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Suppose $X_1,...,X_n $ are independent bernoulli random variables with success parameter $p$, i.e. the common pmf is $f(x)=p^x (1-p) ^{1-x}$ for $x=1$ or $x=0$. I am interested in finding a prediction interval of the weighted sum $$Y=\sum_{i=1}^nw_iX_i,$$ where $w_i>0$ are deterministic weights.

I do not know the underlying probability $p$, but I do have an estimator $\hat{p}$ whose standard deviation $s$ is known. Suppose also that my estimator $\hat{p}$ is unbiased for $p$. I believe it follows that the random variable $\tilde{X}_i$ with success probability $\hat{p}$ is unconditionally distributed the same as $X_i$, take for example $\hat{p}$ to be Beta distributed with parameters that imply a mean of $p$ and calculate $\int f(x|\hat{p}) f_2(\hat{p}) d \hat{p}$, where $f_2$ is the pdf of the beta distribution with implied mean $p$. Therefore, I could simply use

$$\tilde{Y}=\sum_{i=1}^nw_i\tilde{X}_i$$ to construct a prediction interval for $Y$ by simulation of $\tilde{Y}$ or by a central limit theorem approximation for $\tilde{Y}$. While this confidence interval is correct unconditionally, I feel it is quite awkward because it doesnt incorporate that my estimator $\hat{p}$ is quite variable. Is there also a way to incorporate the standard deviation $s$ of my estimator (in case I also know the distribution of $\hat{p}$) to construct a prediction interval for $Y$?

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  • $\begingroup$ Are you looking for a confidence interval on the expectation of $Y$, or for a prediction-interval on $Y$? There is a difference. $\endgroup$ Sep 8, 2022 at 10:13
  • $\begingroup$ @StephanKolassa I am referring to the construction of a random interval $[L, U]$ such that $P(Y \in [L, U])$ is equal to a pre-specified probability. Since $Y$ is not a parameter I think I should technically refer to this as a prediction interval. $\endgroup$
    – Joogs
    Sep 8, 2022 at 10:22
  • $\begingroup$ Yes, that is precisely a prediction interval. Do you want to update your post and the tags? $\endgroup$ Sep 8, 2022 at 10:30

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