Can one construct “original” data from a function of jackknifed data?

Say I have original, uncorrelated data, $$x_i$$, with $$i = 1,2 \ldots N$$. I can jackknife this data set (a simple delete-one) $$\bar{x}_{i} = \frac{1}{N-1}\sum_{j \neq i}x_{j} \quad\quad (1)$$ to construct $$N$$ re-samples. In addition, the re-sampled data can be used to reconstruct the original data set by inverting Eq. (1), $$x_{i} = \sum_{j} \bar{x}_{j} - (N-1) \bar{x}_{i}. \quad\quad (2)$$ Now say I apply some function to my $$N$$ re-samples (the $$\bar{x}_{i}$$s), $$f_{i} = f(\bar{x}_{i})$$. One can use Eq. (2) to reconstruct "original" data from the $$f_i$$ (be replacing $$\bar{x}_{i}$$ with $$f_i$$ in Eq. (2)), and further use that original data to find variances, $$\sigma^2_{i}$$, for each $$f_i$$.

My question, is it acceptable to use Eq. (2) on the $$f_i$$ to construct "original" data?

• I'm not sure I understand your question correctly. It seems to me that you've stated, in the second sentence of your penultimate paragraph that you can reconstruct the original data from the $f_i$ by replacing the $\bar{x_i}$ with $f_i$ in Eq (2), but then you go on to ask just this: if it is acceptable to use Eq. (2) on the $f_i$ to construct the original data. Or am I just not following this correctly? – StatsStudent Apr 19 at 19:19
• @StatsStudent Yeah, I guess I'm putting a lot of meaning into the word "acceptable". I can certainly perform the operation of using Eq. (2) on the $f_i$s, but can the resulting data be used to reliably construct, say, $\sigma^2_i$ for each $f_i$, or is the resulting data too much of a destruction of what the distribution of the original $f$s would look like? Does that help? – kηives Apr 19 at 19:28