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.


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.