Estimation of Population Mean Using a Difference-Type Exponential Imputation Method

Abstract

This paper introduces a difference-type exponential imputation method and the corresponding point estimators. Bias and mean-squared error of the proposed estimators have been determined. It is identified that in addition to Prasad (Mod Assis Statist Appl 12(2):95–106, 2017; Pak J Statist 35(2):97–107, 2019) methods of imputation and the resulting point estimators, a large number of imputation methods and the point estimators are members of the suggested classes of estimators. We note that the expressions of the mean-squared error obtained by Prasad (2017, 2019) of his estimators are not correct, and therefore, we have derived the correct expressions of mean-squared errors of the Prasad’s estimators. It has been found that the proposed classes of estimators are more efficient than Prasad’s (2017, Statist Trans 19(1):159–166, 2018b, 2019). An empirical analysis to see the efficiency of the proposed classes of estimators over other estimators has been carried out.

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Acknowledgements

Authors are thankful to the learned referees, an associate editor and the editor-in-chief for their valuable suggestions regarding improvement of the paper.

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Correspondence to Anurag Gupta.

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Appendix

Appendix

Derivation of Biases and Mean-Squared Errors of the Proposed Classes of Estimators D 1, D 2 and D 3

To derive the biases and MSEs of the suggested classes of estimators d1, d2 and d3 up to first order of approximation, we write\( \bar{y}_{r} = \bar{Y}\left( {1 + e_{or} } \right) \), \( \bar{x}_{r} = \bar{X}\left( {1 + e_{1r} } \right) \), and \( \bar{x}_{n} = \bar{X}\left( {1 + e_{1n} } \right) \). Such that

$$ E\left( {e_{or} } \right) = E\left( {e_{1r} } \right) = E\left( {e_{1n} } \right) = 0 $$

and \( E\left( {e_{0r}^{2} } \right) = \lambda_{1} C_{y}^{2} \), \( E\left( {e_{1r}^{2} } \right) = \lambda_{1} C_{x}^{2} \), \( E\left( {e_{1n}^{2} } \right) = \lambda_{2} C_{x}^{2} \),\( E\left( {e_{or} e_{1r} } \right) = \lambda_{1} \rho C_{y} C_{x}^{{}} \), \( E\left( {e_{or} e_{1n} } \right) = \lambda_{2} \rho C_{y} C_{x}^{{}} \), and \( E\left( {e_{r1} e_{1n} } \right) = \lambda_{2} C_{x}^{2} \), where \( \lambda_{1} = \left( {\frac{1}{r} - \frac{1}{N}} \right) \) and \( \lambda_{2} = \left( {\frac{1}{n} - \frac{1}{N}} \right) \) (see [15]).

Expressing d1, d2 and d3 given by Eqs. (28), (31) and (77), respectively, in terms e’s, we have

$$ d_{1} = \overline{Y} \left[ {k_{1} \left( {1 + e_{or} } \right) - \frac{{k_{2} }}{{R^{ * } }}e_{1r} } \right]\exp \left\{ { - \frac{\delta \theta }{2}e_{1r} \left( {1 + \frac{\theta }{2}e_{1r} } \right)^{ - 1} } \right\} , $$
(96)
$$ d_{2} = \overline{Y} \left[ {k_{1}^{{}} \left( {1 + e_{or} } \right) - \frac{{k_{2}^{*} }}{{R^{*} }}e_{1n} } \right]\exp \left\{ { - \frac{\delta \theta }{2}e_{1n} \left( {1 + \frac{\theta }{2}e_{1n} } \right)^{ - 1} } \right\} , $$
(97)
$$ d_{3} = \bar{Y}\left[ {\alpha_{1} \left( {1 + e_{or}^{{}} } \right) + \alpha_{2} \left( {\frac{1}{{R^{ * } }}} \right)\left( {e_{1n} - e_{1r} } \right)} \right]\exp \left[ { - \frac{\delta \theta }{2}\left( {e_{1n} - e_{1r} } \right)\left\{ {1 + \frac{{\theta_{j} \left( {e_{1n} + e_{1r} } \right)}}{2}} \right\}^{ - 1} } \right] , $$
(98)

where \( R^{ * } = \frac{{\overline{Y} }}{{\overline{X} }} \) and \( \theta = \frac{{p\overline{X} }}{{p\overline{X} + q}} \).

For various choices of p and q, the θ can be changed in θj and Eqs. (96), (97) and (98) can be expressed, respectively, as

$$ d_{1j} = \overline{Y} \left[ {k_{1} \left( {1 + e_{or} } \right) - \frac{{k_{2} }}{{R^{ * } }}e_{1r} } \right]\exp \left\{ { - \frac{{\delta \theta_{j} }}{2}e_{1r} \left( {1 + \frac{{\theta_{j} }}{2}e_{1r} } \right)^{ - 1} } \right\} , $$
(99)
$$ d_{2j} = \overline{Y} \left[ {k_{1}^{*} \left( {1 + e_{or} } \right) - \frac{{k_{2}^{*} }}{{R^{*} }}e_{1n} } \right]\exp \left\{ { - \frac{{\delta \theta_{j} }}{2}e_{1n} \left( {1 + \frac{{\theta_{j} }}{2}e_{1n} } \right)^{ - 1} } \right\} , $$
(100)
$$ d_{3j} = \bar{Y}\left[ {\alpha_{1} \left( {1 + e_{or}^{{}} } \right) + \left( {\frac{{\alpha_{2} }}{{R^{ * } }}} \right)\left( {e_{1n} - e_{1r} } \right)} \right]\exp \left[ { - \frac{{\delta \theta_{j} }}{2}\left( {e_{1n} - e_{1r} } \right)\left\{ {1 + \frac{{\theta_{j} }}{2}\left( {e_{1n} + e_{1r} } \right)} \right\}^{ - 1} } \right] , $$
(101)

where \( \theta_{j} = \frac{{p\bar{X}}}{{\left( {p\bar{X} + q} \right)}} \) (j = 1 to 23) for different choices of p and q; and \( \theta_{1} ,\theta_{2} , \ldots ,\theta_{23} \) are same as defined earlier.

Equations (99), (100) and (101) can be expressed neglecting the terms of e’s having power more than two, we have

$$ \left( {d_{1j} - \overline{Y} } \right) \cong \overline{Y} \left[ {k_{1} \left\{ {1 + e_{or} - \frac{{\delta \theta_{j} }}{2}e_{1r} - \frac{{\delta \theta_{j} }}{2}e_{0r} e_{1r} + \frac{{\delta \left( {\delta + 2} \right)\theta_{j}^{2} }}{8}e_{1r}^{2} } \right\} + \left( {\frac{{k_{2} }}{{R^{ * } }}} \right)\left( {\frac{{\delta \theta_{j} }}{2}e_{{_{1r} }}^{2} - e_{1r} } \right) - 1} \right] , $$
(102)
$$ \left( {d_{2j} - \overline{Y} } \right) \cong \overline{Y} \left[ {k_{1}^{*} \left\{ {1 + e_{or} - \frac{{\delta \theta_{j} }}{2}e_{1n} - \frac{{\delta \theta_{j} }}{2}e_{0r} e_{1n} + \frac{{\delta \left( {\delta + 2} \right)\theta_{j}^{2} }}{8}e_{1n}^{2} } \right\} + \left( {\frac{{k_{2}^{*} }}{{R^{ * } }}} \right)\left( {\frac{{\delta \theta_{j} }}{2}e_{1n}^{2} - e_{1n} } \right) - 1} \right] , $$
(103)
$$ \left( {d_{3j} - \bar{Y}} \right) = \bar{Y}\left[ {\alpha_{1} \left\{ {1 + e_{or}^{{}} + \frac{{\delta \theta_{j} }}{2}\left( {e_{1n} - e_{1r} } \right) + \frac{{\delta \theta_{j} }}{2}\left( {e_{or} e_{1n} - e_{or} e_{1r} } \right) - \frac{{\delta \theta_{j}^{2} \left( {e_{1n}^{2} - e_{1r}^{2} } \right)}}{4} + \frac{{\delta^{2} \theta_{j}^{2} }}{8}\left( {e_{1n} - e_{1r} } \right)^{2} } \right\}} \right. $$
$$ \left. { + \left( {\frac{{\alpha_{2} }}{{R^{ * } }}} \right)\left\{ {\left( {e_{1n} - e_{1r} } \right) + \frac{{\delta \theta_{j} }}{2}\left( {e_{1n} - e_{1r} } \right)^{2} } \right\} - 1} \right] $$
(104)

Taking expectation of both sides of Eq (102), (103) and (104), respectively, we get the biases of the proposed classes of estimators d1j, d2j and d3j, respectively, as

$$ B\left( {d_{1j} } \right) = \overline{Y} \left[ {k_{1} \left\{ {1 + \frac{{\delta \theta_{j} }}{8}\lambda_{1} \left( {\left( {\delta + 2} \right)\theta_{j} C_{x}^{2} - 4\rho C_{y} C_{x} } \right)} \right\} + k_{2} \left( {\frac{{\delta \theta_{j} }}{{2R^{ * } }}} \right)\lambda_{1} C_{x}^{2} - 1} \right] , $$
(105)
$$ B\left( {d_{2j} } \right) = \overline{Y} \left[ {k_{1}^{*} \left\{ {1 + \frac{{\delta \theta_{j} }}{8}\lambda_{2} \left( {\left( {\delta + 2} \right)\theta_{j} C_{x}^{2} - 4\rho C_{y} C_{x} } \right)} \right\} + k_{2}^{*} \left( {\frac{{\delta \theta_{j} }}{{2R^{ * } }}} \right)\lambda_{2} C_{x}^{2} - 1} \right] $$
(106)
$$ B\left( {d_{3j} } \right) = \bar{Y}\left[ {\alpha_{1} \left\{ {1 + \frac{{\delta \theta_{j}^{{}} }}{2}\lambda_{{}}^{ * } \left( {\frac{{\left( {\delta + 2} \right)}}{4}\theta_{j}^{{}} C_{x}^{2} - \rho C_{y} C_{x} } \right)} \right\} + \alpha_{2} \left( {\frac{{\delta \lambda^{ * } }}{{2R^{ * } }}} \right)\theta_{j}^{{}} C_{x}^{2} - 1} \right] $$
(107)

Now, after squaring both sides of Eqs. (102), (103) and (104) and neglecting the terms of e’s having power greater than two, we have

$$ \left( {d_{1j} - \bar{Y}} \right)^{2} = \bar{Y}^{2} \left[ \begin{aligned} 1 + k_{1}^{2} \left\{ {1 + 2e_{or} - \delta \theta_{j} e_{1r} - 2\delta \theta_{j} e_{0r} e_{1r} + e_{or}^{2} + \frac{{\delta \left( {\delta + 1} \right)}}{2}\theta_{j}^{2} e_{1r}^{2} } \right\} + k_{2}^{2} \left( {\frac{1}{{R^{ * 2} }}} \right)e_{1r}^{2} \hfill \\ + 2k_{1} k_{2} \left( {\frac{1}{{R^{ * } }}} \right)\left\{ {\delta \theta_{j} e_{1r}^{2} - e_{0r} e_{1r} - e_{1r} } \right\} - 2k_{1} \left\{ {1 + e_{0r} - \frac{1}{2}\delta \theta_{j} e_{1r} - \frac{1}{2}\delta \theta_{j} e_{or} e_{1r} + \frac{{\delta \left( {\delta + 2} \right)}}{8}\theta_{j}^{2} e_{1r}^{2} } \right\} \hfill \\ - 2k_{2} \left( {\frac{1}{{R^{ * } }}} \right)\left( {\frac{1}{2}\delta \theta_{j} e_{1r}^{2} - e_{1r} } \right) \hfill \\ \end{aligned} \right] $$
(108)
$$ \left( {d_{2j} - \bar{Y}} \right)^{2} = \bar{Y}^{2} \left[ \begin{aligned} 1 + k_{1}^{{ *^{2} }} \left\{ {1 + 2e_{or} - \delta \theta_{j} e_{1n} - 2\delta \theta_{j} e_{0r} e_{1n} + e_{or}^{2} + \frac{{\delta \left( {\delta + 1} \right)}}{2}\theta_{j}^{2} e_{1n}^{2} } \right\} + k_{2}^{ * 2} \left( {\frac{1}{{R^{ * 2} }}} \right)e_{1n}^{2} \hfill \\ + 2k_{1}^{ * } k_{2}^{ * } \left( {\frac{1}{{R^{ * } }}} \right)\left\{ {\delta \theta_{j} e_{1n}^{2} - e_{0r} e_{1n} - e_{1n} } \right\} - 2k_{1}^{ * } \left\{ {1 + e_{0r} - \frac{1}{2}\delta \theta_{j} e_{1n} - \frac{1}{2}\delta \theta_{j} e_{or} e_{1n} + \frac{{\delta \left( {\delta + 2} \right)}}{8}\theta_{j}^{2} e_{1n}^{2} } \right\} \hfill \\ - 2k_{2}^{ * } \left( {\frac{1}{{R^{ * } }}} \right)\left( {\frac{1}{2}\delta \theta_{j} e_{1n}^{2} - e_{1n} } \right) \hfill \\ \end{aligned} \right] $$
(109)
$$ \left( {d_{3j} - \bar{Y}} \right)^{2} = \bar{Y}^{2} \left[ \begin{aligned} 1 + \alpha_{1}^{2} \left\{ {1 + 2e_{or} + \delta \theta_{j} \left( {e_{1n} - e_{1r} } \right) + e_{or}^{2} + \frac{{\delta^{2} \theta_{j}^{2} }}{4}\left( {e_{1n} - e_{1r} } \right)^{2} - 2\delta \theta_{j} \left( {e_{or} e_{1r} - e_{or} e_{1n} } \right)} \right. \hfill \\ \left. { + \frac{{\delta \theta_{j}^{2} }}{2}\left( {e_{1r}^{2} - e_{1n}^{2} } \right) + \frac{{\delta^{2} \theta_{j}^{2} }}{4}\left( {e_{1n} - e_{1r} } \right)^{2} } \right\} + \alpha_{2}^{2} \left( {\frac{1}{{R^{ * 2} }}} \right)\left( {e_{1n} - e_{1r} } \right)^{2} \hfill \\ + 2\alpha_{1} \alpha_{2} \left( {\frac{1}{{R^{ * } }}} \right)\left\{ {\left( {e_{1n} - e_{1r} } \right) + \frac{{\delta \theta_{j} }}{2}\left( {e_{1n} - e_{1r} } \right)^{2} + \left( {e_{or} e_{1n} - e_{or} e_{1r} } \right) + \frac{{\delta \theta_{j} }}{2}\left( {e_{1n} - e_{1r} } \right)^{2} } \right\} \hfill \\ - 2\alpha_{1} \left\{ {1 + e_{or} + \frac{{\delta \theta_{j} }}{2}\left( {e_{1n} - e_{1r} } \right) + \frac{{\delta \theta_{j} }}{2}\left( {e_{or} e_{1n} - e_{or} e_{1r} } \right)} \right\}\left. { + \frac{{\delta \theta_{j}^{2} }}{4}\left( {e_{1r}^{2} - e_{1n}^{2} } \right) + \frac{{\delta^{2} \theta_{j}^{2} }}{8}\left( {e_{1n} - e_{1r} } \right)^{2} } \right\} \hfill \\ - 2\alpha_{2} \left( {\frac{1}{{R^{ * } }}} \right)\left\{ {\left( {e_{1n} - e_{1r} } \right) + \frac{{\delta \theta_{j} }}{2}\left( {e_{1n} - e_{1r} } \right)^{2} } \right\} \hfill \\ \end{aligned} \right] $$
(110)

Taking expectation of both sides of Eqs. (108), (109) and (110), respectively, we get the MSEs of the suggested classes of estimators d1j, d2 j and d3j (j = 1,2,…,23) to the first degree of approximation as

$$ {\text{MSE}}\left( {d_{1j} } \right) = \overline{Y}^{2} \left[ {1 + k_{1}^{2} A_{1j} + k_{2}^{2} A_{2j} + 2k_{1} k_{2} A_{3j} - 2k_{1} A_{4j} - 2k_{2} A_{5j} } \right] , $$
(111)
$$ {\text{MSE}}\left( {d_{2j} } \right) = \overline{Y}^{2} \left[ {1 + k_{1}^{{ *^{2} }} B_{1j} + k_{2}^{{ *^{2} }} B_{2j} + 2k_{1}^{ * } k_{2}^{ * } B_{3j} - 2k_{1}^{ * } B_{4j} - 2k_{2}^{ * } B_{5j} } \right] , $$
(112)
$$ {\text{MSE}}\left( {d_{3j} } \right) = \overline{Y}^{2} \left[ {1 + \alpha_{1}^{2} C_{1j} + \alpha_{2}^{2} C_{2j} + 2\alpha_{1} \alpha_{2} C_{3j} - 2\alpha_{1} C_{4j} - 2\alpha_{2} C_{5j} } \right] , $$
(113)

where \( \left( {A_{1j} ,A_{2j} ,A_{3j} ,A_{4j} ,A_{5j} } \right) \), \( \left( {B_{1j} ,B_{2j} ,B_{3j} ,B_{4j} ,B_{5j} } \right) \) and \( \left( {C_{1j} ,C_{2j} ,C_{3j} ,C_{4j} ,C_{5j} } \right) \) are same as defined earlier. Thus, the results are proved.

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Singh, H.P., Gupta, A. & Tailor, R. Estimation of Population Mean Using a Difference-Type Exponential Imputation Method. J Stat Theory Pract 15, 19 (2021). https://doi.org/10.1007/s42519-020-00151-2

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Keywords

  • Study variable
  • Auxiliary variable
  • Imputation
  • Bias
  • Mean-squared error