The Annals of Probability

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

Peter Hall

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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

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

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60F15: Strong theorems 62G30: Order statistics; empirical distribution functions

Central limit theorem mean mode order statistics spacings strong consistency


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.

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