This paper suggests some estimators for population mean of the study variable in simple random sampling and two-phase sampling using information on an auxiliary variable under second order approximation. Bahl and Tuteja (1991) and Singh et al. (2008) proposed some efficient estimators and studied the properties of the estimators to the first order of approximation. In this paper, we have tried to find out the second order biases and mean square errors of these estimators using information on auxiliary variable based on simple random sampling and two-phase sampling. Finally, an empirical study is carried out to judge the merits of the estimators over others under first and second order of approximation. 1. Introduction Let denote a finite population of distinct and identifiable units. For the estimation of population mean of a study variable , let us consider to be the auxiliary variable that is correlated with study variable , taking the corresponding values of the units. Let a sample of size be drawn from this population using simple random sampling without replacement (SRSWOR) and , ( ) are the values of the study variable and auxiliary variable, respectively, for the th units of the sample. In sampling theory the use of suitable auxiliary information results in considerable reduction in MSE of the ratio estimators. Many authors including Singh and Tailor [1], Kadilar and Cingi [2], Singh et al. [3], and Singh and Kumar [4] suggested estimators using some known population parameters of an auxiliary variable in simple random sampling. These authors studied the properties of the estimators to the first order of approximation. But sometimes it is important to know the behavior of the estimators to the second order of approximation because up to the first order of approximation the behavior of the estimators is almost the same, while the properties for second order change drastically. Hossain et al. [5] and Sharma and Singh [6, 7] studied the properties of some estimators to the second order approximation. Sharma et al. [8, 9] also studied the properties of some estimators under second order of approximation using information on auxiliary attributes. In this paper we have studied properties of some exponential estimators under second order of approximation in simple random sampling and two-phase sampling using information on an auxiliary variable. 2. Some Estimators in Simple Random Sampling For estimating the population mean of , the exponential ratio estimator is given by where and (the notation is used to represents for simple random sampling). The
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