I have the following problem:
If we have a typical random sample $\{X_1, X_2, ...\}$ from some unknown distribution and we want to estimate a quantile we just need to sort our observations and take a specific observation from this sorted sequence.
Let us know assume that we know additionally that $\mathbb{E}[X_i] = 0$ (or any other number).
My question is:
If we have our observations $\{x_1, x_2, ..., x_n\}$, can we substract from them its sample mean (that should be close to $0$ as $\mathbb{E}[X_i] = 0$) and then sort them and take a specific quantile.
Is it mathematically correct?
Can I say sth about my new quantile estimator? Does it have better or worse properties than the standard one?