A Multivariate Exponential Estimator for Vector of Population Means in Two-Phase Sampling
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RESEARCH ARTICLE
A Multivariate Exponential Estimator for Vector of Population Means in Two-Phase Sampling Aamir Sanaullah1
•
Ayesha Ayaz2 • Muhammad Hanif2
Received: 12 October 2017 / Revised: 20 February 2019 / Accepted: 14 June 2019 Ó The National Academy of Sciences, India 2019
Abstract This study is concerned with construction of the generalized multivariate exponential estimator for estimating a population mean vector in the two-phase sampling using multi-auxiliary variables when population information for some auxiliary variables is not available. The optimum conditions which provide the matrix of minimum variance–covariance are obtained for the suggested estimator. Further, a vector of the biases is also provided. Some deduced univariate and multivariate estimators are also shown as special cases of the suggested multivariate exponential estimator. The simulation study is conducted using the artificial symmetric and asymmetric distributions to show that the suggested multivariate estimator performs more efficiently than the existing multivariate estimator. Two real-life examples are used to show the usefulness of the proposed multivariate estimators. Keywords Multivariate exponential estimator Multi-auxiliary variables Two-phase sampling Mean square error Bias Simulation study
& Aamir Sanaullah [email protected] Ayesha Ayaz [email protected] Muhammad Hanif [email protected] 1
Department of Statistics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan
2
Department of Statistics, National College of Business Administration and Economics, Lahore, Pakistan
1 Introduction This article considers the development of a precise multivariate exponential estimator by utilizing the information of some of the auxiliary variables with derivation of the bias and the mean square error when relation between the study and auxiliary variable is not linear. Cochran [1] proposed a ratio method of estimation for estimating the population mean. Chand [2] developed a chain ratio estimator using the information on two auxiliary variables when population mean of one of the auxiliary variables is not available. Srivastava et al. [3], Gupta and Shabbir [4] and Singh et al. [5] extended the work on same lines. Bahl and Tuteja [6] considered the nonlinear relationship between the study and auxiliary variables to build up an exponential-type ratio and product estimators. Yadav et al. [7] worked on mean estimation and projected a chain ratiotype exponential estimator using the information on two auxiliary variables in two-phase sampling and derived the mean square errors (MSEs). Sanaullah et al. [8, 9] provided some generalized exponential chain ratio-type estimators for population mean using the two auxiliary variables when population mean of an auxiliary variable is available. After that, many estimators have been provided by different researchers; for instance, see Singh and Vishwakarma [10], Solanki et al. [11], Khan et al. [12], Singh and Majhi [13], Singh and Solanki [14] and Sanau
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