Let $X$ be a random variable and let $M(X)$ be its median. Using Jensen's inequality, one can prove that
$$|M(X) - \mathbb{E}[X]| \le \sigma $$ which implies that $M(X) = \mathbb{E}[X] + O(\operatorname{Var}(X)^{1/2})$ provided that $Var(X)$ exists. My questions are:
- Can we get a 'higher order' asymptotic expansion for the median in terms of further moments of $X$ ?
- Can we sharpen the above inequality if we assume certain tail behaviors of $X$ such as the tail dies exponentially?
There are no asymptotic expansions for the median in terms of higher moments.
Consider two variables which have all the same moments, like the lognormal variable with pdf $$f(x) = \frac{1}{x \sqrt{2\pi}} \exp\left(\frac{-(\log x)^2}{2}\right)$$ and its perturbation with pdf $$g(x) = f(x)(1 + \sin(2\pi \log x)).$$ Each has $E[X^n]=\exp(n^2/2)$, and they decay at similar rates, but the lognormal variable has median $1$ and its perturbation has median $1.128$.