A method to find an efficient and robust sampling strategy under model uncertainty
Section 2. Optimal strategy under the superpopulation model

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Let $U$ be a finite population of size $N$ with elements labeled $\left\{\mathrm{1,}\mathrm{2,}\dots ,k\mathrm{,}\dots ,N\right\}.$ Let $x_k\mathrm{<mo>=</mo>}\left(x_{1k}\mathrm{,}{x}_{2k}\mathrm{,}\dots \mathrm{,}{x}_{Jk}\right)$ be a known vector of values of $J$ auxiliary variables and $y_k$ the unknown value of a study variable associated to unit $k\in U.$ We are interested in the estimation of the total of $y,$ $t_y\mathrm{<mo>=</mo>}{\displaystyle {\sum }_U{y}_k}.$

Let $\Omega$ be the power set of $U.$ A sample is any subset $s\in \Omega$ and a sampling design is a probability distribution on $\Omega ,$ denoted by $P\left(S\mathrm{<mo>=</mo>}s\right)$ or simply $p\left(s\right).$ Let ${\pi }_k\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{s\ni k}p\left(s\right)}$ be the inclusion probability of $k$ and ${\pi }_{kl}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{s\supset \left\{k\mathrm{,}l\right\}}p\left(s\right)}$ the joint inclusion probability of $k$ and $l.$ A probability sampling design is a sampling design such that ${\pi }_k\mathrm{<mo>></mo>}0$ for all $k\in U.$

An estimator is a real valued function of the sample, ${\widehat{t}}_y\mathrm{<mo>=</mo>}{\widehat{t}}_y\left(S\right).$ By strategy we refer to the couple sampling design and estimator, $\left(p\left(\cdot \right)\mathrm{,}{\widehat{t}}_y\right).$

We consider only probability sampling designs with fixed sample size. As a convenient stepping stone we begin by considering unbiased linear estimators of the form

$${\widehat{t}}_y\mathrm{<mo>=</mo>}\left({\displaystyle \sum _U{z}_k}-{\displaystyle \sum _s\frac{z_k}{{\pi }_k}}\right)+{\displaystyle \sum _s\frac{y_k}{{\pi }_k}}\mathrm{<mo>=</mo>}{\displaystyle \sum _U{z}_k}+{\displaystyle \sum _s\frac{e_k}{{\pi }_k}}(2.1)$$

with $z_k$ arbitrary known constants and $e_k\mathrm{<mo>=</mo>}{y}_k-z_k.$ This estimator is called the difference estimator. The estimator defined in this way is said to be calibrated on $z$ as it satisfies ${\widehat{t}}_z\mathrm{<mo>=</mo>}{\displaystyle {\sum }_U{z}_k}.$ Note that if $z_k\mathrm{<mo>=</mo>}0$ for all $k\in U$ the estimator reduces to ${\widehat{t}}_y\mathrm{<mo>=</mo>}{\displaystyle {\sum }_s{y}_k}/{\pi }_k,$ that is, the Horvitz-Thompson estimator (Horvitz and Thompson, 1952). In later sections we focus on the generalized regression estimator (GREG).

The design Mean Squared Error (MSE) of the difference estimator is

$${\text{MSE}}_{\text{p}}\left({\widehat{t}}_y\right)\mathrm{<mo>=</mo>}{\text{MSE}}_{\text{p}}\left({\displaystyle \sum _s\frac{e_k}{{\pi }_k}}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _U{\displaystyle \sum _U\left({\pi }_{kl}-{\pi }_k{\pi }_l\right)\frac{e_k}{{\pi }_k}\frac{e_l}{{\pi }_l}}}\mathrm{.}(2.2)$$

As mentioned in the introduction, due to the non-existence of an optimal strategy under the design-based approach, often a superpopulation model, ${\xi }_0,$ is proposed and we search for an optimal strategy with respect to the anticipated mean-squared error,

$${\text{MSE}}_{{\xi }_0\text{p}}\left({\widehat{t}}_y\right)\mathrm{<mo>=</mo>}{\text{E}}_{{\xi }_0}{\text{MSE}}_{\text{p}}\left({\widehat{t}}_y\right)\mathrm{<mo>=</mo>}{\text{E}}_{{\xi }_0}{\text{E}}_p\left({\left({\widehat{t}}_y-t_y\right)}^2\right).(2.3)$$

We may assume that the $y$ -values are realizations of the following model, denoted ${\xi }_0,$

$$Y_k\mathrm{<mo>=</mo>}f\left(x_k|{\delta }_1\right)+{\epsilon }_k$$

with

$${\text{E}}_{{\xi }_0}\left({\epsilon }_k\right)\mathrm{<mo>=</mo>}\mathrm{0,}{\text{V}}_{{\xi }_0}\left({\epsilon }_k\right)\mathrm{<mo>=</mo>}{\sigma }_0^{2}g{\left(x_k|{\delta }_2\right)}^2\text{and}{\text{E}}_{{\xi }_0}\left({\epsilon }_k{\epsilon }_l\right)\mathrm{<mo>=</mo>}0\forall k\ne l(2.4)$$

where $\delta \mathrm{<mo>=</mo>}\left({\delta }_1\mathrm{,}{\delta }_2\right)$ is a vector of parameters, $f\mathrm{<mo>:</mo>}{R}^J\to R$ and $g\mathrm{<mo>:</mo>}{R}^J\to R^{+}.$ The random sample $s$ and the errors ${\epsilon }_k$ are assumed to be independent. Following Rosén (2000), the terms $f\left(x_k|{\delta }_1\right)$ and $g\left(x_k|{\delta }_2\right)\mathrm{<mo>></mo>}0$ will be called trend and spread, respectively. The term trend should not in general be understood in a temporal sense, rather it refers to the development of $y$ -values with $x.$

Note that under ${\xi }_0,$ $e_k$ in the difference estimator (2.1) is a random variable that represents the distance between the value of the study variable and $z_k,$ i.e., $e_k\mathrm{<mo>=</mo>}f\left(x_k|{\delta }_1\right)+{\epsilon }_k-z_k.$ Therefore ${\text{E}}_{{\xi }_0}{e}_k\mathrm{<mo>=</mo>}f\left(x_k|{\delta }_1\right)-z_k$ and ${\text{E}}_{{\xi }_0}{e}_k^2\mathrm{<mo>=</mo>}{\left(f\left(x_k|{\delta }_1\right)-z_k\right)}^2+{\sigma }_0^{2}g{\left(x_k|{\delta }_2\right)}^2.$ With some algebra, it can be seen from (2.2) and (2.3) that the anticipated MSE of the difference estimator becomes

$${\text{MSE}}_{{\xi }_0\text{p}}\left({\widehat{t}}_y\right)\mathrm{<mo>=</mo>}{\text{MSE}}_{\text{p}}\left({\displaystyle \sum _s\frac{f\left(x_k|{\delta }_1\right)-z_k}{{\pi }_k}}\right)+{\sigma }_0^2{\displaystyle \sum _U\left(\frac{1}{{\pi }_k}-1\right)g{\left(x_k|{\delta }_2\right)}^2}(2.5)$$

Nedyalkova and Tillé (2008) derive the anticipated MSE in a more general case.

Tillé and Wilhelm (2017) give the anticipated MSE of the Horvitz-Thompson estimator. The second term in (2.5) is the Godambe-Joshi lower bound (e.g., Särndal, Swensson and Wretman, 1992, page 453).

The anticipated MSE in (2.5) is the sum of two positive terms. It is easy to see that if

1. the estimator is calibrated on $z_k\mathrm{<mo>=</mo>}f\left(x_k|{\delta }_1\right)$ the first term vanishes and the anticipated MSE equals the Godambe-Joshi lower bound

$${\text{MSE}}_{{\xi }_0\text{p}}\left({\widehat{t}}_y\right)\mathrm{<mo>=</mo>}{\sigma }_0^2{\displaystyle \sum _U\left(\frac{1}{{\pi }_k}-1\right)g{\left(x_k|{\delta }_2\right)}^2}\mathrm{.}(2.6)$$

1. Furthermore, after imposing the fixed sample size restriction ${\displaystyle {\sum }_U{\pi }_k\mathrm{<mo>=</mo>}n},$ if
2. the design is such that ${\pi }_k\propto g\left(x_k|{\delta }_2\right),$ denoted $\pi \text{ps}\left({\delta }_2\right),$ the second term is minimized and we obtain

$${\text{MSE}}_{{\xi }_0\text{p}}^{\text{opt}}\left({\widehat{t}}_y\right)\mathrm{<mo>=</mo>}{\sigma }_0^2\left(\frac{1}{n}{\left({\displaystyle \sum _{U}g\left(x_k|{\delta }_2\right)}\right)}^2-{\displaystyle \sum _{U}g{\left(x_k|{\delta }_2\right)}^2}\right)\mathrm{.}$$

Conditions 1 and 2 suggest the specific roles of the design and the estimator in the sampling strategy. The estimator should “explain” the trend in the calibration sense of condition 1. The design should “explain” the spread. A strategy that satisfies conditions 1 and 2 simultaneously will be called optimal. In the same sense, any estimator and any design satisfying, respectively, condition 1 and 2, will be called optimal. As this strategy plays an important role in this paper, we will denote it by $\pi \text{ps}\left({\delta }_2\right)-\text{diff}\left({\delta }_1\right).$

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