## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60F15: Strong theorems 62G30: Order statistics; empirical distribution functions

**Keywords**

Central limit theorem mean mode order statistics spacings strong consistency

#### Citation

Hall, Peter. Limit Theorems for Estimators Based on Inverses of Spacings of Order Statistics. Ann. Probab. 10 (1982), no. 4, 992--1003. doi:10.1214/aop/1176993720. https://projecteuclid.org/euclid.aop/1176993720