AISM 54, 667-680

## 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.

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