Model-assisted sample design is minimax for model-based prediction
Section 2. Upper bound for the AV of the BLUP

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Let $U\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\dots \mathrm{,}N\right\}$ denote the finite population. For unit $i\in U,$ the variable of interest is $y_i$ and the $p$ -vector of auxiliary variables is $x_i.$ The sample (of size $n)$ is $s$ and the non-sample set is $r\mathrm{<mo>=</mo>}U-s.$ Auxiliary variables are observed for all $i\in U$ while $y_i$ is observed for $i\in s.$ The aim is to predict $t_y\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{y}_i.$ The probabilities of selection are ${\pi }_i\mathrm{<mo>=</mo>}P\left[i\in s|x_1\mathrm{,}\dots ,x_N\mathrm{,}{y}_1\mathrm{,}\dots \mathrm{,}{y}_N\right]\mathrm{<mo>=</mo>}P\left[i\in s|x_1\mathrm{,}\dots ,x_N\right]\mathrm{<mo>></mo>}0;$ they are assumed to be a function of the population values of $x_i.$ Let $t_x\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{x}_i$ and $t_{xr}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in r}}{x}_i.$

The $n$ by $p$ matrix of sample values of $x,$ which has rows $x_i^T,$ is denoted $X_s.$ The $N-n$ by $p$ matrix of non-sample values of $x$ is $X_r.$ The vector of sample values of $y$ is $y_s.$

The following linear model $M$ is assumed:

$$E_M\left[y_i\right]\mathrm{<mo>=</mo>}{\beta }^T{x}_i(2.1)$$

$${\text{var}}_M\left[y_i\right]\mathrm{<mo>=</mo>}{\sigma }_i^2\mathrm{<mo>=</mo>}{\sigma }^2{v}_i(2.2)$$

$${\text{cov}}_M\left[y_i\mathrm{,}{y}_j\right]\mathrm{<mo>=</mo>}0(2.3)$$

for $i\mathrm{,}j\in U$ with $i\ne j.$ The subscripts $M$ in $E_M,$ ${\text{var}}_M$ and ${\text{cov}}_M$ indicate distributions over repeated realisations of the population values from the model. It is generally assumed that $v_i$ are known, i.e., the error variances are known up to a constant of proportionality. For example, in business surveys, $v_i$ might be a measure of business size, or the square root thereof. The unknown parameters are $\beta$ and ${\sigma }^2.$ The values of $x_i$ are considered to be fixed.

The best linear unbiased predictor (BLUP) (denoted ${\widehat{t}}_y)$ for a generalization of model $M$ is stated in Chapter 2 of Valliant et al. (2000). Its model-based prediction variance is

$${\text{var}}_M\left({\widehat{t}}_y-t_y\right)\mathrm{<mo>=</mo>}{t}_{xr}^T{\left({\displaystyle \sum _{i\in s}}{\sigma }_i^{-2}{x}_i{x}_i^T\right)}^{-1}{t}_{xr}+{\displaystyle \sum _{i\in r}}{\sigma }_i^2\mathrm{.}(2.4)$$

(This can be obtained as a special case of Result 2.2.2 on page 29 of Valliant et al., 2000.)

The anticipated variance is defined as $\text{AV}\left({\widehat{t}}_y-t_y\right)\mathrm{<mo>=</mo>}{E}_M{E}_p{\left({\widehat{t}}_y-t_y\right)}^2$ (Isaki and Fuller, 1982). As ${\widehat{t}}_y$ is model-unbiased, its AV is equal to

$$\text{AV}\mathrm{<mo>=</mo>}{E}_p{\text{var}}_M\left({\widehat{t}}_y-t_y\right)\mathrm{.}$$

Theorem 1 will derive an approximation for this AV. The asymptotic framework is based on the design-based asymptotics of Isaki and Fuller (1982). It is assumed that there is a countably infinite population $i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots .$ A sequence of finite populations $U_t$ is defined by $U_t\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\dots ,N_t\right\}$ where $N_1\mathrm{<mo><</mo>}{N}_2\mathrm{<mo><</mo>}\dots .$ For each $t,$ a sample $s_t$ of size $n_t$ is selected from $U_t$ by arbitrary probability design with probabilities of selection ${\pi }_{i\left(t\right)}\mathrm{<mo>=</mo>}P\left[i\in s_t\right].$ Following (2.7) of Isaki and Fuller (1982), it is assumed that

$$0\mathrm{<mo><</mo>}{\lambda }_1\mathrm{<mo><</mo>}{\pi }_{i\left(t\right)}\mathrm{<mo><</mo>}{\lambda }_2(2.5)$$

for some constants ${\lambda }_1$ and ${\lambda }_2.$ Isaki and Fuller (1982) note that AVs of estimators of total are typically $O\left(n_t\right)$ (which is equivalent to $O(N_t))$ and the totals themselves are also $O\left(n_t\right).$ Population means will be denoted as ${\overline{X}}_t\mathrm{<mo>=</mo>}{N}_t^{-1}{\displaystyle {\sum }_{U_t}}{x}_i$ and the inverse probability weighted estimator of $\overline{X}$ is ${\widehat{\overline{X}}}_{\pi }\mathrm{<mo>=</mo>}{N}_t^{-1}{\displaystyle {\sum }_{s_t}}{\pi }_{i\left(t\right)}^{-1}{x}_i$ (and similarly for $Y$ and other variables).

Two new variables are defined for each unit $i$ by $u_{i\left(t\right)}\mathrm{<mo>=</mo>}{\pi }_{i\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}{x}_i$ (a $p$ -vector) and $v_{i\left(t\right)}\mathrm{<mo>=</mo>}{\pi }_{i\left(t\right)}{\sigma }_i^{-2}{x}_i{x}_i^T$ (a $p$ by $p$ matrix). Their population means are ${\overline{U}}_t$ and ${\overline{V}}_t$ with inverse probability estimators ${\widehat{\overline{U}}}_{t\pi }$ and ${\widehat{\overline{V}}}_{t\pi }.$

Theorem 1. It is assumed that

$$\underset{t\to \infty }{lim}{E}_p\left\{{\widehat{\overline{U}}}_{t\pi }^T{\widehat{\overline{V}}}_{t\pi }^{-1}{\widehat{\overline{U}}}_{t\pi }-{\overline{U}}_t^T{\overline{V}}_t^{-1}{\overline{U}}_t\right\}\mathrm{<mo>=</mo>}0.(2.6)$$

Then

$$E_{p}va{r}_M\left({\widehat{t}}_y-t_y\right)\mathrm{<mo>=</mo>}\widetilde{AV}+o\left(n_t\right)\mathrm{.}(2.7)$$

where

$$\widetilde{AV}\mathrm{<mo>=</mo>}{\displaystyle \sum _U}\left(1-{\pi }_i\right)x_i^T{\left({\displaystyle \sum _U}{\pi }_i{\sigma }_i^{-2}{x}_i{x}_i^T\right)}^{-1}{\displaystyle \sum _U}\left(1-{\pi }_i\right)x_i+{\displaystyle \sum _U}\left(1-{\pi }_i\right){\sigma }_i^2.(2.8)$$

Notes on Theorem 1

• Assumption (2.6) is reminiscent of Result (3.24) of Isaki and Fuller (1982), but there is an important difference. In Isaki and Fuller (1982), unit variables depend only on $i,$ but here $u_{i\left(t\right)}$ and $v_{i\left(t\right)}$ depend on both $i$ and $t$ as they both have a factor ${\pi }_{i\left(t\right)}.$ However, ${\pi }_{i\left(t\right)}$ are bounded by (2.5), so the condition is plausible; it would not be if ${\pi }_{i\left(t\right)}$ could be arbitrarily close to zero.
• It is clear that assumption (2.6) is satisfied if ${\widehat{\overline{U}}}_{t\pi }$ and ${\widehat{\overline{V}}}_{t\pi }$ are consistent in design probability for ${\overline{U}}_t$ and ${\overline{V}}_t,$ and ${\widehat{\overline{V}}}_{t\pi }$ is invertible in a neighbourhood of ${\overline{V}}_t.$ As noted by Isaki and Fuller (1982) in a comment on their condition (3.12), a invertibility requirement of this sort seems reasonable “for any discussion of regression estimation”.

An upper bound for the asymptotic AV over all possible choices of the auxiliary vector $x_i$ will now be derived. This allows for uncertainty about which auxiliary variables will ultimately be included in the model, since this decision is typically only made after data is collected. For example, the full set of variables used in the design might or might not end up being in the model, or spline functions of covariates might be included with knots based on the sample data. Theorem 2 states the upper bound.

Theorem 2. Let $\widetilde{AV}$  be the asymptotic AV defined by (2.8). If ${\displaystyle {\sum }_U}{\pi }_i{\sigma }_i^{-2}{x}_i{x}_i^T$  is invertible and ${\pi }_i\mathrm{<mo>></mo>}0$  for all $i\in U,$  then

$$\widetilde{AV}\le {\displaystyle \sum _U}\left({\pi }_i^{-1}-1\right){\sigma }_i^2(2.9)$$

with strict equality if and only if there exists a $p$ -vector $\lambda$  such that

$$\left({\pi }_i^{-1}-1\right){\sigma }_i^2\mathrm{<mo>=</mo>}{\lambda }^T{x}_i(2.10)$$

for all $i\in U.$

The right hand side of (2.9) is the well known Godambe-Joshi lower bound (Godambe and Joshi, 1965) for the AV of design-unbiased estimators. Here it is an upper bound over model space for model-based BLUPs.

Suppose the total cost of running the survey is ${\displaystyle {\sum }_s}{C}_i$ plus fixed costs, where $C_i$ is the cost associated with surveying unit $i.$ Then the expected cost is $C_E\mathrm{<mo>=</mo>}{\displaystyle {\sum }_U}{C}_i{\pi }_i.$ The sample design which minimizes the upper bound in (2.9) subject to fixed cost is the PPC $\sigma$ design which has

$${\pi }_i\propto {\sigma }_i/\sqrt{C_i}(2.11)$$

(Steel and Clark, 2014 who generalize Särndal et al., 1992, page 452) to allow for unequal costs. Theorem 2 means that (2.11) is a minimax design when there is uncertainty about the form of the model. Note that only the first order inclusion probabilities affect the AV and the bound, but these do not fully specify the design. Samples can be selected using these inclusion probabilities in a variety of ways (Tillé, 2006), including balanced probability sampling (Nedyalkova and Tillé, 2012) which improves the robustness to model mis-specification.

The condition for equality, (2.10), is equivalent to a well known condition for the BLUP to be equal to the generalized regression estimator (formula 3 of Tam, 1988). Tam (1988) argued for the use of sample designs such that (2.10) is satisfied, such as PP $\sigma$ (provided that the model includes an intercept). Nedyalkova and Tillé (2008), building on a result from Royall (1992), showed that PP $\sigma$ is model-based-optimal under equal costs when both $v_i$ and $\sqrt{v_i}$ are linear functions of $x_i,$ a condition called explainable variances. Brewer et al. (1988) noted that (2.10) can also be satisfied if the estimation model includes an instrumental variable, which is a suitable function of the selection probabilities. However, there are many circumstances under which (2.10) is not satisfied, because some auxiliary variables are omitted from the final model, because multiple variables of interest have different variance structures (ruling out PP $\sigma ),$ because there are unequal costs, or because instrumental variables are eschewed due to the loss of efficiency they entail. Theorem 2 shows that the GJLB is an upper bound under these circumstances which are not covered by the results of these authors. The PPC $\sigma$ design in (2.11) is a minimax design in this more general setting.

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