AISM 54, 667-680

© 2002 ISM

## Fisher information in an order statistic and its concomitant

### Z.A. Abo-Eleneen and H.N. Nagaraja

Department of Statistics, Ohio State University, Columbus OH 43210-1247, U.S.A., e-mail: zaher@stat.ohio-state.edu; hnn@stat.ohio-state.edu

(Received July 14, 2000; revised February 9, 2001)

Abstract.
Let $(X,Y)$ have an absolutely continuous distribution
with parameter
$\theta$. We suggest regularity conditions on the parent distribution
that permit the definition of Fisher information (FI) about $\theta$
in an $X$-order statistic and its $Y$-concomitant that are obtained
from a random sample from $(X,Y)$. We describe some general
properties of the FI in such individual pairs. For the
Farlie-Gumbel-Morgenstern parent with dependence parameter $\theta$,
we investigate the properties of this FI, and obtain the asymptotic
relative efficiency of the maximum likelihood estimator of $\theta$
for Type II censored bivariate samples. Assuming $(X,Y)$ is Gumbel
bivariate exponential of second type, and $\theta$ is the mean of $Y$, we
evaluate the FI in the $Y$-concomitant of an $X$-order statistic and
compare it with the FI in a single $Y$-order statistic.

Key words and phrases:
Concomitants of order statistics, Fisher information, Farlie-Gumbel-Morgenstern family, Gumbel Type II bivariate exponential distribution, Type II censoring, maximum likelihood estimator.

**Source**
(TeX , DVI )