## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 5, Number 4 (1977), 779-786.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176343900

Digital Object Identifier
doi:10.1214/aos/1176343900

Mathematical Reviews number (MathSciNet)
MR451528

Zentralblatt MATH identifier
0363.62039

JSTOR
Maritz, J. S.; Wu, Margaret; Stuadte, R. G. A Location Estimator Based on a $U$-Statistic. Ann. Statist. 5 (1977), no. 4, 779--786. doi:10.1214/aos/1176343900. https://projecteuclid.org/euclid.aos/1176343900