A Location Estimator Based on a $U$-Statistic
基于 U 统计量的位置估计量
Let $X_1, \cdots, X_n$ be i.i.d. $F$, and estimate the median of $F$ by the median $T_\beta$ of $\beta X_i + (1 - \beta)X_j, i \neq j$, where $\beta$ is a fixed positive constant. Then $T_\beta$ is the solution of a $U$-statistic equation from which its asymptotic normality is readily derived. The asymptotic relative efficiency of $T_\beta$ is computed for a few cdfs $F$ and seen to be reasonably high for unintuitive choices such as $\beta = .9, \beta = 2$, and also to be…