## The Annals of Statistics

### Estimates of Location: A Large Deviation Comparison

Gerald L. Sievers

#### Abstract

This paper considers the estimation of a location parameter $\theta$ in a one-sample problem. The asymptotic performance of a sequence of estimates $\{T_n\}$ is measured by the exponential rate of convergence to 0 of $\max \{P_\theta(T_n < \theta - a), P_\theta(T_n > \theta + a)\}, \text{say} e(a).$ This measure of asymptotic performance is equivalent to one considered by Bahadur (1967). The optimal value of $e(a)$ is given for translation invariant estimates. Some computational methods are reviewed for determining $e(a)$ for a general class of estimates which includes $M$-estimates, rank estimates and Hodges-Lehmann estimates. Finally, some numerical work is presented on the asymptotic efficiencies of some standard estimates of location for normal, logistic and double exponential models.

#### Article information

Source
Ann. Statist., Volume 6, Number 3 (1978), 610-618.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344205

Digital Object Identifier
doi:10.1214/aos/1176344205

Mathematical Reviews number (MathSciNet)
MR494631

Zentralblatt MATH identifier
0395.62034

JSTOR