## The Annals of Statistics

### Estimators Related to $U$-Processes with Applications to Multivariate Medians: Asymptotic Normality

#### Abstract

If a criterion function $g(x_1, \ldots, x_m; \theta)$ depends on $m > 1$ samples, then a natural estimator of $\arg \max P^mg(x_1, \ldots, x_m; \theta)$ is the $\arg \max$ of a $U$-process. It is observed that, under suitable conditions, these estimators are asymptotically normal. This is then applied to prove asymptotic normality of Liu's simplical median and of Oja's medians in $\mathbb{R}^d$.

#### Article information

Source
Ann. Statist., Volume 22, Number 3 (1994), 1460-1477.

Dates
First available in Project Euclid: 11 April 2007

https://projecteuclid.org/euclid.aos/1176325637

Digital Object Identifier
doi:10.1214/aos/1176325637

Mathematical Reviews number (MathSciNet)
MR1311984

Zentralblatt MATH identifier
0827.62023

JSTOR
Arcones, Miguel A.; Chen, Zhiqiang; Gine, Evarist. Estimators Related to $U$-Processes with Applications to Multivariate Medians: Asymptotic Normality. Ann. Statist. 22 (1994), no. 3, 1460--1477. doi:10.1214/aos/1176325637. https://projecteuclid.org/euclid.aos/1176325637