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