题号:14639    题型:解答题    来源:2024年丘成桐大学生数学竞赛(概率与统计)-无答案
Consider a random sample of size $n$, and write the data as an $r=r_n$ by $c=c_n$ matrix, $\left\{X_{i j}: i=1, \cdots, r_n ; j=1, \cdots, c_n\right\}$ with $n=r_n c_n$. To spec ify notation, $\left\{X_{i j}\right\}$ 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
$$
\hat{\beta}_n=\min _j\left\{\max _i\left\{X_{i j}\right\}\right\} .
$$
(a) What is the condition on $r_n$ when $n \rightarrow \infty$ 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 $r_n\left(\hat{\beta}_n-\beta\right)$ converges in distribution, and find the li miting distribution function.
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