Model-assisted sample design is minimax for model-based prediction
Section 4. Discussion

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The Godambe-Joshi lower bound is shown here to be an upper bound for the AVs of BLUPs based on a correct model. Simulation MSEs of BLUPs based on an adaptively chosen spline or linear model are consistently less than the GJLB or just above it, even when variances are mis-specified. The MSEs are well below the bound if an important design variable does not figure in the model.

The upper bound result relies on the BLUP being model-unbiased. BLUPs based on a badly mis-specified model had MSEs well above the bound in the simulation study. Choosing a working model with minimum BIC out of a class including spline models avoided this problem.

Once data are available, model-based inference conditions on the sample selected. Probability sampling nevertheless has many advantages even though it is not the basis of inference (e.g., Särndal et al., 1992; Valliant et al., 2013). At the design stage, the AV is then the most relevant objective, because it averages over all the possible samples which may then be selected. The upper bound derived here is relevant at the design stage because there would usually be considerable uncertainty about the form of the model which will ultimately be adopted (there may be exceptions when there are historical or related data to support specification of a model or where one is willing to trust that the true model lies in a class of polynomial or other specific alternative models).

A sensible strategy in practice would be:

1. Set ${\pi }_i$ so as to give low values for the upper bound ${\displaystyle {\sum }_{i\in U}}\left({\pi }_i^{-1}-1\right)v_i$ where the model variances are proportional to $v_i$ (or a weighted combination of the upper bounds for multiple variables of interest) while also respecting cost and practical considerations. If there is a single variable of interest, and unit costs are proportional to $C_i,$ then ${\pi }_i\propto \sqrt{v_i/C_i}$ is recommended as it is a minimax strategy.
2. Once the sample is selected and data are available, choose a regression model based on this data.
3. Estimate population totals using the BLUPs under the selected model.
4. This may or may not result in condition (2.10) being satisfied, depending on the costs $C_i$ and the auxiliary variables selected in the final model.

All optimal sample design results, whether model-based, design-based or model-assisted, rely on knowledge of the relative unit or stratum residual variances. This appears to be unavoidable. This paper helps when the form of the mean model is not known in advance by giving an upper bound over the space of models for the mean. There does not appear to be a correspondingly useful bound over possible variance models, so the form of the variance model must be guessed or assumed.

Acknowledgements

I would like to thank Stephen Haslett for his encouragement of methodological research amongst competing priorities. Thorough reviews by two referees and an associate editor substantially improved the paper. This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI Australia), an NCRIS enabled capability supported by the Australian Government.

Appendix

Proof of Theorem 1

From (2.4),

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

Making use of the definitions of ${\widehat{\overline{U}}}_{\pi }$ and ${\widehat{\overline{V}}}_{\pi },$ and assumption (2.6), the first term of (A.1) becomes

$$\begin{array}{ll}{E}_p\left\{t_{xr}^T{\left({\displaystyle \sum _{i\in s}}{\sigma }_i^{-2}{x}_i{x}_i^T\right)}^{-1}{t}_{xr}\right\}\hfill & =E_p\left\{{\left(N_t\overline{X}-N_t{\widehat{\overline{U}}}_{\pi }\right)}^T{\left(N_t{\widehat{\overline{V}}}_{\pi }\right)}^{-1}\left(N_t\overline{X}-N_t{\widehat{\overline{U}}}_{\pi }\right)\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}_t{E}_p\left\{{\left(\overline{X}-{\widehat{\overline{U}}}_{\pi }\right)}^T{\widehat{\overline{V}}}_{\pi }^{-1}\left(\overline{X}-{\widehat{\overline{U}}}_{\pi }\right)\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}_t\left\{{\left(\overline{X}-\overline{U}\right)}^T{\overline{V}}^{-1}\left(\overline{X}-\overline{U}\right)+o\left(1\right)\right\}.\text{(A}\text{.2)}\hfill \end{array}$$

The result follows immediately from (A.1) and (A.2).

Lemma 1: Let $a_1\mathrm{,}\dots \mathrm{,}{a}_n$ and $b_1\mathrm{,}\dots \mathrm{,}{b}_n$ be scalars where $b_i\mathrm{<mo>></mo>}0$ for all $i.$ Let $x_1\mathrm{,}\dots \mathrm{,}{x}_n$ be $p$ -vectors. Then

$${\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{a}_i{x}_i\right)}^T{\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{b}_i{x}_i{x}_i^T\right)}^{-1}\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{a}_i{x}_i\right)\le {\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{a}_i^2/b_i\text{(A}\text{.3)}$$

provided the matrix inverse exists. Equality in (A.3) obtains if and only if

$$a_i{b}_i^{-1}\mathrm{<mo>=</mo>}{\lambda }^T{x}_i\text{(A}\text{.4)}$$

for all $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}n$ for some $p$ -vector $\lambda .$

Proof of Lemma 1

Let $b\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n}{b}_i.$ Let $X$ be a discrete random variable taking on the values $a_i/b_i.$ Let $Y$ be a discrete random variable taking on the values $x_i,$ for $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}n.$ Let $P\left[Y\mathrm{<mo>=</mo>}{x}_i\mathrm{,}X\mathrm{<mo>=</mo>}{a}_i/b_i\right]\mathrm{<mo>=</mo>}{b}_i/b$ for $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}n.$ Write $M_1\le M_2$ if $M_1-M_2$ is negative semi-definite for any matrices $M_1$ and $M_2.$ Theorem 1 of Tripathi (1999) states that for any random vectors $X$ and $Y,$

$$E\left[X{Y}^T\right]{\left\{E\left[Y{Y}^T\right]\right\}}^{-1}E\left[Y{X}^T\right]\le E\left[X{X}^T\right]\text{(A}\text{.5)}$$

provided the matrix inverse exists. With my definition of $X$ and $Y,$ (A.5) becomes

$${\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{b}_i{b}^{-1}{a}_i{b}_i^{-1}{x}_i^T{\left\{{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{b}_i{b}^{-1}{x}_i{x}_i^T\right\}}^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{b}_i{b}^{-1}{a}_i{b}_i^{-1}{x}_i\le {\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{b}_i{b}^{-1}{a}_i^2{b}_i^{-2}\text{(A}\text{.6)}$$

which leads directly to (A.3). Tripathi (1999) states that the equality is sharp if

$$X^T{\lambda }_1+Y^T{\lambda }_2\mathrm{<mo>=</mo>}0\text{(A}\text{.7)}$$

with probability 1 for some ${\lambda }_1$ of the same dimension as $X$ and ${\lambda }_2$ of the same dimension as $Y.$ Here, (A.7) becomes

$$a_i{b}_i^{-1}{\lambda }_1\mathrm{<mo>=</mo>}{x}_i^T{\lambda }_2$$

for all $i,$ which is equivalent to (A.4).

Proof of Theorem 2

Let $a_i\mathrm{<mo>=</mo>}1-{\pi }_i$  and $b_i\mathrm{<mo>=</mo>}{\pi }_i{\sigma }_i^{-2}.$ From Lemma 1,

$$\begin{array}{ll}\widetilde{\text{AV}}\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _U}\left(1-{\pi }_i\right)x_i^T{\left({\displaystyle \sum _U}{\pi }_i{\sigma }_i^{-2}{x}_i{x}_i^T\right)}^{-1}{\displaystyle \sum _U}\left(1-{\pi }_i\right)x_i^T+{\displaystyle \sum _U}\left(1-{\pi }_i\right){\sigma }_i^2\hfill \\ \hfill & \le {\displaystyle \sum _U}{\left(1-{\pi }_i\right)}^2{\pi }_i^{-1}{\sigma }_i^2+{\displaystyle \sum _U}\left(1-{\pi }_i\right){\sigma }_i^2\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _U}\left(1-{\pi }_i\right){\pi }_i^{-1}{\sigma }_i^2\left(1-{\pi }_i+{\pi }_i\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _U}\left({\pi }_i^{-1}-1\right){\sigma }_i^2\hfill \end{array}$$

with strict equality if and only if

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

for some vector $\lambda .$

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