Aust.N.Z.J.Stat. 43(1), 2001, 33–46 MEDIAN ESTIMATION USING DOUBLE SAMPLING SARJINDER SINGH 1∗ ,ANWAR H. JOARDER 2 AND D.S. TRACY 3 UniversityofSaskatchewan, K.F.U.ofPetroleumandMinerals and UniversityofWindsor Summary This paper proposes a general class of estimators for estimating the median in double sampling. The position estimator, stratification estimator and regression type estimator attain the minimum variance of the general class of estimators. The optimum values of the first-phase and second-phase sample sizes are also obtained for the fixed cost and the fixed variance cases. An empirical study examines the performance of the double sampling strategies for median estimation. Finally, an extension of the methods of Chen & Qin (1993) and Kuk & Mak (1994) is considered for the double sampling strategy. Keywords: auxiliary information; cost aspects; empirical study; median estimation; two-phase sampling; variance. 1. Introduction In survey sampling, statisticians often find that they are dealing with variables, such as income, expenditure, etc., that have highly skewed distributions. In such situations, the me- dian is regarded as a more appropriate measure of location than the mean. Kuk & Mak (1989) suggest estimators of the median using auxiliary information in survey sampling. Francisco & Fuller (1991) also deal with the problem of estimation of the median as part of the estimation of a finite population distribution function. A brief description of the motivation of Kuk & Mak (1989) is given below. Let Y i and X i , i = 1, 2,...,N, denote the values of the population units for the study variable Y and the auxiliary variable X, respectively. Further let y i and x i , i = 1, 2,...,n, denote the values of the units included in a sample S n of size n drawn by simple random sampling without replacement (SRSWOR). Assuming that the median M X of variable X is known, Kuk & Mak (1989) proposed a ratio estimator ˆ M (r) Y = ˆ M Y M X ˆ M X , (1) where ˆ M Y and ˆ M X are the sample estimators of M Y and M X , respectively. Suppose that y (1) ,y (2) ,...,y (n) are the y values of sample units in ascending order. Further, let t be an Received May 1998; revised January 2000; accepted January 2000. ∗ Author to whom correspondence should be addressed. 1 Dept of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N 5E6, Canada. e-mail: sarjinder@yahoo.com 2 Dept of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. 3 Dept of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada N9B 3P4. Acknowledgments. The authors thank an Associate Editor, a referee, the Editor and the Technical Editor for valuable comments that improved this paper. Professor D.S. Tracy passed away on 27 December 1998 at the airport, New Delhi, India. The opinions and results discussed in this paper are the authors’ and not necessarily of their institute(s). This work was done while the first author was at the Australian Bureau of Statistics, Canberra, Australia. c  Australian Statistical Publishing Association Inc. 2001. Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden MA 02148, USA