# Bootstrap validity for non plug-in estimators

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.