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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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 ψwj’s are given in Table 11.
Appendix 2
Data Statistics: Population of redwood trees
Stratum(h) | \( N_{h} \) | \( \rho_{yxh} \) | \( \bar{Y}_{h} \) | \( \bar{X}_{h} \) | \( S_{yh}^{2} \) | \( S_{xh}^{2} \) | \( C_{yh}^{{}} \) | \( C_{xh}^{{}} \) | \( \beta_{1} \left( {x_{h} } \right) \) | \( \beta_{2} \left( {x_{h} } \right) \) | \( M\left( {x_{h} } \right) \) |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 800 | 0.867 | 0.143 | 0.248 | 0.175 | 0.748 | 2.934 | 3.476 | 4.366 | 24.718 | 3.500 |
2 | 800 | 0.900 | 0.101 | 0.181 | 0.118 | 0.461 | 3.402 | 3.748 | 4.145 | 20.695 | 2.500 |
\( N \) | \( \rho_{yx} \) | \( \bar{Y} \) | \( \bar{X} \) | \( S_{y}^{2} \) | \( S_{x}^{2} \) | \( C_{y}^{{}} \) | \( C_{x}^{{}} \) | \( \beta_{1} \left( x \right) \) | \( \beta_{2} \left( x \right) \) | \( M\left( x \right) \) | |
---|---|---|---|---|---|---|---|---|---|---|---|
1600 | 0.876 | 0.121 | 0.213 | 0.147 | 0.606 | 3.147 | 3.641 | 4.399 | 25.117 | 7.000 |
Data statistics: transformed population of redwood trees
Stratum(h) | \( N_{h} \) | \( \rho_{{w_{y} w_{x} h}} \) | \( \bar{W}_{yh} \) | \( \bar{W}_{xh} \) | \( S_{wyh}^{2} \) | \( S_{wxh}^{2} \) | \( C_{wyh}^{{}} \) | \( C_{wxh}^{{}} \) | \( \beta_{1} \left( {w_{xh} } \right) \) | \( \beta_{2} \left( {w_{xh} } \right) \) | \( M\left( {w_{xh} } \right) \) |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 800 | 0.880 | 0.143 | 0.248 | 0.160 | 0.634 | 2.801 | 3.219 | 3.491 | 15.579 | 3.00 |
2 | 800 | 0.879 | 0.102 | 0.181 | 0.111 | 0.420 | 3.231 | 3.576 | 3.609 | 14.957 | 2.00 |
\( N \) | \( \rho_{{w_{y} w_{x} }} \) | \( \bar{Y} \) | \( \bar{X} \) | \( S_{wy}^{2} \) | \( S_{wx}^{2} \) | \( C_{wy}^{{}} \) | \( C_{wx}^{{}} \) | \( \beta_{1} \left( {w_{x} } \right) \) | \( \beta_{2} \left( {w_{x} } \right) \) | \( M\left( {w_{x} } \right) \) | |
---|---|---|---|---|---|---|---|---|---|---|---|
1600 | 0.881 | 0.122 | 0.214 | 0.136 | 0.528 | 3.001 | 3.390 | 3.604 | 16.159 | 6.000 |
Data statistics: artificial population
Stratum (h) | \( N_{h} \) | \( \rho_{yxh} \) | \( \bar{Y}_{h} \) | \( \bar{X}_{h} \) | \( S_{yh}^{2} \) | \( S_{xh}^{2} \) | \( C_{yh}^{{}} \) | \( C_{xh}^{{}} \) | \( \beta_{1} \left( {x_{h} } \right) \) | \( \beta_{2} \left( {x_{h} } \right) \) | \( M\left( {x_{h} } \right) \) |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 200 | 0.916 | 1.140 | 1.455 | 19.736 | 28.982 | 3.863 | 3.700 | 5.348 | 34.970 | 20.50 |
2 | 200 | 0.941 | 2.380 | 2.645 | 73.523 | 88.672 | 3.602 | 3.560 | 5.150 | 32.721 | 38.00 |
\( N \) | \( \rho_{yx} \) | \( \bar{Y} \) | \( \bar{X} \) | \( S_{y}^{2} \) | \( S_{x}^{2} \) | \( C_{y}^{{}} \) | \( C_{x}^{{}} \) | \( \beta_{1} \left( x \right) \) | \( \beta_{2} \left( x \right) \) | \( M\left( x \right) \) | |
---|---|---|---|---|---|---|---|---|---|---|---|
400 | 0.935 | 1.765 | 2.050 | 46.892 | 59.035 | 6.847 | 5.068 | 5.756 | 42.136 | 76.000 |
Data statistics: transformed artificial population
Stratum(h) | \( N_{h} \) | \( \rho_{{w_{y} w_{x} h}} \) | \( \bar{W}_{yh} \) | \( \bar{W}_{xh} \) | \( S_{wyh}^{2} \) | \( S_{wxh}^{2} \) | \( C_{wyh}^{{}} \) | \( C_{wxh}^{{}} \) | \( \beta_{1} \left( {w_{xh} } \right) \) | \( \beta_{2} \left( {w_{xh} } \right) \) | \( M\left( {w_{xh} } \right) \) |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 200 | 0.992 | 1.141 | 1.453 | 8.523 | 14.392 | 2.569 | 2.540 | 2.144 | 5.614 | 5.775 |
2 | 200 | 0.718 | 2.772 | 2.645 | 67.505 | 36.496 | 2.963 | 2.322 | 2.201 | 6.357 | 9.971 |
\( N \) | \( \rho_{{w_{y} w_{x} }} \) | \( \bar{Y} \) | \( \bar{X} \) | \( S_{wy}^{2} \) | \( S_{wx}^{2} \) | \( C_{wy}^{{}} \) | \( C_{wx}^{{}} \) | \( \beta_{1} \left( {w_{x} } \right) \) | \( \beta_{2} \left( {w_{x} } \right) \) | \( M\left( {w_{x} } \right) \) | |
---|---|---|---|---|---|---|---|---|---|---|---|
400 | 0.994 | 1.752 | 2.049 | 18.842 | 25.687 | 2.476 | 2.472 | 2.445 | 7.945 | 19.400 |
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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DOI: https://doi.org/10.1007/s10651-019-00438-z