# How can I make a prediction interval for a future response (not its mean) in regression by using bootstrap?

I'd like to know how I can use bootstrap to predict the confidence interval for a future response (not for its mean) no matter what theorical model and error distribution are, I know I can train the upper and lower bootstrap limits for the known observations by using some regression model I've even done that but I'd like a more full bootstrap regarding approach and I don't know if that approach makes a confidence interval for either the response or its mean.

Take:

$$y_i=f(x_i)+\epsilon_i, i\in\{1,...,n\}$$

I'd like to get a prediction confidence interval for an unknown observation $$y_k$$ with $$k \notin \{1,...,n\}$$ but how would I do that? how would I retrain the model in an observation which I haven't even seen this in order to get its confidence interval?

• Terminology: An interval for an observation is not a confidence interval. It might be a prediction interval (but possibly not -- see the details at the link): en.wikipedia.org/wiki/Prediction_interval Commented Jan 14, 2022 at 3:39
• It is attractive to sample from the model residuals for creating future responses. There are several possible approaches. E.g., do you want prediction intervals to be conditional on the data or not? BTW, you still need to eradicate the many occurrences of "confidence" from this post, because it is not about confidence intervals at all.
– whuber
Commented Jan 14, 2022 at 16:38