An acquaintance of mine has been using this wrong inference formula for years: given
- a i.i.d. sample $\mathbf{X}={X_1,\dots,X_N}$ for a continuous RV $X$,
- sample mean $\bar{X}=\frac{\sum X_i}{N}$ and sample standard deviation $\bar{\sigma}=\frac{\sum \left(X_i-\bar{X}\right)^2}{N-1}$
estimate the 0.95-quantile $q_{0.95}$ as
$$q_{0.95} = \bar{X} + 2 \bar{\sigma}$$
(which is not even a decent point estimate - you should at the very least use $q_{0.95} = \bar{X} + 1.645\bar{\sigma}$).
What are the correct confidence intervals for a generic $q$, in the three cases:
- we know nothing on $X$ (apart from the fact that it's continuous), and we look for an exact (non-asymptotic) answer. I think this answer for the median could be modified for a generic quantile
- as before, but an asymptotic solution is fine. I guess there should be at least a couple answers here...one for quantiles which aren't close to 0 or 1, and one for quantiles which are. Maybe one based on normal approximation and one based on Poisson?
- finally, we assume $X$ to have a Gaussian distribution with unknown mean and variance.