In the MIT OpenCourseWare notes for 18.05 Introduction to Probability and Statistics, Spring 2014 (currently available [here](https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014/readings/MIT18_05S14_Reading24.pdf)), it states:

> The bootstrap percentile method is appealing due to its simplicity. However it depends on
the bootstrap distribution of $\bar{x}^{*}$ based on a **particular** sample being a good approximation to
the true distribution of $\bar{x}$. Rice says of the percentile method, "Although this direct equation
of quantiles of the bootstrap sampling distribution with confidence limits may seem initially
appealing, it’s rationale is somewhat obscure."[2] In short, **don’t use the bootstrap
percentile method**. Use the empirical bootstrap instead (we have explained both in the
hopes that you won’t confuse the empirical bootstrap for the percentile bootstrap).

> [2] John Rice, *Mathematical Statistics and Data Analysis*, 2nd edition, p. 272

After a bit of searching online, this is the only quote I've found which outright states that the percentile bootstrap should not be used. 

What I recall reading from the text *Principles and Theory for Data Mining and Machine Learning* by Clarke et al. is that the main justification for bootstrapping is the fact that 
$$\dfrac{1}{n}\sum_{i=1}^{n}\hat{F}_n(x) \overset{p}{\to} F(x)$$
where $\hat{F}_n$ is the empirical CDF. (I don't recall details beyond this.)

> Is it true that the percentile bootstrap method should not be used? If so, what alternatives are there for when $F$ isn't necessarily known (i.e., not enough information is available to do a parametric bootstrap)?

**Edit**: Because clarification has been requested, the **empirical bootstrap** refers to the following procedure:

> When forming a  $100(1-\alpha)$% confidence interval for $\theta$ in the form $\hat{\theta} \pm c \cdot \text{se}$, where $\text{se}$ is the standard error, gather estimates of $\text{se}$ using bootstrapping and use $\text{se}_{\alpha/2}$, $\text{se}_{1-\alpha/2}$, with subscripts denoting percentiles of the bootstrap estimates.

> Paraphrased from Section 11.2 of *Computer Age Statistical Inference* by Efron and Hastie (2016).

The MIT notes linked above do something similar to this, but not exactly: they compute $\delta_1 = (\hat{\theta}^{*}-\hat{\theta})_{\alpha/2}$ and $\delta_2 =  (\hat{\theta}^{*}-\hat{\theta})_{1-\alpha/2}$ with $\hat{\theta}^{*}$ the bootstrapped estimates of $\theta$ and $\hat{\theta}$ the full-sample estimate of $\theta$, and the resulting estimated confidence interval would be $[\hat{\theta}-\delta_1, \hat{\theta} - \delta_2]$. 

In essence, the main idea is this: empirical bootstrapping estimates an amount proportional to the **difference between the point estimate and the actual parameter**, i.e., $\hat{\theta}-\theta$, and uses this difference to come up with the lower and upper CI bounds.

The **percentile bootstrap** refers to the following: 

> To form a $100(1-\alpha)$% confidence interval for $\theta(\mathbf{x})$, let $\hat{\theta}(\mathbf{x})$ be a statistic for $\theta(\mathbf{x})$, resample to compute $\hat{\theta}(\mathbf{x}^{*})$ with $\mathbf{x}^{*}$ a resampling of the same size as $\mathbf{x}$, and use $[\hat{\theta}(\mathbf{x}^{*})_{\alpha/2}, \hat{\theta}(\mathbf{x}^{*})_{1-\alpha/2}]$ as the confidence interval for $\theta(\mathbf{x})$. 

> Paraphrased from Section 11.2 of *Computer Age Statistical Inference* by Efron and Hastie (2016).
 
In this situation, we use bootstrapping to **compute estimates of the parameter of interest and take the percentiles of these estimates for the confidence interval**.

Please let me know if anything I wrote above is unclear or incorrect.