American Journal of Mathematics and Statistics 2014, 4(2): 65-71 DOI: 10.5923/j.ajms.20140402.03 Estimation for Domains in Stratified Sampling Design in the Presence of Nonresponse E. P. Clement 1,* , G. A. Udofia 2 , E. I. Enang 2 1 Department of Mathematics and Statistics, University of Uyo, Uyo, Nigeria 2 Department of Mathematics, Statistics and Computer Science, University of Calabar, Calabar, Nigeria Abstract An analytical approach for finding the best sampling design subject to a cost constraint is developed. We consider stratified random sampling design when elements of the inclusion probabilities are not equal but are in same stratum and proposed estimators of totals for domains of study under nonresponse in the context of calibration estimation. We derived optimum stratum sample sizes for a given set of unit costs for the sample design and compared empirically the relative performances of the proposed calibration estimators with a corresponding global estimator. Analysis and evaluation are presented. Keywords Calibration estimation, Domain estimation, GREG-estimator, Optimum allocation, Sampling design 1. Introduction In sample survey, separate estimates of a parameter may be required for subpopulations into which a population is divided without separately sampling from these subpopulations. Such subpopulations are called domains of study [1]. The method of estimating the domain parameters is called domain estimation. [2] first considered in detail some of the problems associated with the estimation of domain totals, means and proportions in the case of a single-stage simple random sampling. He noted that the variance of an estimator of a domain parameter is increased by the fact that the number of the domain elements, and hence the number of those elements that can fall in a random sample of a fixed size, is unknown before the start of the survey. [3] gave a derivation of Yates’ results in multi-stage sampling. [3] paper is one of the first attempts to unify the theory of domain estimation. Hartley provided the theory for a number of sample designs where domain estimation was of interest. His paper mostly discussed estimations that did not make use of auxiliary information. He did, however, consider the case of ratio estimation where population totals were known for the domains. [4] extended Yates’ results to double sampling for probability proportional to size (PPS) when information on the size, X, of each sampling unit is unknown. [5] proposed an empirical Bayes estimation of domain means under nested * Corresponding author: epclement@yahoo.com (E. P. Clement) Published online at http://journal.sapub.org/ajms Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved error linear regression model with measurement errors in the covariates. The problem of allocation of resources when domains of study are of primary interest is discussed by [6]. However, despite these vast extensions of Yates results, the phenomenon of nonresponse and its problems in domain estimation have not been adequately addressed. In many human surveys, information is in most cases not obtained from all the units in the survey even after some call-backs. An estimate obtained from such incomplete data may be misleading especially when the respondents differ from the non-respondents because the estimate can be biased. Nonresponse always exists when surveying human populations as people hesitate to respond in surveys; and increases notably while studying sensitive issues like family size. Nonresponse as an aspect in almost every type of sample survey creates problems for estimation which cannot simply be eliminated by increasing sample size. The phenomenon of nonresponse in a sample survey reduces the precision of parameters estimates and increases bias in estimates resulting in larger mean square error, thus ultimately reducing their efficiency. An important technique to address these problems is by calibration. Calibration as a tool for reweighting for nonresponse was first introduced by [7] for the estimation of finite population characteristics like means, ratios and totals. This calibration approach requires the formulation of suitable auxiliary variables. The calibration approach provides a unified treatment of the use of auxiliary information in surveys with nonresponse. In the presence of powerful auxiliary information, the calibration approach meets the objectives of reducing both the sampling error and the nonresponse error. In survey sampling many authors, such as [7-11] defined