I have a model whose estimation is a function (finite number of points) over an intervalle. I am looking at the sum of each estimated point of the function : $\hat{\theta} := \sum_{i=0}^k \hat{f}(i)$ which is my parameter of interest.
Now, if I want a $95\%$ CI using resampling, I see two ways to do so:
First, I can do $P$ resampling, compute $P$ estimated function $\hat{f}_j$ and then for each function compute $\hat{\theta}_j$ as the sum of this function at each point. A $95\%$ CI could then be computed by taking the $2.5\%$ and $97.5\%$ quantiles of the sample $(\hat{\theta}_j)$
An other way to do is to do $P$ resampling and compute $P$ estimated functions $\hat{f}_j$. Then, for each point $i \in \{0, \dots k\}$ we can compute the $95\%$ confidence interval $[l_i, u_i]$ from the sample $(\hat{f}_j(i))_{0 \leq j \leq P}$. We then have an $97.5\%$ upper function ($f(i) = u_i$) as well as a $2.5\%$ lower function ($f(i) = l_i$). A $95\%$ CI for $\theta$ could then be computed as the sum at each point of the lower function and upper function.
Those two methods yield very differents CI and I was wondering which one is correct?
My opinion is that the first method is correct but I can't see why the second would not be.
