Chain ratio and product estimators for population mode using two-phase sampling scheme


  • Sanjay Kumar Department of Statistics, Central University of Rajasthan, Bandarsindri, Kishangarh, Ajmer, Rajasthan, India
  • Akhilesh Yadav Department of Statistics, Central University of Rajasthan, Bandarsindri, Kishangarh, Ajmer, Rajasthan, India



Mode estimation, Auxiliary information, Cost aspects, Mean square error, Confidence interval, Simulation study


Background: For skewed datasets, the mode is utilized as a more appropriate measure of location. We formed chain ratio and product estimators for the population mode using two types of auxiliary information under the two-phase sampling scheme.

Methods: Expressions for biases and mean square errors for the formed estimators up to the first order of approximation are obtained. The confidence intervals of the estimators are obtained and the aspects related to fixed cost and fixed variance are also studied. A simulation study is performed to support the theoretical outcomes. A real dataset is also provided which was collected from the department of agriculture, United States.

Results: The simulation results for the fixed first-phase sample size  and the second-phase sample size  show that the mean square error is 0.035, the bias is 0.012, the confidence interval is 2.67-3.15, and the cost of the survey under the fixed variance is ₹ 194.67 of the proposed estimator , which is lower than 0.039, 0.055, 2.70-3.20, and ₹ 247.37 of the ratio estimator  and 0.078, 0.041, 2.57-3.31, and ₹ 268.73 of the naive estimator .

Conclusions: The simulation results show that the proposed estimator  performs better than the ratio estimator  and the naive estimator .


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Chand L. Some ratio type estimators based on two or more auxiliary variables. Available at: https://www. Accessed on February 2023.

Doodson AT. Relation of the mode, median and mean in frequency curves. Biometrika. 1917;11(4):425-9.

Gross S. Proceedings of the Section on Survey Research Methods Median estimation in sample surveys. Alexandria: American Statistical Association publication. 2012.

Guha, S, Chandra, H. Improved chain-ratio type estimator for population total in double sampling. Mathemat Popul Studies. 2020;27(4):216-31.

Kuk AY, Mak TK. Median estimation in the presence of auxiliary information. J Royal Stat Soc. 1989;51(2): 261-9.

Kumar S, Tiwari, N. Generalized naive ratio and product based estimators for estimating population mode in simple random sampling. Int J Stat Economics. 2019;20(1):57-79.

Lamichhane R, Singh S. Estimation of mode using two-phase sampling. Commu Stat Simul Comput. 2016:45(7):2586-97.

Neyman J. Contribution to the theory of sampling human populations. J Am Stat Assoc. 1938:33(201): 101-16.

Dataset from the Department of Agriculture, United States. Available at: Accessed on 01 July 2022.






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