@ballardw I think the OP meants that the CI for a quantile is often wider for extreme quantiles than for quantiles near the median. But as you and others have mentioned, it depends on the distribution.
@AndersS The book by Hahn and Meeker (1991) has lots f informationa bout confidence intervals, including for quantiles, I think. THe SAS doc fgor PROC UNIVARIATE discusses CIs for quantiles. If you assume the data are normal, you can use the CIPCTLNORMAL optin to estimate CIs. In that situation (normal data), it is true that the CIs are narrower for the median than for extreme quantiles. See the example https://go.documentation.sas.com/doc/en/pgmsascdc/9.4_3.5/procstat/procstat_univariate_examples10.htm
When you don't know distribution oif the data, you can use CIPCTLDF option to get distribution-free intervals. The formulas are at https://go.documentation.sas.com/doc/en/pgmsascdc/9.4_3.5/procstat/procstat_univariate_details14.htm#procstat.univariate.clpctl
Regarding "which quantiles are useful," it is true that many works report 5%, 10%, 25%, 50%, 75%, 90%, and 95%. However, this blog shows that for bootstrap computations you can need 2.5% and 97.5%, too https://blogs.sas.com/content/iml/2016/08/10/bootstrap-confidence-interval-sas.html
The blog at https://blogs.sas.com/content/iml/2017/05/24/definitions-sample-quantiles.html
have many ways to compute quantiles. They are all simlar: form the ECDF, use it to estiamte the CDF (stepwise, piecewise, cubic spline, etc) and then use inverse interpolation to estimate the quantiles. As you say, slope of ECDF determine accruacy of estimate, but big slope can be anywhere, not just at median.
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