The Annals of Probability

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

Peter Hall

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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


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