A note on regression estimation with unknown population size 3. Alternative regression estimatorA note on regression estimation with unknown population size 3. Alternative regression estimator

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We now consider an alternative estimator that does not use the population size $\left(N\right)$  information. Rather, it uses the known inclusion probabilities ${\pi }_i$  provided that they are known for each unit in the population. Given that ${\displaystyle {\sum }_{i\in U}}{\pi }_i\mathrm{<mo>=</mo>}n,$  we can use $z_i\mathrm{<mo>=</mo>}{\left({\pi }_i\mathrm{,}{x}_i^{*{\rm T}}\right)}^{{\rm T}}$  as auxiliary data in the model

$$y_i\mathrm{<mo>=</mo>}{z}_i^{{\rm T}}\beta +e_i\mathrm{,}$$

where $e_i\stackrel{\text{ind}}{\sim }\left(\mathrm{0,}{\sigma }^2{\pi }_i\right).$ This means that the incorporation of the variance structure $c_i$ of the error in the regression vector is given by $c_i\mathrm{<mo>=</mo>}{d}_i/{\sigma }^2.$ The resulting estimator is given by

$${\widehat{Y}}_{\text{KREG}}\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(Z-{\widehat{Z}}_{\pi }\right)}^{{\rm T}}{\widehat{B}}_{\text{KREG}}\mathrm{,}(3.1)$$

with $Z\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{z}_i,$ $\widehat{Z}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}{d}_i{z}_i$ and

$${\widehat{B}}_{\text{KREG}}\mathrm{<mo>=</mo>}{\left({\displaystyle {\displaystyle \sum }}_{i\in s}{c}_i{d}_i{z}_i{z}_i^{{\rm T}}\right)}^{-1}{\displaystyle {\displaystyle \sum }}_{i\in s}{c}_i{d}_i{z}_i{y}_i\mathrm{.}(3.2)$$

This estimator corresponds exactly to the one given by Isaki and Fuller (1982).

Remark 3.1 By construction,

$${\displaystyle \sum _{i\in s}}{d}_i^2\left(y_i-z_i^{{\rm T}}{\widehat{B}}_{\text{KREG}}\right)z_i\mathrm{<mo>=</mo>}0.$$

Since ${\pi }_i$  is a component of $z_i,$  we have ${\displaystyle {\sum }_{i\in s}}{d}_i\left(y_i-z_i^{{\rm T}}{\widehat{B}}_{\text{KREG}}\right)\mathrm{<mo>=</mo>\; 0},$  this leads to

$${\widehat{Y}}_{\text{KREG}}\mathrm{<mo>=</mo>}{Z}^{{\rm T}}{\widehat{B}}_{\text{KREG}}\mathrm{.}$$

Thus, ${\widehat{Y}}_{\text{KREG}}$  is the best linear unbiased predictor of $Y\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}{y}_i$  under the model

$$y_i\mathrm{<mo>=</mo>}{\pi }_i{\beta }_1+x_i^{*{\rm T}}{\beta }_2+e_i\mathrm{,}$$

where $e_i\sim \left(\mathrm{0,}{\sigma }^2{\pi }_i\right).$

Note that ${\widehat{B}}_{\text{KREG}}$ can be expressed as ${\widehat{B}}_{\text{GREG}}$ by setting $c_i\mathrm{<mo>=</mo>}{d}_i/{\sigma }^2$ and $x_i\mathrm{<mo>=</mo>}{z}_i.$ Thus, the proposed regression estimator can be viewed as a special case of GREG estimator. Using the argument similar to (2.12), we obtain

$${\widehat{Y}}_{\text{KREG}}-Y\cong {\displaystyle \sum _{i\in s}}{d}_i{E}_i^{*}-{\displaystyle \sum _{i\in U}}{E}_i^{*}\mathrm{,}(3.3)$$

where $E_i^{*}\mathrm{<mo>=</mo>}{y}_i-z_i^{{\rm T}}{B}_{\text{KREG}}$ and

$$B_{\text{KREG}}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in U}}{c}_i{z}_i{z}_i^{{\rm T}}\right)}^{-1}{\displaystyle \sum _{i\in U}}{c}_i{z}_i{y}_i\mathrm{.}$$

The proposed estimator is approximately unbiased and its asymptotic variance

$$V\left\{{\displaystyle \sum _{i\in s}}{d}_i\left(y_i-z_i^{{\rm T}}{B}_{\text{KREG}}\right)\right\}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in U}}{\displaystyle \sum _{j\in U}}{\Delta }_{ij}\frac{E_i^{*}}{{\pi }_i}\frac{E_j^{*}}{{\pi }_j}$$

is often smaller than the asymptotic variance of Singh and Raghunath (2011)’s estimator.

The optimal version of ${\widehat{Y}}_{\text{KREG}}$ uses $z_i\mathrm{<mo>=</mo>}{\left({\pi }_i\mathrm{,}{x}_i^{*{\rm T}}\right)}^{{\rm T}}$ as auxiliary data. It is given by

$${\widehat{Y}}_{\text{KOPT}}\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(Z-{\widehat{Z}}_{\pi }\right)}^{{\rm T}}{\widehat{B}}_{\text{KOPT}}\mathrm{,}(3.4)$$

where ${\widehat{B}}_{\text{KOPT}}$ is obtained by substituting $x_i$ by $z_i$ in equation (2.10).

Remark 3.2 For fixed-size sampling designs, we have $V_p\left({\displaystyle {\sum }_{i\in s}}{d}_i{\pi }_i\right)\mathrm{<mo>=</mo>\; 0.}$  In this case, the optimal regression coefficient vector $B_{\text{KOPT}}\mathrm{<mo>=</mo>}{V}_p{\left({\widehat{Z}}_{\pi }\right)}^{-1}{\text{Cov}}_p\left({\widehat{Z}}_{\pi }\mathrm{,}{\widehat{Y}}_{\pi }\right)$  cannot be computed because the variance-covariance matrix $V_p\left({\widehat{Z}}_{\pi }\right)$  is not invertible. Thus, the optimal estimator with $z_i\mathrm{<mo>=</mo>}{\left({\pi }_i\mathrm{,}{x}_i^{*{\rm T}}\right)}^{{\rm T}}$  reduces to the optimal estimator (2.9) only using $x_i^{*}.$

Remark 3.3 For random-size sampling designs, $V_p\left({\displaystyle {\sum }_{i\in s}}{d}_i{\pi }_i\right)\ge 0.$  In this case, all of the components of $z_i\mathrm{<mo>=</mo>}{\left({\pi }_i\mathrm{,}{x}_i^{*{\rm T}}\right)}^{{\rm T}}$  can be used in the design-optimal regression estimator (2.9).

A difficulty with using the optimal estimator ${\widehat{Y}}_{\text{KOPT}}$ is that it requires the computation of the joint inclusion probabilities ${\pi }_{ij}\text{}:$ these may be difficult to compute for certain sampling designs. An estimator that does not require the computation of the joint inclusion probabilities is obtained by assuming that ${\pi }_{ij}\mathrm{<mo>=</mo>}{\pi }_i{\pi }_j.$ We refer to this estimator as the pseudo-optimal estimator, ${\widehat{Y}}_{\text{POPT}}.$ It is given by

$${\widehat{Y}}_{\text{POPT}}\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(Z-{\widehat{Z}}_{\pi }\right)}^{{\rm T}}{\widehat{B}}_{\text{POPT}}\mathrm{,}(3.5)$$

where

$${\widehat{B}}_{\text{POPT}}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in s}}{c}_i{d}_i{z}_i{z}_i^{{\rm T}}\right)}^{-1}{\displaystyle \sum _{i\in s}}{c}_i{d}_i{z}_i{y}_i$$

and

$$c_i\mathrm{<mo>=</mo>}{d}_i-1.$$

In general, the pseudo-optimal estimator ${\widehat{Y}}_{\text{POPT}}$ should yield estimates that are quite close to those produced by ${\widehat{Y}}_{\text{KREG}}$ when the sampling fraction is small. Note that ${\widehat{Y}}_{\text{POPT}}$ is exactly equal to the optimal estimator ${\widehat{Y}}_{\text{KOPT}}$ in the case of Poisson sampling. In this sampling design the inclusion probabilities of units in the sample are independent. The approximate design variance for ${\widehat{Y}}_{\text{KREG}},$ ${\widehat{Y}}_{\text{KOPT}}$ and ${\widehat{Y}}_{\text{POPT}}$ have the same form as the one given in equation (2.3) with the $E_i’\text{s}$ respectively given by $y_i-z_i^{{\rm T}}{\widehat{B}}_{\text{KREG}},$ $y_i-z_i^{{\rm T}}{\widehat{B}}_{\text{KOPT}}$ and $y_i-z_i^{{\rm T}}{\widehat{B}}_{\text{POPT}}.$

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