A note on regression estimation with unknown population size 2. Regression estimatorsA note on regression estimation with unknown population size 2. Regression estimators

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Under general regularity conditions (Isaki and Fuller 1982; Montanari 1987), an approximation to the regression estimator (1.1) is

$${\tilde{Y}}_{\text{REG}}\mathrm{<mo> = </mo>}{\widehat{Y}}_{\pi }+{\left(X-{\widehat{X}}_{\pi }\right)}^{{\rm T}}B\mathrm{,}(2.1)$$

where $B$ is the limit in probability of $\widehat{B}$ when both the sample and the population sizes tend to infinity. For large samples, the variance of regression estimator (1.1) can be studied via (2.1). Note that ${\tilde{Y}}_{\text{REG}}$ is unbiased under the sampling plan $p\left(s\right)$ and can be re-expressed as:

$${\tilde{Y}}_{\text{REG}}\mathrm{<mo>=</mo>}{X}^{{\rm T}}B+{\displaystyle \sum _{i\in s}}{d}_i{E}_i\mathrm{,}(2.2)$$

where $E_i\mathrm{<mo>=</mo>}{y}_i-x_i^{{\rm T}}B.$

The design variance for ${\widehat{Y}}_{\text{REG}}$ can be approximated by

$${\text{AV}}_p\left({\widehat{Y}}_{\text{REG}}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in U}}{\displaystyle \sum _{j\in U}}{\Delta }_{ij}\frac{E_i}{{\pi }_i}\frac{E_j}{{\pi }_j}\mathrm{,}(2.3)$$

where ${\Delta }_{ij}\mathrm{<mo>=</mo>}{\pi }_{ij}-{\pi }_i{\pi }_j$ and ${\pi }_{ij}$ is the second order inclusion probability for units $i$ and $j.$ Both the model-assisted (Särndal, Swensson and Wretman 1992) and the optimal-variance (Montanari 1987) approaches can be used to estimate $B.$ They both yield approximately unbiased estimators. In the case of the model-assisted approach, the basic properties (bias and variance terms) are valid even when the model is not correctly specified. Under the optimal-variance approach no assumption is made on the variable of interest.

The model-assisted estimator of Särndal et al. (1992) assumes a working model between the variable of interest $\left(y\right)$ and the auxiliary variables $\left(x\right).$ The working model is denoted by $m\mathrm{<mo>: </mo>}{y}_i\mathrm{<mo> = </mo>}{x}_i^{{\rm T}}\beta +{\epsilon }_i$ where $\beta$ is a vector of $p$ unknown parameters, $E_m\left({\epsilon }_i|x_i\right)\mathrm{<mo>=</mo>\; 0},$ $V_m\left({\epsilon }_i|x_i\right)\mathrm{<mo>=</mo>}{\sigma }_i^2,$ and ${\text{Cov}}_m\left({\epsilon }_i\mathrm{,}{\epsilon }_j|x_i\mathrm{,}{x}_j\right)\mathrm{<mo>=</mo>\; 0,}i\ne j.$ Under this approach, $B$ in equation (2.1) is the ordinary least squares estimator of $\beta$ in the population and it is given by

$$B_{\text{GREG}}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in U}}{c}_i{x}_i{x}_i^{{\rm T}}\right)}^{-1}\left({\displaystyle \sum _{i\in U}}{c}_i{x}_i{y}_i\right)\mathrm{,}(2.4)$$

where $c_i\mathrm{<mo>=</mo>}{\sigma }_i^{-2}.$ This yields the following estimator for the total $Y$

$${\widehat{Y}}_{\text{GREG}}\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(X-{\widehat{X}}_{\pi }\right)}^{{\rm T}}{\widehat{B}}_{\text{GREG}}\mathrm{,}(2.5)$$

where

$${\widehat{B}}_{\text{GREG}}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in s}}{c}_i{d}_i{x}_i{x}_i^{{\rm T}}\right)}^{-1}\left({\displaystyle \sum _{i\in s}}{c}_i{d}_i{x}_i{y}_i\right)\mathrm{.}(2.6)$$

The optimal estimator of Montanari (1987), obtained by minimizing the design variance of

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

is

$${\tilde{Y}}_{\text{OPT}}\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(X-{\widehat{X}}_{\pi }\right)}^{{\rm T}}{B}_{\text{OPT}}\mathrm{,}(2.7)$$

where

$$\begin{array}{ll}{B}_{\text{OPT}}\hfill & \mathrm{<mo>=</mo>}{\left\{V\left({\widehat{X}}_{\pi }\right)\right\}}^{-1}\text{Cov}\left({\widehat{X}}_{\pi }\mathrm{,}{\widehat{Y}}_{\pi }\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in U}}{\displaystyle \sum _{j\in U}}{\Delta }_{ij}\frac{x_i}{{\pi }_i}\frac{x_j^{{\rm T}}}{{\pi }_j}\right)}^{-1}\left({\displaystyle \sum _{i\in U}}{\displaystyle \sum _{j\in U}}{\Delta }_{ij}\frac{x_i}{{\pi }_i}\frac{y_j}{{\pi }_j}\right)\mathrm{.}\hfill \end{array}(2.8)$$

The optimal estimator for the total $Y$ is estimated by

$${\widehat{Y}}_{\text{OPT}}\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(X-{\widehat{X}}_{\pi }\right)}^{{\rm T}}{\widehat{B}}_{\text{OPT}}\mathrm{,}(2.9)$$

where

$${\widehat{B}}_{\text{OPT}}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in s}}{\displaystyle \sum _{j\in s}}\frac{{\Delta }_{ij}}{{\pi }_{ij}}\frac{x_i}{{\pi }_i}\frac{x_j^{{\rm T}}}{{\pi }_j}\right)}^{-1}\left({\displaystyle \sum _{i\in s}}{\displaystyle \sum _{j\in s}}\frac{{\Delta }_{ij}}{{\pi }_{ij}}\frac{x_i}{{\pi }_i}\frac{y_j}{{\pi }_j}\right)\mathrm{.}(2.10)$$

Note that the computation of the regression vectors requires that the first component that defines them is invertible. We can ensure this by reducing the number of auxiliary variables that are input into the regression if not much loss in efficiency of the resulting regression estimator is incurred. If, on the other hand, there is a significant loss in efficiency, then we can invert these singular matrices using generalised inverses.

As mentioned in the introduction, not all population totals may be known for each component of the auxiliary vector $x.$ The regression normally uses the auxiliary variables for which a corresponding population total is known. Decomposing $x_i$ as ${\left(\mathrm{1,}{x}_i^{*{\rm T}}\right)}^{{\rm T}}$ where $x_i^{*}\mathrm{<mo>=</mo>}{\left(x_{2i}\mathrm{,}\dots \mathrm{,}{x}_{pi}\right)}^{{\rm T}},$ Singh and Raghunath (2011) proposed a GREG-like estimator that assumes that the regression is based on an intercept and the variable $x^{*},$ even though only the population total of the $x^{*}$ is known.

For the case that $N$ is not known and that the population total of $x^{*}$ is known, their estimator is

$${\widehat{Y}}_{\text{SREG}}\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}{\widehat{B}}_{2,\text{GREG}}\mathrm{,}(2.11)$$

where $X^{*}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{x}_i^{*}$ and ${\widehat{X}}_{\pi }^{*}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}{d}_i{x}_i^{*}.$ The regression vector of estimated coefficients ${\widehat{B}}_{2,\text{GREG}}$ is obtained from ${\widehat{B}}_{\text{GREG}}\mathrm{<mo>=</mo>}{\left({\widehat{B}}_{1,GREG}\mathrm{,}{\widehat{B}}_{2,\text{GREG}}^{{\rm T}}\right)}^{{\rm T}}$ given by (2.6). The approximate design variance for ${\widehat{Y}}_{\text{SREG}}$ takes the same form as equation (2.3), with $E_i\mathrm{<mo>=</mo>}{y}_i-x_i^{*{\rm T}}{B}_{2,\text{GREG}},$ where

$$B_{\mathrm{2,}\text{GREG}}\mathrm{<mo>=</mo>}{\left\{{\displaystyle \sum _{i\in U}}{c}_i\left(x_i^{*}-{\overline{X}}_N^{*}\right){\left(x_i^{*}-{\overline{X}}_N^{*}\right)}^{{\rm T}}\right\}}^{-1}{\displaystyle \sum _{i\in U}}{c}_i\left(x_i^{*}-{\overline{X}}_N^{*}\right)y_i$$

and ${\overline{X}}_N^{*}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{x}_i^{*}/N.$

The properties of (2.11) can be obtained by noting that

$$\begin{array}{ll}{\widehat{Y}}_{\text{SREG}}-Y\hfill & \mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }-Y+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}{\widehat{B}}_{2,\text{GREG}}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }-Y+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}{B}_{2,\text{GREG}}+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}\left({\widehat{B}}_{2,\text{GREG}}-B_{2,\text{GREG}}\right)\mathrm{.}\hfill \end{array}$$

Since ${\widehat{B}}_{\mathrm{2,}\text{GREG}}-B_{\mathrm{2,}\text{GREG}}\mathrm{<mo>=</mo>}{O}_p\left(n^{-1/2}\right)$ under some regularity conditions discussed in Fuller (2009, Chapter 2), the last term is of smaller order. Thus, ignoring the smaller order terms, we get the following approximation

$${\widehat{Y}}_{\text{SREG}}-Y\cong {\displaystyle \sum _{i\in s}}{d}_i{E}_i-{\displaystyle \sum _{i\in U}}{E}_i\mathrm{,}(2.12)$$

where $E_i\mathrm{<mo>=</mo>}{y}_i-x_i^{*{\rm T}}{B}_{\mathrm{2,}\text{GREG}}.$ Thus, ${\widehat{Y}}_{\text{SREG}}$ is approximately design-unbiased. The asymptotic variance can be computed using

$$V\left\{{\displaystyle \sum _{i\in s}}{d}_i{E}_i-{\displaystyle \sum _{i\in U}}{E}_i\right\}\mathrm{<mo>=</mo>}E\left\{{\left({\displaystyle \sum _{i\in s}}{d}_i{E}_i-{\displaystyle \sum _{i\in U}}{E}_i\right)}^2\right\}\mathrm{.}$$

As we can see, the asymptotic variance can be quite large unless ${\displaystyle {\sum }_{i\in U}}{E}_i\mathrm{<mo>=</mo>\; 0.}$

Remark 2.1 If $y_i\mathrm{<mo>=</mo>}a+b{x}_i,$  we have ${\widehat{Y}}_{\text{SREG}}-Y\mathrm{<mo>=</mo>}\left({\widehat{N}}_{\pi }-N\right)a$  and this implies that $V\left({\widehat{Y}}_{\text{SREG}}\right)\mathrm{<mo>=</mo>}{a}^{2}V\left({\widehat{N}}_{\pi }\right).$  This means that if $V\left({\widehat{N}}_{\pi }\right)\mathrm{<mo>></mo>\; 0},$  we can artificially increases $a^{2}V\left({\widehat{N}}_{\pi }\right),$  the variance of ${\widehat{Y}}_{\text{SREG}},$  by choosing large values of $a.$

Note that the optimal regression estimator using $x^{*}\mathrm{<mo>=</mo>}{\left(x_2\mathrm{,}\dots \mathrm{,}{x}_p\right)}^{{\rm T}}$ is also approximately design unbiased because

$$\begin{array}{ll}{\widehat{Y}}_{\text{OPT}}^{*}-Y\hfill & \mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }-Y+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}{\widehat{B}}_{\text{OPT}}^{*}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }-Y+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}{B}_{\text{OPT}}^{*}+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}\left({\widehat{B}}_{\text{OPT}}^{*}-B_{\text{OPT}}^{*}\right)\mathrm{,}\hfill \end{array}$$

where $B_{\text{OPT}}^{*}$ is obtained by replacing $x_i$ by $x_i^{*}$ in equation (2.8). Since ${\widehat{B}}_{\text{OPT}}^{*}-B_{\text{OPT}}^{*}\mathrm{<mo>=</mo>}{O}_p\left(n^{-1/2}\right)$ under some regularity conditions discussed in Fuller (2009, Chapter 2), ignoring the smaller order terms we get

$${\widehat{Y}}_{\text{OPT}}^{*}-Y\cong {\widehat{Y}}_{\pi }-Y+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}{B}_{\text{OPT}}^{*}\mathrm{.}$$

The asymptotic variance of ${\widehat{Y}}_{\text{OPT}}^{*}$ is smaller than the one associated with ${\widehat{Y}}_{\text{SREG}}.$ The reason for this is that the optimal estimator minimizes the asymptotic variance among the class of estimators of the form

$${\widehat{Y}}_B\mathrm{<mo>=</mo>}{\widehat{Y}}_{\pi }+{\left(X^{*}-{\widehat{X}}_{\pi }^{*}\right)}^{{\rm T}}\widehat{B}(2.13)$$

indexed by $\widehat{B}.$

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