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.
Similar content being viewed by others
References
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics, Wiley, New York.
Balasubramanian, K. and Beg, M. I. (1997). Concomitants of order statistics in Morgenstern type bivariate exponential distribution, J. Appl. Statist. Sci., 5, 233–245.
David, H. A. and Nagaraja, H. N. (1998). Concomitants of order statistics, Handbook of Statistics, Vol. 16 (eds. N. Balakrishnan and C. R. Rao) 487–513, Elsevier, Amsterdam.
Gumbel, E. J. (1960). Bivariate exponential distributions, J. Amer. Statist. Assoc., 55, 698–707.
Hutchinson, T. P. and Lai, C. D. (1990). Continuous Bivariate Distributions, Emphasising Applications, Rumsby Scientific, Adelaide, Australia.
Lehmann, E. L. (1999). Elements of Large-sample Theory, Springer, New York.
Mehrotra, K. G., Johnson, R. A. and Bhattacharyya, G. K. (1979). Exact Fisher information for censored samples and the extended hazard rate functions, Comm. Statist. Theory Methods, 15, 1493–1510.
Nagaraja, H. N. (1983). On the information contained in order statistics, Tech. Report, No. 278, Department of Statistics, The Ohio State University, Columbus, Ohio.
Nagaraja, H. N. (1994). Tukey's linear sensitivity and order statistics, Ann. Inst. Statist. Math., 46, 757–768.
Nelsen, R. B. (1999). An Introduction to Copulas, Lecture Notes in Statist., 139, Springer, New York.
Park, S. (1996). Fisher information in order statistics, J. Amer. Statist. Assoc., 91, 385–390.
Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed., Wiley, New York.
Scaria, J. and Nair, U. (1999). On concomitants of order statistics from Morgenstern family, Biometrical J., 41, 483–489.
Smith, M. D. and Moffatt, P. G. (1999). Fisher's information on the correlation coefficient in bivariate logistic models, Australian & New Zealand Journal of Statistics, 41, 315–330.
Tukey, J. W. (1965). Which part of the sample contains the information?, Proc. Nat. Acad. Sci., U.S.A., 53, 127–134.
Zheng, G. and Gastwirth, J. L. (2000). Where is the Fisher information in an ordered sample?, Statist. Sinica, 10, 1267–1280.
Author information
Authors and Affiliations
About this article
Cite this article
Abo-Eleneen, Z.A., Nagaraja, H.N. Fisher Information in an Order Statistic and its Concomitant. Annals of the Institute of Statistical Mathematics 54, 667–680 (2002). https://doi.org/10.1023/A:1022479514859
Issue Date:
DOI: https://doi.org/10.1023/A:1022479514859