Short question: I would like to know if there is theoretical validity for using bootstrap for estimators that are not based on the plug-in principle. Most sources I studied so far work with plug-in estimators.

Full picture: Let $F$ be the cdf, $\hat{F}$ the empirical cdf obtained from an i.i.d. sample $X = (X_1,\ldots,X_n)$, and $(\Theta, d)$ be a metric space. My quantity of interest is $\theta = t(F)$. If I understood correctly, the plug-in estimator would be $\hat{\theta} = t(\hat{F})$. However, I am interest in another estimator $\hat{\gamma} = g(\hat{F})$, which also has the purpose of estimating $\theta$.

I know that $\hat{\gamma}$ is strongly consistent. Fixing the sample size $n$, I would like to assess the convergence computing

$$ E\left[d\left(\hat{\gamma}, \theta\right) | X\sim F\right] \quad. $$

Following the same recipe as for the plug-in (approximating $F$ by $\hat{F}$ and using simulation to compute the expectation over $\hat{F}$), one would arrive at

$$ \frac{1}{B}\sum_{j=1}^B d\left(\hat{\gamma_i^*}, \hat{\gamma} \right) \quad,$$

where $\hat{\gamma_i^*} = g(\hat{F_i^*})$, $\hat{F_i^*}$ being the empirical cdf of the i-th bootstrap sample $X_i^*$.

My major doubt is the fact that I substituted $\theta$ by $\hat{\gamma}$. For the plug-in estimators, swapping $\theta$ by $\hat{\theta}$ is justifiable since we are approximating $F$ by $\hat{F}$. I do not know if this is justifiable for non plug-in estimators. It seems to me that I swapped them ad hoc.



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