Consider a random sample of size $n$, and write the data as an $r=r_n$ by $c=c_n$ matrix, $\{X_{ij}:i=1,\cdots ,r_n;j=1,\cdots ,c_n\}$ with $n=r_n{c}_n$. To spec ify notation, $\left\{X_{ij}\right\}$ are i.i.d. with c.d.f. $\mathrm{F}(\mathrm{x})$ and continuous de nsity $f(x)$. Let $\beta$ denote the median, i.e., $F(\beta )=0.5$. Define an estimator by
$${\hat{\beta }}_n=\underset{j}{min}\left\{\underset{i}{max}\left\{X_{ij}\right\}\right\}.$$
(a) What is the condition on $r_n$ when $n\to \mathrm{\infty }$ for median-un biasedness, i.e., $\beta$ is also the median for the distribution of ${\hat{\beta }}_n$ ?
(b) We further assume $F$ is differentiable in an open neighbo rhood of $\beta$ and has a positive derivative at $\beta$. For $r_n$ in (a), sh ow that $r_n({\hat{\beta }}_n-\beta )$ converges in distribution, and find the li miting distribution function.