Bootstrapping with quantiles of data instead of SD*z?

I have recently been bootstrapping the confidence intervals of a neural network model estimated to data.

I execute the following psudo-code, which seems similar to previous bootstraps I have done:

1. Take all observations of data $$x_i$$, estimate $$\hat{y} = \hat{\theta}(x_i)$$
2. Resample $$N$$ observations from the set of $$x_i$$ with replacement. Estimate $$\tilde{y} = \tilde{\theta}(x_i)$$. Repeat until we have $$M$$ different versions of $$\tilde{y}_m$$
3. Calculate the standard deviation of everything in $$\tilde{y}_M$$ as $$\sigma_{\tilde{y}}$$ and then the confidence interval of $$\hat{y}$$ is $$\hat{y} \pm z * \sigma_{\tilde{y}}$$ where z is chosen to be an appropriate value from a standard normal distribution, usually 1.96 for 95% CI.

At the same time, I see other approaches that use the quantiles of $$\tilde{y}_M$$ in order to construct intervals of some sort, such as here and here. They do this instead of the $$\hat{y} \pm z * \sigma_{\tilde{y}}$$ I have come to expect. The intuition for such an argument is very strong and seems valid - but the approach is unfamiliar. What's going on with the use of quantiles? Is something else being calculated (prediction intervals, etc.) that are similar in flavor but are not the same?

• Davison and Hinkley's book Bootstrap Methods and their Application presents a number of approaches to bootstrapping (not exhaustive but both parametric and nonparametric cases are discussed) and it does discuss bootstrap prediction intervals and nonparametric regression models; this might be helpful as a reference, for all that the various pieces are not all in the same places, since the relevant concepts should carry across. Commented May 12, 2022 at 1:50

• I'm a little confused - ARMAX modeling? I am asking about when it is appropriate to use $\pm z*SD$ vs quantiles to estimate confidence intervals. Commented May 11, 2022 at 15:17