Efficient Estimation of Mean in Two-Phase Sampling when Measurement Error and Non-Response are Simultaneously Present

Abstract

In the present paper, we suggest the generalized classes of efficient estimators for finite population mean estimation in two-phase sampling if non-response and measurement errors are simultaneously present. For each generalized class of estimators, the expressions for the bias and mean square error have been derived to compare the performance of the proposed generalized classes of estimators with some modified estimators in two-phase sampling including modified ratio-type estimator, modified exponential-type ratio estimator, and modified generalized estimator. A simulation study based on different simulated populations by changing the correlation between the auxiliary variable and the study variable is carried out to illustrate the performance of the proposed generalized classes of estimators by changing the correlation from low to high. Finally, it is shown that the proposed generalized class of estimators performs better than the aforementioned modified estimators when correlation is moderate or high.

Appendices

Appendix A: (Description on Different Simulated Populations used for Simulation Results)

X is assumed to follow a normal distribution in each of the three different simulated populations, and Y is simulated using the model $Y = \beta_{yx} X + e$; further description of population parameters is shown separately for each population.

• Simulated Population 1:

  $X \sim N\left( {0.5,\,1} \right)$, $Y = 0.2041241X + e$, $e \sim N\left( {0,\,\,1} \right)$, $U \sim N\left( {0,\,\,9} \right)$, $V \sim N\left( {0,\,\,9} \right)$, $\rho_{YX} = 0.20$, $N = 50000$, $W_{2} = 30\% ,\,40\% ,\,\,60\%$, and $k = 2,\,3$.

• Simulated Population 2:

  $X \sim N\left( {0.5,\,1} \right)$, $Y = 0.5773503X + e$, $e \sim N\left( {0,\,\,1} \right)$, $U \sim N\left( {0,\,\,9} \right)$, $V \sim N\left( {0,\,\,9} \right)$, $\rho_{YX} = 0.50$, $N = 50000$, $W_{2} = 30\% ,\,40\% ,\,\,60\%$, and $k = 2,\,3$.

• Simulated Population 3:

  $X \sim N\left( {0.5,\,1} \right)$, $Y = 0.9801961X + e$, $e \sim N\left( {0,\,\,1} \right)$, $U \sim N\left( {0,\,\,9} \right)$, $V \sim N\left( {0,\,\,9} \right)$, $\rho_{YX} = 0.70$, $N = 50000$, $W_{2} = 30\% ,\,40\% ,\,\,60\%$, and $k = 2,\,3$.

Appendix B

See Appendix Tables

Table 1 Theoretical MSE (in braces) values, and empirical MSE values of all estimators using correlation $\rho_{yx}$ = 0.70

1

Table 2 Theoretical MSE (in braces) values, and empirical MSE values of all estimators using correlation $\rho_{yx}$ = 0.50

2

Table 3 Theoretical MSE (in braces) values, and empirical MSE values of all estimators using correlation $\rho_{yx}$ = 0.20

3

Appendix C

In order for derivations of different variances, covariance and MSEs assuming the joint presence of non-response and measurement error, some of the important derivations and expectations are reproduced following the Azeem [24], Cochran [1], and Hansen and Hurwitz [5].

Let us consider $W_{Y}^{*} = \sum\limits_{i = 1}^{n} {\left( {Y_{i}^{{*^{\prime\prime}}} - \overline{Y}} \right)}$, $W_{X}^{*} = \sum\limits_{i = 1}^{n} {\left( {X_{i}^{{*^{\prime\prime}}} - \overline{X}} \right)}$, $W_{X} = \sum\limits_{i = 1}^{n} {\left( {X^{\prime}_{i} - \overline{X}} \right)}$, $W_{U}^{*} = \sum\limits_{i = 1}^{n} {U_{i}^{*} }$, and $W_{V}^{*} = \sum\limits_{i = 1}^{n} {V_{i}^{*} }$.

Further, we consider.

$e_{0}^{*} = \frac{1}{{n\overline{Y}}}\left( {W_{Y}^{*} + W_{U}^{*} } \right)$, or alternatively $e_{0}^{*} = \frac{{\overline{y}^{{*^{\prime\prime}}} - \overline{Y}}}{{\overline{Y}}}$,

$e_{1}^{*} = \frac{1}{{n\overline{X}}}\left( {W_{X}^{*} + W_{V}^{*} } \right)$, or alternatively $e_{1}^{*} = \frac{{\overline{x}^{{*^{\prime\prime}}} - \overline{X}}}{{\overline{X}}}$,

and $e^{\prime}_{1} = \frac{{W_{X} }}{{n^{\prime}\overline{X}}}$, or alternatively $e^{\prime}_{1} = \frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}$.

Now consider, $W_{Y}^{*} + W_{U}^{*} = \sum\limits_{i = 1}^{n} {\left( {Y_{i}^{{*^{\prime\prime}}} - \overline{Y}} \right)} + \sum\limits_{i = 1}^{n} {U_{i}^{*} }$,

$W_{Y}^{*} + W_{U}^{*} = \sum\limits_{i = 1}^{n} {\left( {Y_{i}^{{*^{\prime\prime}}} - \overline{Y}} \right)} + \sum\limits_{i = 1}^{n} {\left( {y_{i}^{{*^{\prime\prime}}} - Y_{i}^{{*^{\prime\prime}}} } \right)}$,

Or alternatively.

$\frac{{W_{Y}^{*} + W_{U}^{*} }}{n} = \frac{1}{n}\sum\limits_{i = 1}^{n} {y_{i}^{{*^{\prime\prime}}} } - \overline{Y}$.

Now we divide the n into three components, $n_{1}$ (respondents of the first attempt), $r$(selected for the interview) and $n_{2} - r$(not selected for the interview) such that $n = n_{1} + (n_{2} - r) + r$. Therefore, we rewrite the previous expression as.

$\frac{{W_{Y}^{*} + W_{U}^{*} }}{n} = \frac{1}{n}\left( {n_{1} \frac{{\sum\limits_{i = 1}^{{n_{1} }} {y_{i}^{{*^{\prime\prime}}} } }}{{n_{1} }} + \left( {n_{2} - r} \right)\frac{{\sum\limits_{{i = n_{1} + 1}}^{{n_{1} + n_{2} - r}} {y_{i}^{{*^{\prime\prime}}} } }}{{n_{2} - r}} + r\frac{{\sum\limits_{{i = n_{1} + n_{2} - r + 1}}^{n} {y_{i}^{{*^{\prime\prime}}} } }}{r}} \right) - \overline{Y}$.

$\frac{{W_{Y}^{*} + W_{U}^{*} }}{n} = \frac{1}{n}\left( {n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{{*^{\prime\prime}}} + \left( {n_{2} - r} \right)\overline{y}_{{\left( {n_{2} - r} \right)}}^{{*^{\prime\prime}}} + r\overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} } \right) - \overline{Y}$.

Or alternatively can be given by.

$\frac{{W_{Y}^{*} + W_{U}^{*} }}{n} = \frac{1}{n}\left( {n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{{*^{\prime\prime}}} + \left( {n_{2} - r} \right)\overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} + r\overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} } \right) - \overline{Y}$, since $\overline{y}_{\left( r \right)}^{{*^{\prime\prime}}}$ and $\overline{y}_{{\left( {n_{2} - r} \right)}}^{{*^{\prime\prime}}}$ are the unbiased estimators for the same population of non-respondents, i.e.$E\left( {\overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} } \right) = E\left( {\overline{y}_{{\left( {n_{2} - r} \right)}}^{{*^{\prime\prime}}} } \right) = \overline{Y}_{2}$. For more details, see Azeem [25], Hansen and Hurwitz [5], and Cochran [1].

$\frac{{W_{Y}^{*} + W_{U}^{*} }}{n} = \frac{1}{n}\left( {n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{{*^{\prime\prime}}} + n_{2} \overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} } \right) - \overline{Y}$.

$\frac{{W_{Y}^{*} + W_{U}^{*} }}{n} = \overline{y}^{{*^{\prime\prime}}} - \overline{Y}$.

$$ {\text{Or}}\;{\text{alternatively}}\,\overline{y}^{{*^{\prime\prime}}} = \frac{{n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{{*^{\prime\prime}}} + n_{2} \overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} }}{n} = \overline{Y} + \frac{{W_{Y}^{*} + W_{U}^{*} }}{n} $$

(C-1)

In order to show $\overline{y}^{{*^{\prime\prime}}}$ is an unbiased estimator, take expectations of $\overline{y}^{{*^{\prime\prime}}}$,

$E\left( {\overline{y}^{{*^{\prime\prime}}} } \right) = E\left( {\frac{{n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{{*^{\prime\prime}}} + n_{2} \overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} }}{n}} \right)$,

Or alternatively we consider.

$E_{1} E_{2} \left( {\overline{y}^{{*^{\prime\prime}}} } \right) = E_{1} E_{2} \left( {\frac{{n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{{*^{\prime\prime}}} + n_{2} \overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} }}{n}} \right)$,

$= E_{1} \left( {\frac{{n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{*\prime \prime } + E_{2} \left( {n_{2} \overline{y}_{\left( r \right)}^{*\prime \prime } } \right)}}{n}} \right)$,

$= E_{1} \left( {\frac{{n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{*\prime \prime } + n_{2} \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } }}{n}} \right)$,$\because E_{2} \left( {\overline{y}_{\left( r \right)}^{{*^{\prime\prime}}} } \right) = \overline{y}_{{\left( {n_{2} } \right)}}^{{*^{\prime\prime}}}$.

or.

$= E_{1} \left( {w_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{*\prime \prime } + w_{2} \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } } \right)$, where $w_{1} = \frac{{n_{1} }}{n}$, and $w_{2} = \frac{{n_{2} }}{n}$.

$E\left( {\overline{y}^{{*^{\prime\prime}}} } \right) = W_{1} \overline{Y}_{1} + W_{2} \overline{Y}_{2}$,

Hence $E(\overline{y}^{*\prime \prime } ) = \frac{{\sum\limits_{i = 1}^{N} {Y_{i} } }}{N} = \overline{Y}$, and $\overline{y}^{*}$ is an unbiased estimator of $\overline{Y}$. Now in order to get an expression for variance of $\overline{y}^{*}$, we proceed as

$$ Var\left( {\overline{y}^{*\prime \prime } } \right) = E\left( {\overline{y}^{*\prime \prime } - \overline{Y}} \right)^{2} = E\left( {\frac{{n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{*\prime \prime } + n_{2} \overline{y}_{\left( r \right)}^{*\prime \prime } }}{n} - \overline{Y}} \right)^{2} $$

(C-2)

or.

$Var\left( {\overline{y}^{*\prime \prime } } \right) = \frac{1}{{n^{2} }}E\left( {n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{*\prime \prime } + n_{2} \overline{y}_{\left( r \right)}^{*\prime \prime } - n\overline{Y}} \right)^{2}$.

Since $\overline{y}\prime \prime = \frac{1}{n}\left( {n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{*\prime \prime } + n_{2} \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } } \right)$ therefore alternatively we can consider $n_{1} \overline{y}_{{\left( {n_{1} } \right)}}^{*} = n\overline{y} - n_{2} \overline{y}_{{\left( {n_{2} } \right)}}^{*}$, see Cochran [1]. Now.

$= \frac{1}{{n^{2} }}E\left( {\left( {n\overline{y}\prime \prime - n_{2} \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } } \right) + n_{2} \overline{y}_{\left( r \right)}^{*\prime \prime } - n\overline{Y}} \right)^{2}$,

or

$$ Var\left( {\overline{y}^{*\prime \prime } } \right) = \frac{1}{{n^{2} }}E_{1} E_{2} \left( {n\left( {\overline{y}\prime \prime - \overline{Y}} \right) + n_{2} \left( {\overline{y}_{\left( r \right)}^{*\prime } - \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } } \right)} \right)^{2} $$

(C-3)

Eq. (C-3) is given by

$$ Var(\overline{y}^{*\prime \prime } ) = \left( {E_{1} \left( {\overline{y}\prime \prime - \overline{Y}} \right)^{2} + E_{1} \left( {\frac{{n_{2} }}{n}} \right)^{2} E_{2} \left( {\overline{y}_{\left( r \right)}^{*\prime \prime } - \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime } } \right)^{2} } \right) $$

(C-4)

because $E_{1} \frac{{n_{2} }}{n}\left( {\overline{y}\prime \prime - \overline{Y}} \right)E_{2} \left( {\overline{y}_{\left( r \right)}^{*\prime \prime } - \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } } \right) = 0$.

From Eq. (C-4), we consider.

$E_{1} \left( {\frac{{n_{2} }}{n}} \right)^{2} E_{2} \left( {\overline{y}_{\left( r \right)}^{*\prime \prime } - \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } } \right)^{2}$ = $E_{1} \left( {\frac{{n_{2} }}{n}} \right)^{2} \frac{{n_{2} - r}}{{n_{2} - 1}}\frac{{s_{Y\left( 2 \right)}^{2} + s_{U\left( 2 \right)}^{2} }}{r}$,

or.

 = $E_{1} \left( {\frac{{n_{2} }}{n}} \right)^{2} \frac{{n_{2} - r}}{{n_{2} - 1}}\frac{{n_{2} - 1}}{{n_{2} }}\frac{{S_{Y\left( 2 \right)}^{2} + S_{U\left( 2 \right)}^{2} }}{r}$ since $\left( {s_{Y\left( 2 \right)}^{2} + s_{U\left( 2 \right)}^{2} } \right) = \frac{{n_{2} - 1}}{{n_{2} }}\left( {S_{Y\left( 2 \right)}^{2} + S_{U\left( 2 \right)}^{2} } \right)$.

$= E_{1} \left( {\frac{{n_{2} }}{{n^{2} }}} \right)\frac{{n_{2} - r}}{r}\left( {S_{Y\left( 2 \right)}^{2} + S_{U\left( 2 \right)}^{2} } \right)$,since $k = \frac{{n_{2} }}{r}$.

$= \frac{{nN_{2} }}{{n^{2} N}}(k - 1)\left( {S_{Y\left( 2 \right)}^{2} + S_{U\left( 2 \right)}^{2} } \right)$,where $E_{1} (n_{2} ) = \frac{{nN_{2} }}{N}$.

$$ E_{1} \left( {\frac{{n_{2} }}{n}} \right)^{2} E_{2} \left( {\overline{y}_{\left( r \right)}^{*\prime \prime } - \overline{y}_{{\left( {n_{2} } \right)}}^{*\prime \prime } } \right)^{2} = \frac{{W_{2} (k - 1)}}{n}\left( {S_{Y\left( 2 \right)}^{2} + S_{U\left( 2 \right)}^{2} } \right) = \theta^{*} \left( {S_{Y\left( 2 \right)}^{2} + S_{U\left( 2 \right)}^{2} } \right) $$

(C-5)

From Eq. (C-4), we consider

$$ E_{1} \left( {y^{\prime \prime } - \overline{Y}} \right)^{2} = \frac{N - n}{N}\frac{{\left( {S_{Y}^{2} + S_{U}^{2} } \right)}}{n} = \left( {\frac{1}{n} - \frac{1}{N}} \right)\left( {S_{Y}^{2} + S_{U}^{2} } \right) = \lambda^{\prime \prime } \left( {S_{Y}^{2} + S_{U}^{2} } \right) $$

(C-6)

Now, substituting Eq. (C-5) and Eq. (C-6) in Eq. (C-4) the variance of $\overline{y}^{*\prime \prime }$ is given by.

${\text{Var}}(\overline{y}^{*\prime \prime } ) = \left( {\lambda^{\prime \prime } \left( {S_{Y}^{2} + S_{U}^{2} } \right) + \theta^{*} \left( {S_{Y\left( 2 \right)}^{2} + S_{U\left( 2 \right)}^{2} } \right)} \right)$.

Or alternatively can be given by.

$E\left( {e_{0}^{*} } \right)^{2} = E_{1} E_{2} \left( {\frac{{\overline{y}^{*\prime \prime } - \overline{Y}}}{{\overline{Y}}}} \right)^{2} = E_{1} E_{2} \left( {\frac{{W_{Y}^{*} + W_{U}^{*} }}{{\overline{Y}n}}} \right)^{2}$,

or also can be given by

$$ E\left( {e_{0}^{*} } \right)^{2} = \frac{1}{{\overline{Y}^{2} }}Var(\overline{y}^{*\prime \prime } ) = V_{02}^{\prime \prime } $$

(C-7)

Similarly, one can get.

$Var(\overline{x}^{*\prime \prime } ) = \lambda^{\prime \prime } \left( {S_{X}^{2} + S_{V}^{2} } \right) + \theta^{*} \left( {S_{X\left( 2 \right)}^{2} + S_{V\left( 2 \right)}^{2} } \right)$,

alternatively is given by.

$E\left( {e_{1}^{*} } \right)^{2} = E_{1} E_{2} \left( {\frac{{\overline{x}^{*\prime \prime } - \overline{X}}}{{\overline{X}}}} \right)^{2} = E_{1} E_{2} \left( {\frac{{W_{X}^{*} + W_{V}^{*} }}{{\overline{X}n}}} \right)^{2}$,

$$ E\left( {e_{1}^{*} } \right)^{2} = \frac{1}{{\overline{X}^{2} }}Var(\overline{x}^{{*^{\prime\prime}}} ) = V_{02}^{\prime \prime } $$

(C-8)

Similarly

$$ E\left( {e_{0}^{*} e_{1}^{*} } \right) = E_{1} E_{2} \left( {\frac{{\overline{x}^{*\prime \prime } - \overline{X}}}{{\overline{X}}}} \right)\left( {\frac{{\overline{y}^{*\prime \prime } - \overline{Y}}}{{\overline{Y}}}} \right) = \frac{1}{{\overline{X}\overline{Y}}}Cov(\overline{x}^{*\prime \prime } ,\overline{y}^{*\prime \prime } ) = V\prime \prime_{11} $$

(C-9)

where $Cov(\overline{x}^{*\prime \prime } ,\overline{y}^{*\prime \prime } ) = \lambda \prime \prime S_{YX} + \theta^{*} S_{YX(2)}$.

Similarly.

$E\left( {e^{\prime}_{1} } \right)^{2} = E^{\prime}\left( {\frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}} \right)^{2} = E^{\prime}\left( {\frac{{W_{X} }}{{\overline{X}n}}} \right)^{2} = \frac{1}{{\overline{X}^{2} }}Var(\overline{x}^{\prime}) = \frac{1}{{\overline{X}^{2} }}\lambda^{\prime}S_{X}^{2} = V^{\prime}_{20}$,

and

$$ E\left( {e^{\prime}_{1} e_{1}^{*} } \right) = E^{\prime}E_{1} E_{2} \left( {\frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}} \right)\left( {\frac{{\overline{x}^{{*^{\prime\prime}}} - \overline{X}}}{{\overline{X}}}} \right) = \frac{1}{{\overline{X}^{2} }}Var(\overline{x}^{\prime}) = V^{\prime}_{20} $$

(C-10)

and

$$ E\left( {e^{\prime}_{1} e_{0}^{*} } \right) = E^{\prime}E_{1} E_{2} \left( {\frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}} \right)\left( {\frac{{\overline{y}^{{*^{\prime\prime}}} - \overline{Y}}}{{\overline{Y}}}} \right) = \frac{1}{{\overline{X}\overline{Y}}}Cov(\overline{x}^{\prime},\overline{y}^{{*^{\prime\prime}}} ) = V^{\prime}_{11} $$

(C-11)

When non-response and measurement errors are present only on study variable, we get Eq. (C-1)-( C-11) given below.

Let us consider $W_{Y}^{*} = \sum\limits_{i = 1}^{n} {\left( {Y_{i}^{{*^{\prime\prime}}} - \overline{Y}} \right)}$, $W_{X} = \sum\limits_{i = 1}^{n} {\left( {X_{i} - \overline{X}} \right)}$, $W_{X} = \sum\limits_{i = 1}^{n} {\left( {X^{\prime}_{i} - \overline{X}} \right)}$, and $W_{U}^{*} = \sum\limits_{i = 1}^{n} {U_{i}^{*} }$.

Further we consider.

$e_{0}^{*} = \frac{1}{{n\overline{Y}}}\left( {W_{Y}^{*} + W_{U}^{*} } \right)$, or alternatively $e_{0}^{*} = \frac{{\overline{y}^{{*^{\prime\prime}}} - \overline{Y}}}{{\overline{Y}}}$,

$e_{1} = \frac{{W_{X} }}{{n\overline{X}}}$, or alternatively $e_{1} = \frac{{\overline{x} - \overline{X}}}{{\overline{X}}}$,

and $e^{\prime}_{1} = \frac{{W_{X} }}{{n^{\prime}\overline{X}}}$, or alternatively $e^{\prime}_{1} = \frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}$.

Now consider, $W_{Y}^{*} + W_{U}^{*} = \sum\limits_{i = 1}^{n} {\left( {Y_{i}^{{*^{\prime\prime}}} - \overline{Y}} \right)} + \sum\limits_{i = 1}^{n} {U_{i}^{*} }$,

$W_{Y}^{*} + W_{U}^{*} = \sum\limits_{i = 1}^{n} {\left( {Y_{i}^{{*^{\prime\prime}}} - \overline{Y}} \right)} + \sum\limits_{i = 1}^{n} {\left( {y_{i}^{{*^{\prime\prime}}} - Y_{i}^{{*^{\prime\prime}}} } \right)}$,

Or alternatively.

$\frac{{W_{Y}^{*} + W_{U}^{*} }}{n} = \frac{1}{n}\sum\limits_{i = 1}^{n} {y_{i}^{{*^{\prime\prime}}} } - \overline{Y}$.

Now we get.

$Var(\overline{x}) = \lambda^{\prime \prime } S_{X}^{2}$,

alternatively is given by

$$ E\left( {e_{1} } \right)^{2} = E_{1} E_{2} \left( {\frac{{\overline{x} - \overline{X}}}{{\overline{X}}}} \right)^{2} = E_{1} E_{2} \left( {\frac{{W_{X} }}{{\overline{X}n}}} \right)^{2} = \frac{1}{{\overline{X}^{2} }}Var(\overline{x}) = V_{20} $$

(C-12)

Similarly

$$ E\left( {e_{0}^{*} e_{1} } \right) = E_{1} E_{2} \left( {\frac{{\overline{x} - \overline{X}}}{{\overline{X}}}} \right)\left( {\frac{{\overline{y}^{{*^{\prime\prime}}} - \overline{Y}}}{{\overline{Y}}}} \right) = \frac{1}{{\overline{X}\overline{Y}}}Cov(\overline{x},\overline{y}^{{*^{\prime\prime}}} ) = V^{\prime\prime}_{11} $$

(C-13)

where ${\text{Cov}}(\overline{x}^{*\prime \prime } ,\overline{y}^{*\prime \prime } ) = \lambda^{\prime \prime } S_{YX} + \theta^{*} S_{YX(2)}$.

Similarly.

$E\left( {e^{\prime}_{1} } \right)^{2} = E^{\prime}\left( {\frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}} \right)^{2} = E^{\prime}\left( {\frac{{W_{X} }}{{\overline{X}n}}} \right)^{2} = \frac{1}{{\overline{X}^{2} }}Var(\overline{x}^{\prime}) = \frac{1}{{\overline{X}^{2} }}\lambda^{\prime}S_{X}^{2} = V^{\prime}_{20}$,

and

$$ E\left( {e^{\prime}_{1} e_{1}^{*} } \right) = E^{\prime}E_{1} E_{2} \left( {\frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}} \right)\left( {\frac{{\overline{x}^{{*^{\prime\prime}}} - \overline{X}}}{{\overline{X}}}} \right) = \frac{1}{{\overline{X}^{2} }}Var(\overline{x}^{\prime}) = V^{\prime}_{20} $$

(C-14)

and

$$ E\left( {e^{\prime}_{1} e_{0}^{*} } \right) = E^{\prime}E_{1} E_{2} \left( {\frac{{\overline{x}^{\prime} - \overline{X}}}{{\overline{X}}}} \right)\left( {\frac{{\overline{y}^{{*^{\prime\prime}}} - \overline{Y}}}{{\overline{Y}}}} \right) = \frac{1}{{\overline{X}\overline{Y}}}Cov(\overline{x}^{\prime},\overline{y}^{{*^{\prime\prime}}} ) = V^{\prime}_{11} $$

(C-15)