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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.

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Definitely generate the bootstrap interval for $\theta$ based on the distribution of bootstrapped $\hat{\theta}^{(r)}$. That's because your goal is obtaining an unbiased estimate of the sampling distribution of $\hat{\theta}$. The reason why the second approach probably won't work is:

$$\text{var}(\hat{\theta}) = \text{var} \left(\sum_{i=1}^k \hat{f}(j)\right) \ne \sum_{i=1}^k \text{var}\left(\hat{f}(i)\right)$$

Except when the pointwise observations of the functional are independent, which is usually not the case, but your specific situation may be different.

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