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.
Publié le : 1982-11-14
Classification:
Central limit theorem,
mean,
mode,
order statistics,
spacings,
strong consistency,
60F05,
60F15,
62G30
@article{1176993720,
author = {Hall, Peter},
title = {Limit Theorems for Estimators Based on Inverses of Spacings of Order Statistics},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 992-1003},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993720}
}
Hall, Peter. Limit Theorems for Estimators Based on Inverses of Spacings of Order Statistics. Ann. Probab., Tome 10 (1982) no. 4, pp. 992-1003. http://gdmltest.u-ga.fr/item/1176993720/