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Dec 8, 2021 at 18:49 comment added Geoffrey Johnson Yes, I suggest replacing the word "confidence" with "prediction" in your question. Here is a thread on prediction intervals. If the prediction target is drawn from an approximately normal distribution you can create a Wald-type prediction interval as well. If you are predicting a subject-level observation from a non-normal distribution the link above discusses a paper with simple and effective methods. There are many methods for prediction and the intuition to your approach may be effective.
Dec 8, 2021 at 5:20 comment added Davi Américo I think wald statistics would not be useful I think I would have to have so many observations to a single observation $x_i$ but the question is about how to solve the given problem (minimize $L$). Regardless, I'd think about booststrap but the computional spending would be a lot. I was figuring about just remove the two first terms in $L$ they dont seem be useful.
Dec 8, 2021 at 3:06 comment added Davi Américo Yeah a predictive confidence interval do you think I be better updating this?
Dec 8, 2021 at 2:28 history answered Geoffrey Johnson CC BY-SA 4.0