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 θ. We suggest regularity conditions on the parent distribution that permit the definition of Fisher information (FI) about θ 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 θ, we investigate the properties of this FI, and obtain the asymptotic relative efficiency of the maximum likelihood estimator of θ for Type II censored bivariate samples. Assuming (X, Y) is Gumbel bivariate exponential of second type, and θ 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.