be $\hat \sigma_{pred}^2 = Var(\hat y_{new}) + \hat \sigma^2$ where $Var(\hat y_{new})$ is the variance of the predictions and $\hat \sigma^2$ is the MSE from the random noise. Since we say they are a random independent sample we get no covariance.

So would the prediction interval end up like this? $$\hat y_{new} \pm t_{n-2}\hat \sigma_{pred}$$ $$\hat y_{new} \pm t_{n-2}\sqrt{Var(\hat y_{new}) + \hat \sigma^2}$$

Or should i use the standard error? $$s.e = \sqrt\frac{Var(\hat y_{new}) + \hat \sigma^2}{n} $$ $$\hat y_{new} \pm t_{n-2}s.e$$

I'm a bit confused with why i should use the standard error and not the first interval, also i'm not sure where the $n-2$ from the t-student comes from.


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