How well does bootstrapping approximate the sampling distribution of an estimator? Having recently studied bootstrap, I came up with a conceptual question that still puzzles me:
You have a population, and you want to know a population attribute, i.e. $\theta=g(P)$, where I use $P$ to represent population. This $\theta$ could be population mean for example. Usually you can't get all the data from the population. So you draw a sample $X$ of size $N$ from the population. Let's assume you have i.i.d. sample for simplicity. Then you obtain your estimator $\hat{\theta}=g(X)$. You want to use $\hat{\theta}$ to make inferences about $\theta$, so you would like to know the variability of  $\hat{\theta}$. 
First, there is a true sampling distribution of $\hat{\theta}$. Conceptually, you could draw many samples (each of them has size $N$) from the population. Each time you will have a realization of $\hat{\theta}=g(X)$ since each time you will have a different sample. Then in the end, you will be able to recover the true distribution of $\hat{\theta}$. Ok, this at least is the conceptual benchmark for estimation of the distribution of $\hat{\theta}$. Let me restate it: the ultimate goal is to use various method to estimate or approximate the true distribution of $\hat{\theta}$.
Now, here comes the question. Usually, you only have one sample $X$ that contains $N$ data points. Then you resample from this sample many times, and you will come up with a bootstrap distribution of $\hat{\theta}$. My question is: how close is this bootstrap distribution to the true sampling distribution of $\hat{\theta}$? Is there a way to quantify it?
 A: Bootstrap is based on the convergence of the empirical cdf to the true cdf, that is,
$$\hat{F}_n(x) = \frac{1}{n}\sum_{i=1}^n\mathbb{I}_{X_i\le x}\qquad X_i\stackrel{\text{iid}}{\sim}F(x)$$ converges (as $n$ goes to infinity) to $F(x)$ for every $x$. Hence convergence of the bootstrap distribution of $\hat{\theta}(X_1,\ldots,X_n)=g(\hat{F}_n)$  is driven by this convergence which occurs at a rate $\sqrt{n}$ for each $x$, since $$\sqrt{n}\{\hat{F}_n(x)-F(x)\}\stackrel{\text{dist}}{\longrightarrow}\mathsf{N}(0,F(x)[1-F(x)])$$ even though this rate and limiting distribution does not automatically transfer to $g(\hat{F}_n)$. In practice, to assess the variability of the approximation, you can produce a bootstrap evaluation of the distribution of $g(\hat{F}_n)$ by double-bootstrap, i.e., by bootstrapping bootstrap evaluations.
As an update, here is an illustration I use in class:

where the lhs compares the true cdf $F$ with the empirical cdf $\hat{F}_n$ for $n=100$ observations and the rhs plots $250$ replicas of the lhs, for 250 different samples, in order to measure the variability of the cdf approximation. In the example I know the truth and hence I can simulate from the truth to evaluate the variability. In a realistic situation, I do not know $F$ and hence I have to start from $\hat{F}_n$ instead to produce a similar graph.
Further update: Here is what the tube picture looks like when starting from the empirical cdf:

A: In Information Theory the typical way to quantify how "close" one distribution to another is to use KL-divergence 
Let's try to illustrate it with a highly skewed long-tail dataset - delays of plane arrivals in the Houston airport (from hflights package). Let $\hat \theta$ be the mean estimator. First, we find the sampling distribution of $\hat \theta$, and then the bootstrap distribution of $\hat \theta$ 
Here's the dataset: 

The true mean is 7.09 min. 
First, we do a certain number of samples to get the sampling distribution of $\hat \theta$, then we take one sample and take many bootstrap samples from it.
For example, let's take a look at two distributions with the sample size 100 and 5000 repetitions. We see visually that these distributions are quite apart, and the KL divergence is 0.48. 

But when we increase the sample size to 1000, they start to converge (KL divergence is 0.11)

And when the sample size is 5000, they are very close (KL divergence is 0.01)

This, of course, depends on which bootstrap sample you get, but I believe you can see that the KL divergence goes down as we increase the sample size, and thus bootstrap distribution of $\hat \theta$ approaches sample distribution $\hat \theta$ in terms of KL Divergence. To be sure, you can try to do several bootstraps and take the average of the KL divergence.
Here's the R code of this experiment: https://gist.github.com/alexeygrigorev/0b97794aea78eee9d794
