Table 1 Empirical exhibition of theoretical conditions given in Sect. “Some relevant existing estimators”
From: Improved exponential type ratio estimator in double sampling for stratification
Sr. No. | Comparison | Conditions | Population1 | Population 2 |
|---|---|---|---|---|
1 | \(\begin{gathered} MSE\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overline{Y}}_{{C{\text{Re}} }}^{ds} } \right) \hfill \\ < MSE\left( {\overline{y}_{ds} } \right) \hfill \\ \end{gathered}\) | \(\begin{gathered} \left( {1 - f} \right)\left( {R_{2}^{2} S_{z}^{2} - 4R_{2} S_{yz} } \right) < \hfill \\ \sum\limits_{{}}^{{}} {W_{h} \left( {\frac{1}{{\nu_{h} }} - 1} \right)\left( {8R_{1} S_{yxh} - 4R_{1}^{2} S_{xh}^{2} } \right)} \hfill \\ \end{gathered}\) | − 35,207.30 < − 24,822.30 | − 45,841.40 < 577,546.89 |
2 | \(\begin{gathered} MSE\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overline{Y}}_{{C{\text{Re}} }}^{ds} } \right) \hfill \\ < MSE\left( {\overline{y}_{R}^{ds} } \right) \hfill \\ \end{gathered}\) | \(S_{z}^{2} < \frac{{4S_{yz} }}{{R_{2} }}\) | 31,692.05 < 1,15,137.80 | 34.84 < 148.77 |
3 | \(\begin{gathered} MSE\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overline{Y}}_{{C{\text{Re}} }}^{ds} } \right) \hfill \\ < MSE\left( {\hat{\overline{Y}}_{{\text{Re}}}^{ds} } \right) \hfill \\ \end{gathered}\) | \(\begin{gathered} \left( {1 - f} \right)\left( {R_{2}^{2} S_{z}^{2} - 4R_{2} S_{yz} } \right) < \hfill \\ \sum\limits_{{}}^{{}} {W_{h} \left( {\frac{1}{{\nu_{h} }} - 1} \right)\left( {8R_{1} S_{yxh} - 3R_{1}^{2} S_{xh}^{2} - R_{1} S_{xh}^{2} \beta_{yxh} } \right)} \hfill \\ \end{gathered}\) | − 35,207.3 0 < − 11,535.13 | − 45,841.40 < 716,948.13 |
4 | \(\begin{gathered} MSE(\hat{\overline{Y}}_{{C{\text{Re}} }}^{ds} ) \hfill \\ < \;MSE(\hat{\overline{Y}}_{{\text{Re}}}^{*} ) \hfill \\ \end{gathered}\) | \(\begin{gathered} \left( {1 - f} \right)\;\left( {R_{2}^{2} S_{z}^{2} - 4R_{2} S_{yz} } \right)\; \le \; \hfill \\ 4\;\left[ {\sum\limits_{h = 1}^{L} {W_{h} \left( {\frac{1}{{\nu_{h} }} - 1} \right)\;\left( {R_{1}^{2} S_{xh}^{2} \left( {\frac{{g^{2} }}{4} - 1} \right) + R_{1} S_{yxh} \left( {2 - g} \right)} \right)} } \right] \hfill \\ \end{gathered}\) | − 35,207.32 < − 34,653.76 | − 45,841.44 < 8842.96 |
5 | \(\begin{gathered} MSE(\hat{\overline{Y}}_{{C{\text{Re}} }}^{ds} ) \hfill \\ < \;MSE(\hat{\overline{Y}}_{Rd}^{*} ) \hfill \\ \end{gathered}\) | \(\begin{gathered} \left( {1 - f} \right)\;\left( {R_{2}^{2} S_{z}^{2} - 4R_{2} S_{yz} } \right)\; \le \; \hfill \\ 4\;\left[ {\sum\limits_{h = 1}^{L} {W_{h} \left( {\frac{1}{{\nu_{h} }} - 1} \right)\;\left( {R_{1}^{2} S_{xh}^{2} \left( {g^{2} - 1} \right) + 2R_{1} S_{yxh} \left( {1 - g} \right)} \right)} } \right] \hfill \\ \end{gathered}\) | − 35,207.32 < − 26,379.66 | * |
6 | \(\begin{gathered} MSE(\hat{\overline{Y}}_{{C{\text{Re}} }}^{ds} ) \hfill \\ < \;MSE(\hat{\overline{Y}}_{Rpd}^{*} ) \hfill \\ \end{gathered}\) | \(\begin{gathered} \left( {1 - f} \right)\;\left( {R_{2}^{2} S_{z}^{2} - 4R_{2} S_{yz} } \right)\; \le \hfill \\ \;4\;\left[ {\sum\limits_{h = 1}^{L} {W_{h} \left( {\frac{1}{{\nu_{h} }} - 1} \right)\;\left( \begin{gathered} R_{1} S_{xh}^{2} \left( {g^{2} - 1} \right) \hfill \\ + 2R_{1} S_{yxh} \left( {1 - g} \right) + g^{2} R_{2}^{2} S_{zh}^{2} \hfill \\ + 2gR_{2} S_{yzh} - 2g^{2} R_{1} R_{2} S_{xzh} \hfill \\ \end{gathered} \right)} } \right] \hfill \\ \end{gathered}\) | − 35,207.32 < 54,665.01 | * |
