Generalized Exponential-Type Estimators for Population Mean Taking Two Auxiliary Variables for Unknown Means in Stratified Sampling with Sub-sampling the Non-Respondents

  • Aamir Sanaullah
  • Muhammad Noor ul Amin
  • Muhammad Hanif
  • Nursel Koyuncu
Original Paper


In this paper, we have considered stratified two-phase sampling design with sub-sampling the non-respondents in the presence of non-response for estimating population mean considering the information of two auxiliary variables. The proposed estimators are the exponential function of two auxiliary variables when means of the two auxiliary variables are not known in prior. Further the proposed estimators are provided with their generalized form. The bias and mean square error expressions of the proposed estimators have been derived in two different cases of non-response. The conditions for which proposed estimators are more efficient as compared to some other estimators, have also been discussed in each case of non-response. It is shown that the proposed estimators are more efficient as compared to Hansen and Hurwitz (J Am Stat Assoc 41:517–529, 1946) unbiased estimator and Tabasum and Khan (Assam Stat Rev 20(1):73–83, 2006) two-phase ratio and product estimators modified to the stratified sampling. An empirical study has also been carried out to demonstrate the performances of the estimators.


Ratio estimator Product estimator Exponential estimator Non-response Two-phase stratified sampling 


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Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Aamir Sanaullah
    • 1
  • Muhammad Noor ul Amin
    • 1
  • Muhammad Hanif
    • 2
  • Nursel Koyuncu
    • 3
  1. 1.Department of StatisticsCOMSATS Institute of Information TechnologyLahorePakistan
  2. 2.Department of StatisticsNational College of Business Administration and EconomicsLahorePakistan
  3. 3.Department of StatisticsHacettepe UniversityBeytepeTurkey

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