A class of estimators in two-phase sampling with subsampling the non-respondents Giancarlo Diana a , Pier Francesco Perri b,⇑ a Department of Statistical Sciences, University of Padova, Via Cesare Battisti 241, 35121 Padova, Italy b Department of Economics, Statistics and Finance, University of Calabria, Via P. Bucci, 87036 Arcavacata di Rende, Italy article info Keywords: Asymptotic variance Auxiliary information Efficiency Regression type-estimator abstract Motivated by Singh and Kumar [14], we introduce in this paper a general class of estima- tors for the population mean of a study variable when two auxiliary variables are used in the presence of non-response. The minimum asymptotic variance bound of the estimators belonging to the class is determined and the optimality of Singh–Kumar estimators dis- cussed. The best estimator in the class is analytically found in accordance with the auxil- iary information used and the efficiency gain that can be achieved upon competitive estimators is shown by an empirical study. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction Refusals to cooperate or respond are common in survey practice. To cope with this persistent problem, survey statisti- cians usually consider and adapt the non-respondents sub-sampling scheme introduced by Hansen and Hurwitz [6] to a wide range of practical situations. One topic which is widely debated in sampling theory is the estimation of the population mean (  Y ) of a target variable in the presence of non-respondents when auxiliary information is available. Along these lines, Okafor and Lee [9] proposed some two-phase sampling ratio and regression estimators for  Y when the mean of an auxiliary characteristic is unknown and there is non-response on the study variable. Similarly, Singh and Kumar [12,13] proposed ra- tio-cum-product and regression-type estimators, while Khare and Sinha [8] considered the problem of estimating  Y using multi-auxiliary variables with known population means in presence of non-response on the study variable and of informa- tion on the auxiliary variable which may or may not be missing. Finally, Singh and Kumar [14] considered two auxiliary variables, one or both with unknown population mean and used two-phase sampling procedures in the presence of non- response for the estimation of  Y . Inspired by this study, in the present work we discuss the estimation of  Y in a unified framework using two-phase sam- pling when data may not have been collected for all the sampled units due to the problem of non-response. We consider two auxiliary variables and introduce a general class of estimators for which the estimator that attains the minimum asymptotic variance bound is a regression-type estimator. We show how the class can be adapted to a number of situations which make use of a different amount of auxiliary information. It is shown that the estimators proposed by Singh and Kumar [14], though members of the class, are not optimum since their efficiency can be readily improved upon by more suitable estimators which employ the same amount of information. 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.03.106 ⇑ Corresponding author. E-mail address: pierfrancesco.perri@unical.it (P.F. Perri). Applied Mathematics and Computation 219 (2013) 10033–10043 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc