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November, 1982 Limit Theorems for Estimators Based on Inverses of Spacings of Order Statistics
Peter Hall
Ann. Probab. 10(4): 992-1003 (November, 1982). DOI: 10.1214/aop/1176993720


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.


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Peter Hall. "Limit Theorems for Estimators Based on Inverses of Spacings of Order Statistics." Ann. Probab. 10 (4) 992 - 1003, November, 1982.


Published: November, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0516.60024
MathSciNet: MR672299
Digital Object Identifier: 10.1214/aop/1176993720

Primary: 60F05
Secondary: 60F15 , 62G30

Keywords: central limit theorem , ‎mean‎ , Mode , order statistics , spacings , strong consistency

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 4 • November, 1982
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