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A Multivariate Exponential Estimator for Vector of Population Means in Two-Phase Sampling

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Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

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.

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Author information

Authors and Affiliations

  1. Department of Statistics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

    Aamir Sanaullah

  2. Department of Statistics, National College of Business Administration and Economics, Lahore, Pakistan

    Ayesha Ayaz & Muhammad Hanif

Authors
  1. Aamir Sanaullah
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  2. Ayesha Ayaz
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  3. Muhammad Hanif
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Corresponding author

Correspondence to Aamir Sanaullah.

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Appendices

Appendix 1

1.1 Simulation Study for Symmetric Data

See Tables 1, 2, 3 and 4.

Table 4 The MSE values of the univariate estimator for the symmetric data
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1.2 Simulation Study for Asymmetric Data

Tables 5, 6, 7 and 8.

Table 5 Variance–covariance matrices of multivariate estimators for asymmetric data under Model I
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Table 6 Variance–covariance matrices of multivariate estimators for asymmetric data under Model II
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Table 7 Determinants of the variance–covariance matrices for asymmetric data
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Table 8 The MSE values of the univariate estimator for asymmetric data
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Appendix 2

2.1 Empirical Results Based on Real-Life Examples

See Tables 9, 10, 11, 12, 13, 14 and 15.

Table 9 Source of real data sets
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Table 10 Description of variables
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Table 11 Variance–covariance matrices of the two real data sets
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Table 12 Variance-covariance matrices of multivariate estimators using real data set-I
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Table 13 Variance–covariance matrices of multivariate estimators using real data set-II
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Table 14 Determinants of the variance–covariance matrices for real data set-I and data set-II
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Table 15 The MSE values of the univariate estimator using real data set-I and data set-II
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Sanaullah, A., Ayaz, A. & Hanif, M. A Multivariate Exponential Estimator for Vector of Population Means in Two-Phase Sampling. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 647–659 (2020). https://doi.org/10.1007/s40010-019-00633-4

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  • DOI: https://doi.org/10.1007/s40010-019-00633-4

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