Computational Statistics & Data Analysis 51 (2007) 2559 – 2572 www.elsevier.com/locate/csda Quantile estimation in two-phase sampling María del Mar Rueda a , ∗ , Antonio Arcos a , Juan Francisco Muñoz a , Sarjinder Singh b a Department of Statistics and O.R., University of Granada, 18071 Granada, Spain b Department of Statistics, St. Could State University, 720 FourthAvenue South, St. Cloud, MN 56301-4498, USA Received 12 January 2005; received in revised form 3 January 2006; accepted 3 January 2006 Available online 24 January 2006 Abstract The estimation of quantiles in two-phase sampling with arbitrary sampling design in each of the two phases is investigated. Several ratio and exponentiation type estimators that provide the optimum estimate of a quantile based on an optimum exponent  are proposed. Properties of these estimators are studied under large sample size approximation and the use of double sampling for stratification to estimate quantiles can also be seen. The real performance of these estimators will be evaluated for the three quartiles on the basis of data from two real populations using different sampling designs. The simulation study shows that proposed estimators can be very satisfactory in terms of relative bias and efficiency. © 2006 Elsevier B.V.All rights reserved. Keywords: Auxiliary information; Finite population quantiles; Two-phase sampling; Stratified random sampling 1. Introduction The problem of estimating a population mean in the presence of an auxiliary variable has been widely discussed in the finite population sampling literature. However, for the problem of estimating a population median, the situation is quite different and only recently has this problem been discussed. Rao et al. (1990) proposed ratio and difference estimators for the median using a design-based approach. Kuk and Mak (1989) proposed two estimators for which it was only necessary to know the values of the median of the auxiliary variable for the whole population. More recently, Rueda et al. (1998) and Rueda and Arcos (2001) proposed confidence intervals for quantiles based on ratio and difference estimators of the distribution function. In Rueda et al. (2003, 2004) the population information is used through a quantile of the auxiliary variable with the same or different order as that of the quantile of the main variable considered for estimation using difference type estimators. The above estimators are based on prior knowledge of the median Q x (0.5) of the auxiliary characteristic. In many cases Q x (0.5) may not be known, and it may be seen that taking the sample selection in two phases is an attractive solution. Two-phase sampling is a good compromise for surveys in which no prior knowledge is available about the population. A key to successful two-phase sampling is the creation of a highly informative frame for the part of the population ∗ Corresponding author. Departamento de Estadística e I.O., Facultad de Ciencias, Avda. Fuentenueva, Universidad de Granada, 18071, Granada, Spain. Tel.: +34 958240494; fax: +34 958243267. E-mail addresses: mrueda@ugr.es (M. del Mar Rueda), arcos@ugr.es (A. Arcos), jfmunoz@ugr.es (J.F. Muñoz), sarjinder@yahoo.com (S. Singh). 0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.01.002