Estimation of rare and clustered population mean using stratified adaptive cluster sampling

Abstract

For many clustered populations, the prior information on an initial stratification exists but the exact pattern of the population concentration may not be predicted. Under this situation, the stratified adaptive cluster sampling (SACS) may provide more efficient estimates than the other conventional sampling designs for the estimation of rare and clustered population parameters. For practical interest, we propose a generalized ratio estimator with the single auxiliary variable under the SACS design. The expressions of approximate bias and mean squared error (MSE) for the proposed estimator are derived. Numerical studies are carried out to compare the performances of the proposed generalized estimator over the usual mean and combined ratio estimators under the conventional stratified random sampling (StRS) using a real population of redwood trees in California and generating an artificial population by the Poisson cluster process. Simulation results show that the proposed class of estimators may provide more efficient results than the other estimators considered in this article for the estimation of highly clumped population.

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Acknowledgements

The first author is thankful to the Higher Education Commission (HEC) of Pakistan (1-8/HEC/HRD/2017/8248) for awarding him an International Research fellowship. The first author is grateful to Professor Sat N. Gupta, Head of Mathematics and Statistics Department, The University of North Carolina, Greensboro, for his valuable suggestions. The Authors would like to acknowledge the comments and recommendations made by the Editors and Reviewers on earlier versions of the manuscript, which resulted in substantial improvement of the original version and presentation of the article.

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Correspondence to Muhammad Nouman Qureshi.

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Handling Editor: Pierre Dutilleul.

Appendices

Appendix 1

Different special forms of generalized ratio estimator may be obtained by using different values of scalar constants aj,st and dj,st. The expressions of approximate biases, MSE, and their associated ψwjs are given in Table 11.

Table 11 Special cases of generalized ratio estimator in SACS

Appendix 2

Data Statistics: Population of redwood trees

Stratum(h)\( N_{h} \)\( \rho_{yxh} \)\( \bar{Y}_{h} \)\( \bar{X}_{h} \)\( S_{yh}^{2} \)\( S_{xh}^{2} \)\( C_{yh}^{{}} \)\( C_{xh}^{{}} \)\( \beta_{1} \left( {x_{h} } \right) \)\( \beta_{2} \left( {x_{h} } \right) \)\( M\left( {x_{h} } \right) \)
18000.8670.1430.2480.1750.7482.9343.4764.36624.7183.500
28000.9000.1010.1810.1180.4613.4023.7484.14520.6952.500
 \( N \)\( \rho_{yx} \)\( \bar{Y} \)\( \bar{X} \)\( S_{y}^{2} \)\( S_{x}^{2} \)\( C_{y}^{{}} \)\( C_{x}^{{}} \)\( \beta_{1} \left( x \right) \)\( \beta_{2} \left( x \right) \)\( M\left( x \right) \)
 16000.8760.1210.2130.1470.6063.1473.6414.39925.1177.000

Data statistics: transformed population of redwood trees

Stratum(h)\( N_{h} \)\( \rho_{{w_{y} w_{x} h}} \)\( \bar{W}_{yh} \)\( \bar{W}_{xh} \)\( S_{wyh}^{2} \)\( S_{wxh}^{2} \)\( C_{wyh}^{{}} \)\( C_{wxh}^{{}} \)\( \beta_{1} \left( {w_{xh} } \right) \)\( \beta_{2} \left( {w_{xh} } \right) \)\( M\left( {w_{xh} } \right) \)
18000.8800.1430.2480.1600.6342.8013.2193.49115.5793.00
28000.8790.1020.1810.1110.4203.2313.5763.60914.9572.00
 \( N \)\( \rho_{{w_{y} w_{x} }} \)\( \bar{Y} \)\( \bar{X} \)\( S_{wy}^{2} \)\( S_{wx}^{2} \)\( C_{wy}^{{}} \)\( C_{wx}^{{}} \)\( \beta_{1} \left( {w_{x} } \right) \)\( \beta_{2} \left( {w_{x} } \right) \)\( M\left( {w_{x} } \right) \)
 16000.8810.1220.2140.1360.5283.0013.3903.60416.1596.000

Data statistics: artificial population

Stratum (h)\( N_{h} \)\( \rho_{yxh} \)\( \bar{Y}_{h} \)\( \bar{X}_{h} \)\( S_{yh}^{2} \)\( S_{xh}^{2} \)\( C_{yh}^{{}} \)\( C_{xh}^{{}} \)\( \beta_{1} \left( {x_{h} } \right) \)\( \beta_{2} \left( {x_{h} } \right) \)\( M\left( {x_{h} } \right) \)
12000.9161.1401.45519.73628.9823.8633.7005.34834.97020.50
22000.9412.3802.64573.52388.6723.6023.5605.15032.72138.00
 \( N \)\( \rho_{yx} \)\( \bar{Y} \)\( \bar{X} \)\( S_{y}^{2} \)\( S_{x}^{2} \)\( C_{y}^{{}} \)\( C_{x}^{{}} \)\( \beta_{1} \left( x \right) \)\( \beta_{2} \left( x \right) \)\( M\left( x \right) \)
 4000.9351.7652.05046.89259.0356.8475.0685.75642.13676.000

Data statistics: transformed artificial population

Stratum(h)\( N_{h} \)\( \rho_{{w_{y} w_{x} h}} \)\( \bar{W}_{yh} \)\( \bar{W}_{xh} \)\( S_{wyh}^{2} \)\( S_{wxh}^{2} \)\( C_{wyh}^{{}} \)\( C_{wxh}^{{}} \)\( \beta_{1} \left( {w_{xh} } \right) \)\( \beta_{2} \left( {w_{xh} } \right) \)\( M\left( {w_{xh} } \right) \)
12000.9921.1411.4538.52314.3922.5692.5402.1445.6145.775
22000.7182.7722.64567.50536.4962.9632.3222.2016.3579.971
 \( N \)\( \rho_{{w_{y} w_{x} }} \)\( \bar{Y} \)\( \bar{X} \)\( S_{wy}^{2} \)\( S_{wx}^{2} \)\( C_{wy}^{{}} \)\( C_{wx}^{{}} \)\( \beta_{1} \left( {w_{x} } \right) \)\( \beta_{2} \left( {w_{x} } \right) \)\( M\left( {w_{x} } \right) \)
 4000.9941.7522.04918.84225.6872.4762.4722.4457.94519.400

Appendix 3: nomenclature

\( S_{{w_{x} h}}^{2} = \left( {N_{h}^{{}} - 1} \right)^{ - 1} \sum\nolimits_{i = 1}^{{N_{h} }} {\left( {w_{xhi} - \bar{W}_{xh} } \right)^{2} } \) :

Variance in hth stratum

\( \beta_{1} \left( {w_{xh} } \right) = \frac{{N_{h} \sum\nolimits_{i = 1}^{{N_{h} }} {\left( {w_{xhi} - \bar{W}_{xh} } \right)}^{3} }}{{\left( {N_{h} - 1} \right)\left( {N_{h} - 2} \right)S_{{w_{x} h}}^{3} }} \) :

Coefficient of skewness in hth stratum

\( \beta_{2} \left( {w_{xh} } \right) = \frac{{N_{h} \left( {N_{h} + 1} \right)\sum\nolimits_{i = 1}^{{N_{h} }} {\left( {w_{xhi} - \bar{W}_{xh} } \right)}^{3} }}{{\left( {N_{h} - 1} \right)\left( {N_{h} - 2} \right)\left( {N_{h} - 3} \right)S_{{w_{x} h}}^{4} }} - \frac{{2\left( {N_{h} - 1} \right)^{2} }}{{\left( {N_{h} - 2} \right)\left( {N_{h} - 3} \right)}} \) :

Coefficient of kurtosis in hth stratum

\( M_{xh} = M\left( {x_{h} } \right) \) :

Maximum value in hth stratum

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Qureshi, M.N., Kadilar, C. & Hanif, M. Estimation of rare and clustered population mean using stratified adaptive cluster sampling. Environ Ecol Stat 27, 151–170 (2020). https://doi.org/10.1007/s10651-019-00438-z

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Keywords

  • Auxiliary information
  • Clustered populations
  • Hansen–Hurwitz estimation
  • Percentage relative efficiency
  • Stratification