Abstract
Estimation of the quantile μ + κσ of an exponential distribution with parameters (μ, σ) is considered under an arbitrary strictly convex loss function. For κ obeying a certain condition, the inadmissibility of the best affine equivariant procedure is established by exhibiting a better estimator. The LINEX loss is studied in detail. For quadratic loss, sufficient conditions are given for a scale equivariant estimator to dominate the best affine equivariant one and, when κ exceeds a lower bound specified below, a new minimax estimator is identified.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Basu, A. P. and Ebrahimi, N. (1991). Bayesian approach to life testing and reliability estimation using asymmetric loss function, J. Statist. Plann. Inference, 29, 21-31.
Brewster, J. F. (1974). Alternative estimators for the scale parameter of the exponential distribution with unknown location, Ann. Statist., 2, 553-557.
Brewster, J. F. and Zidek, J. V. (1974). Improving on equivariant estimators, Ann. Statist., 2, 21-38.
Brown, L. D. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters, Ann. Math. Statist., 39, 29-48.
Elfessi, A. (1997). Estimation of a linear function of the parameters of an exponential distribution from doubly censored samples, Statist. Probab. Lett., 36, 251-259.
Elfessi, A. and Pal, N. (1991). On location and scale parameters of exponential distributions with censored observations, Comm. Statist. Theory Methods, 20, 1579-1592.
Huang, S. Y. (1995). Empirical Bayes testing procedures in some nonexponential families using asymmetric linex loss function, J. Statist. Plann. Inference, 46, 293-309.
Kubokawa, T. (1994). A unified approach to improving equivariant estimators, Ann. Statist., 22, 290-299.
Kuo, L. and Dey, D. K. (1990). On the admissibility of the linear estimators of the Poisson mean using LINEX loss function, Statist. Decisions, 8, 201-210.
Lehmann, E. L. (1986). Testing Statistical Hypotheses, 2nd ed., Wiley, New York.
Madi, M. T. and Leonard, T. (1996). Bayesian estimation for shifted exponential distributions, J. Statist. Plann. Inference, 55, 345-351.
Madi, M. T. and Tsui, K. W. (1990). Estimation of the common scale of several shifted exponential distributions with unknown locations, Comm. Statist. Theory Methods, 19, 2295-2313.
Parsian, A. (1990). On the admissibility of an estimator of a normal mean vector under a LINEX loss function, Ann. Inst. Statist. Math., 42, 657-669.
Parsian, A. and Sanjari Farsipour, N. (1993). On the admissibility and inadmissibility of estimators of scale parameters using an asymmetric loss function, Comm. Statist. Theory Methods, 22, 2877-2901.
Parsian, A. and Sanjari Farsipour, N. (1997). Estimation of parameters of exponential distribution in the truncated space using asymmetric loss function, Statist. Papers, 38, 423-443.
Parsian, A., Sanjari Farsipour, N. and Nematollahi, N. (1993). On the minimaxity of Pitman type estimator under a LINEX loss function, Comm. Statist. Theory Methods, 22, 97-113.
Rukhin, A. L. (1986). Admissibility and minimaxity results in the estimation problem of exponential quantiles, Ann. Statist., 14, 220-237.
Rukhin, A. L. and Strawderman, W. E. (1982). Estimating a quantile of an exponential distribution, J. Amer. Statist. Assoc., 77, 159-162.
Rukhin, A. L. and Zidek, J. V. (1985). Estimation of linear parametric functions for several exponential samples, Statist. Decisions, 3, 225-238.
Sadooghi-Alvandi, S. M. (1990). Estimation of the parameter of a Poisson distribution using a LINEX loss function, Austral. J. Statist., 32, 393-398.
Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Ann. Inst. Statist. Math., 16, 155-160.
Varian, H. L. (1975). A Bayesian approach to real estate assesment, Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage (eds. Stephen E. Fienberg and Arnold Zellner), 195-208, North-Holland, Amsterdam.
Zellner, A. (1986). Bayesian estimation and prediction using asymmetric loss functions, J. Amer. Statist. Assoc., 81, 446-451.
Zidek, J. V. (1973). Estimating the scale parameter of the exponential distribution with unknown location, Ann. Statist., 1, 264-278.
Author information
Authors and Affiliations
About this article
Cite this article
Petropoulos, C., Kourouklis, S. Estimation of an Exponential Quantile under a General Loss and an Alternative Estimator under Quadratic Loss. Annals of the Institute of Statistical Mathematics 53, 746–759 (2001). https://doi.org/10.1023/A:1014648819462
Issue Date:
DOI: https://doi.org/10.1023/A:1014648819462