A Location Estimator Based on a $U$-Statistic

Abstract

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 remarkably constant for $\beta > 1$. Moreover, the influence curves and breakdown points of $\{T_\beta: \beta > 0\}$ are derived and indicate that the good robustness properties of the Hodges-Lehmann estimator $(\beta = \frac{1}{2})$ are shared by the entire class. Monte Carlo estimates of the variance of $T_\beta$ for sample sizes $n = 10, 20$, and 40 indicate that some of these estimators perform as well as those discussed in the Princeton Robustness Study when the underlying $F$ is double-exponential or Cauchy.