## The Annals of Probability

### Limit Theorems for Estimators Based on Inverses of Spacings of Order Statistics

Peter Hall

#### Abstract

Let $X_{n1} < X_{n2} < \cdots < X_{nn}$ denote the order statistics of an $n$-sample from the distribution with density $f$. We prove the strong consistency and asymptotic normality of estimators based on the series $(\frac{1}{2}) \sum^{n-k}_1 (X_{n,r+k} + X_{nr})/(X_{n,r+k} - X_{nr})^p \text{and} \sum^{n-k}_1 (X_{n,r+k} - X_{nr})^{-p}$, where $k > 2p > 0$ are fixed constants. These series may be used to estimate functionals of $f$. The ratio of the series was introduced by Grenander (1965) as an estimator of a location parameter, and he established weak consistency. In recent years several authors have examined such estimators using Monte Carlo experiments, but the lack of an asymptotic theory has prevented a more detailed discussion of their properties.

#### Article information

Source
Ann. Probab., Volume 10, Number 4 (1982), 992-1003.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993720

Digital Object Identifier
doi:10.1214/aop/1176993720

Mathematical Reviews number (MathSciNet)
MR672299

Zentralblatt MATH identifier
0516.60024

JSTOR