Is the sample quantile unbiased for the true quantile?

Question

I would like to find a way to show whether the sample quantile is an unbiased estimator of the true quantiles. Let $F$ be strictly increasing with density function $f$. I will define the $p$-th quantile for $0<p<1$ as $Q(p)=F^{-1}(p)$ and the sample quantile as $$\hat{F}_n^{-1}(p)=\inf\{x:\hat{F}_n(x)\geq p\},$$ where $\hat{F}_n(x)$ is the empirical distribution function, given by $$\hat{F}_n(x)=\frac{1}{n}\sum_{i=1}^n I(X_i \leq x).$$ Based on literature I have read, I expect the sample quantile to be biased, but I am having trouble figuring out how to take the expected value of $\hat{F}_n^{-1}(p)$, particularly since it is defined as the infimum of a set. I do know that the expected value of the empirical distribution function is $F(x)$. Any help or references that could guide me would be greatly appreciated!

Answer 1

$\hat{F}_n^{-1}(p)$ is the smallest value $x$ such that at least $p$ fraction of the sample points satisfy $X_i \leq x$. In other words, at least $np$ of the sample points satisfy $X_i \leq x$, and since $np$ may not be an integer we can actually say at least $\lceil np \rceil$. Thus $\hat{F}_n(p)^{-1}=x$ if and only if at least $\lceil np \rceil$ of the sample points satisfy $X_i \leq x$, and there exists $X_i$ such that $X_i=x$ (otherwise we'd be able to shrink $x$ a little and still have $\hat{F}_n(x)>p$).

This is still in a bit too complicated a form to take the expected value, but it may help to look at a small case, say $n=1$. In this case, if $p>0$ then $\hat{F}_n^{-1}(p)=x$ if and only if $X_1=x$. In other words, $\hat{F}_n^{-1}(p)=X_i$ for all $p>0$. This is not an unbiased estimator of $Q(p)$.