Small area benchmarked estimation under the basic unit level model when the sampling rates are non‑negligible
Section 3. Benchmarked estimators

• Previous
• Next

We now proceed to develop benchmarked estimators of the small area means ${\overline{Y}}_i$ using unit level model (2.2) or augmented versions of it. We assume that a reliable direct estimator ${\widehat{Y}}_w={\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j\in s_i}{w}_{ij}{y}_{ij}}}$ of the population total $Y$ is available, where $Y={\displaystyle {\sum }_{i=1}^m{Y}_i},$ and $Y_i=N_i{\overline{Y}}_i$ is the total of small area $i.$ Let ${\widehat{\overline{Y}}}_i$ be the model-based small area estimator of ${\overline{Y}}_i.$ It is desirable to ensure that the aggregated values of ${\widehat{\overline{Y}}}_i,$ agree with the reliable estimator ${\widehat{Y}}_w.$ The small area means estimators ${\widehat{\overline{Y}}}_i,$ $i=1,\dots ,m,$ are said to be benchmarked to ${\widehat{Y}}_w$ if

$${\displaystyle \sum _{i=1}^m{N}_i{\widehat{\overline{Y}}}_i}={\widehat{Y}}_w.(3.1)$$

Let ${\widehat{Y}}_w$ be a GREG estimator with weights calibrated at the population level on a vector of auxiliary variables $x_{ij}^{*}.$ This estimator is analogous to the combined regression estimator if one views the small areas as strata. The vector of auxiliary variables $x_{ij}^{*}$ may or may not be the same as $x_{ij}.$ We distinguish two cases in this context: $x_{ij}\subseteq x_{ij}^{*}$ and $x_{ij}\not\subset x_{ij}^{*}.$ The first case, $x_{ij}\subseteq x_{ij}^{*},$ implies that all the components of $x_{ij}$ also belong to $x_{ij}^{*},$ and that $x_{ij}^{*}$ may or may not have additional components that are different from those contained in $x_{ij}.$ The second case, $x_{ij}\not\subset x_{ij}^{*},$ implies that some of the components of $x_{ij}$ do not appear in $x_{ij}^{*}.$ We assume that the first component of both vectors $x_{ij}$ and $x_{ij}^{*}$ are equal to one, as they represent an intercept term.

For a given sample $s,$ auxiliary data $x_{ij}^{*}$ and basic design weights $d_{ij}=1/{\pi }_{ij},$ the GREG estimator of the population total $Y$ is given by

$${\widehat{Y}}^{\text{GREG}}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{w}_{ij}^{\text{GREG}}{y}_{ij}}},$$

where the GREG weights $w_{ij}^{\text{GREG}}$ are given by

$$w_{ij}^{\text{GREG}}=d_{ij}\left(1+{\left(X^{*}-{\widehat{X}}^{*\text{HT}}\right)}^T{\left({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{d}_{ij}{x}_{ij}^{*}{x}_{ij}^{*T}}}\right)}^{-1}{x}_{ij}^{*}\right).(3.2)$$

In equation (3.2), $X^{*}={\displaystyle {\sum }_{i=1}^m{X}_i^{*}},$ where $X_i^{*}={\displaystyle {\sum }_{j=1}^{N_i}{x}_{ij}^{*}}$ represents the known small area total, whereas ${\widehat{X}}^{*\text{HT}}={\displaystyle {\sum }_{i=1}^m{\widehat{X}}_i^{*\text{HT}}}$ and ${\widehat{X}}_i^{*\text{HT}}={\displaystyle {\sum }_{j\in s_i}{d}_{ij}{x}_{ij}^{*}}$ represent respectively the direct design-based Horvitz-Thompson estimators of $X^{*}$ and $X_i^{*}.$ Note that

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{w}_{ij}^{\text{GREG}}{x}_{ij}^{*}}}=X^{*}.(3.3)$$

Using the GREG weights $w_{ij}^{\text{GREG}},$ estimators of $N_i$ and $X_i$ are given by

$${\widehat{N}}_i^{\text{GREG}}={\displaystyle \sum _{j\in s_i}{w}_{ij}^{\text{GREG}}}\text{and}{\widehat{X}}_i^{\text{GREG}}={\displaystyle \sum _{j\in s_i}{w}_{ij}^{\text{GREG}}{x}_{ij}}.(3.4)$$

The small area estimates ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ and ${\widehat{\overline{Y}}}_i^{\text{YR}}$ given respectively by (2.11) and (2.13), do not satisfy the benchmarking equation (3.1) for ${\widehat{Y}}_w={\widehat{Y}}^{\text{GREG}}\text{:}$ that is the total estimates ${\widehat{Y}}^{\text{EBLUP}}={\displaystyle {\sum }_{i=1}^m{N}_i{\widehat{\overline{Y}}}_i^{\text{EBLUP}}},$ and ${\widehat{Y}}^{\text{YR}}={\displaystyle {\sum }_{i=1}^m{N}_i{\widehat{\overline{Y}}}_i^{\text{YR}}}$ do not match the GREG estimator ${\widehat{Y}}^{\text{GREG}}.$ We need to adjust ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ and ${\widehat{\overline{Y}}}_i^{\text{YR}}$ so that the sum of these modified small area estimators add up to ${\widehat{Y}}^{\text{GREG}}$ when they are summed over all the $m$ small areas.

A very simple modification to the ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}’s$ and ${\widehat{\overline{Y}}}_i^{\text{YR}}’s$ is called ratio benchmarking. It consists of multiplying each ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ and ${\widehat{\overline{Y}}}_i^{\text{YR}}$ by the common adjustment factors ${\widehat{Y}}^{\text{GREG}}/{\displaystyle {\sum }_{i=1}^m{N}_i{\widehat{\overline{Y}}}_i^{\text{EBLUP}}}$ and ${\widehat{Y}}^{\text{GREG}}/{\displaystyle {\sum }_{i=1}^m{N}_i{\widehat{\overline{Y}}}_i^{\text{YR}}}$ respectively, leading to the ratio benchmarked estimators

$${\widehat{\overline{Y}}}_{ib}^{\text{EBRat}}={\widehat{\overline{Y}}}_i^{\text{EBLUP}}\frac{{\widehat{Y}}^{\text{GREG}}}{{\displaystyle {\sum }_{i=1}^m{N}_i{\widehat{\overline{Y}}}_i^{\text{EBLUP}}}}\text{and}{\widehat{\overline{Y}}}_{ib}^{\text{YRat}}={\widehat{\overline{Y}}}_i^{\text{YR}}\frac{{\widehat{Y}}^{\text{GREG}}}{{\displaystyle {\sum }_{i=1}^m{N}_i{\widehat{\overline{Y}}}_i^{\text{YR}}}}.(3.5)$$

It readily follows that both ${\widehat{\overline{Y}}}_{ib}^{\text{EBRat}}$ and ${\widehat{\overline{Y}}}_{ib}^{\text{YRat}}$ satisfy equation (3.1) with ${\widehat{Y}}_w={\widehat{Y}}^{\text{GREG}}.$ In equation (3.5) and hereafter the subscript $b$ denotes that the estimators are benchmarked to ${\widehat{Y}}^{\text{GREG}}.$

Note that the ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}’s$ and ${\widehat{\overline{Y}}}_i^{\text{YR}}’s$ in equation (3.5) are multiplied by the same factor regardless of their precision and ignoring the particular small area characteristics, such as the variability of the units within a small area, or the small area sample size. Consequently, the resulting benchmarked estimators, ${\widehat{\overline{Y}}}_{ib}^{\text{EBRat}}$ and ${\widehat{\overline{Y}}}_{ib}^{\text{YRat}},$ based on this simple procedure, are just proportional modifications of estimators ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ and ${\widehat{\overline{Y}}}_i^{\text{YR}}$ respectively, to obtain the desired concordance. This limitation can be avoided by using the small area model (2.2) to construct the benchmarked estimators. 

We now proceed to show how model (2.2) can be used to obtain estimators benchmarked to ${\widehat{Y}}^{\text{GREG}}.$ In Sections 3.1 and 3.2 we adapt the procedures in Stefan and Hidiroglou (2020) for obtaining benchmarked estimators to the case of non‑negligible sampling rates. In Sections 3.3 and 3.4 we introduce two restricted benchmarked estimators based on the procedure proposed by Ugarte et al. (2009). The benchmarked estimators of Sections 3.1 and 3.2 rely on the assumption that $x_{ij}\subseteq x_{ij}^{*},$ whereas the estimators of Sections 3.3 and 3.4 can be computed for any vector $x_{ij}$ or $x_{ij}^{*}.$

3.1   Augmented EBLUP benchmarked estimators

The GREG weights $w_{ij}^{\text{GREG}}$ should be used in the estimation to achieve benchmarking to ${\widehat{Y}}^{\text{GREG}}.$ A possible way that $w_{ij}^{\text{GREG}}$ can be incorporated in the estimation is by augmenting the small area model (2.2) with a suitable auxiliary variable that is a function of $w_{ij}^{\text{GREG}}.$ This procedure is based on the augmented model approach used by Wang et al. (2008), whereby estimates obtained using the FH area-level model could be forced to add up to specified totals. Stefan and Hidiroglou (2020) adapted the Wang et al. (2008) approach under the basic unit-level model and for negligible sampling rates. They showed that benchmarking to ${\widehat{Y}}^{\text{GREG}}$ could be obtained by augmenting model (2.2) with the GREG weights $w_{ij}^{\text{GREG}}.$ We extend Stefan and Hidiroglou (2020) to the case when the sampling rates are non‑negligible. For this case, benchmarking to ${\widehat{Y}}^{\text{GREG}}$ is achieved by augmenting model (2.2) with $q_{ij}=w_{ij}^{\text{GREG}}-1.$ This leads to the augmented model given by

$$y_{ij}=x_{ij}^T{\beta }_{1a}+q_{ij}{\beta }_{2a}+v_{ia}+e_{ija},i=1,\dots ,m;j\in s_i.(3.6)$$

The random effects $v_{ia}$ are assumed to be i.i.d. $N\left(0,{\sigma }_{va}^2\right)$ and independent of the unit errors $e_{ija},$ and the $e_{ija}’s$ are assumed to be i.i.d. $N\left(0,{\sigma }_{ea}^2\right).$ The EBLUP estimators of ${\beta }_a={\left({\beta }_{1a}^T,{\beta }_{2a}\right)}^T$ and $v_{ia}$ in (3.6) are respectively denoted by ${\widehat{\beta }}_a={\left({\widehat{\beta }}_{1a}^T,{\widehat{\beta }}_{2a}\right)}^T$ and ${\widehat{v}}_{ia}.$ We can now spell Result 1 for ${\widehat{\beta }}_a$ and ${\widehat{v}}_{ia}.$

Result 1. The EBLUP estimators ${\widehat{\beta }}_a$ and ${\widehat{v}}_{ia}$ based on model (3.6) obey the following equation

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{y}_{ij}}}+{\left({\displaystyle \sum _{i=1}^m{x}_{ir}}\right)}^T{\widehat{\beta }}_{1a}+{\displaystyle \sum _{i=1}^m{q}_{iw}{\widehat{\beta }}_{2a}}+{\displaystyle \sum _{i=1}^m\left({\widehat{N}}_i^{\text{GREG}}-n_i\right){\widehat{v}}_{ia}}={\widehat{Y}}^{\text{GREG}},(3.7)$$

where $q_{iw}={\displaystyle {\sum }_{j\in s_i}{q}_{ij}^2}={\displaystyle {\sum }_{j\in s_i}{(w_{ij}^{\text{GREG}}-1)}^2}.$

Proof: See Appendix A.

It follows from equation (3.7) that small area estimators benchmarked to ${\widehat{Y}}^{\text{GREG}}$ are given by 

$${\widehat{\overline{Y}}}_{iab}^{\text{EBLUP}}=\frac{1}{N_i}\left[{\displaystyle \sum _{j\in s_i}{y}_{ij}}+x_{ir}^T{\widehat{\beta }}_{1a}+q_{iw}{\widehat{\beta }}_{2a}+\left({\widehat{N}}_i^{\text{GREG}}-n_i\right){\widehat{v}}_{ia}\right].(3.8)$$

The subscript $a$ indicates that ${\widehat{\overline{Y}}}_{iab}^{\text{EBLUP}}$ is based on an augmented small area model.

3.2   You-Rao benchmarked estimators

The procedure proposed by You and Rao (2002) can be used with any survey weights $w_{ij}.$ However, there is no guarantee that the resulting YR estimator will be benchmarked to ${\widehat{Y}}^{\text{GREG}}.$ When the sampling rates are negligible, Stefan and Hidiroglou (2020) obtained benchmarked estimators with the You and Rao’s (2002) procedure based on the weights $w_{ij}=w_{ij}^{\text{GREG}}$ of the GREG estimator. When the sampling rates are non‑negligible, we now show that the weights $w_{ij}=w_{ij}^{\text{GREG}}-1$ lead to YR benchmarked estimators.

Let ${\widehat{\beta }}^{\text{YR}}$ and ${\widehat{v}}^{\text{YR}}={\left({\widehat{v}}_1^{\text{YR}},\dots ,{\widehat{v}}_m^{\text{YR}}\right)}^T$ be YR estimators of $\beta$ and $v$ respectively with $w_{ij}$ replaced by $w_{ij}^{\text{GREG}}-1.$ Using ${\widehat{\beta }}^{\text{YR}},$ ${\widehat{v}}_i^{\text{YR}}$ and the $N_i-n_i$ estimates ${\widehat{y}}_{ij}^{\text{YR}}=x_{ij}^T{\widehat{\beta }}^{\text{YR}}+{\widehat{v}}_i^{\text{YR}}$ for $j\in r_i,$ a YR estimator, denoted as ${\widehat{\overline{Y}}}_i^{\text{YR}},$ can be computed with equation (2.13). However, ${\widehat{\overline{Y}}}_i^{\text{YR}}$ is not benchmarked to ${\widehat{Y}}^{\text{GREG}}$ even if it uses the weights $w_{ij}^{\text{GREG}}-1.$ The original YR procedure leads to a self-benchmarked estimator in a limited number of cases.

To achieve the benchmark to ${\widehat{Y}}^{\text{GREG}},$ a YR modified estimator, denoted as ${\widehat{\overline{Y}}}_{ib}^{\text{YR}},$ is defined as follows:

$${\widehat{\overline{Y}}}_{ib}^{\text{YR}}=\frac{1}{N_i}\left[{\displaystyle \sum _{j\in s_i}{y}_{ij}}+x_{ir}^T{\widehat{\beta }}^{\text{YR}}+\left({\widehat{N}}_i^{\text{GREG}}-n_i\right){\widehat{v}}_i^{\text{YR}}\right].(3.9)$$

The following proves that ${\widehat{\overline{Y}}}_{ib}^{\text{YR}}$ defined by (3.9) benchmarks to ${\widehat{Y}}^{\text{GREG}}.$

Result 2. Let ${\widehat{\beta }}^{\text{YR}}$ and ${\widehat{v}}^{\text{YR}}={\left({\widehat{v}}_1^{\text{YR}},\dots ,{\widehat{v}}_m^{\text{YR}}\right)}^T$ be respectively the YR estimators of $\beta$ and $v,$ constructed with weights $w_{ij}^{\text{GREG}}-1.$ Then, $\left({\widehat{\beta }}^{\text{YR}},{\widehat{v}}^{\text{YR}}\right)$ satisfy the following equation:

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{y}_{ij}}}+{\displaystyle \sum _{i=1}^m{x}_{ir}^T{\widehat{\beta }}^{\text{YR}}}+{\displaystyle \sum _{i=1}^m\left({\widehat{N}}_i^{\text{GREG}}-n_i\right){\widehat{v}}_i^{\text{YR}}}={\widehat{Y}}^{\text{GREG}}.$$

Proof: See Appendix A.

Given $x_{ij}^{*},$ the weights $w_{ij}^{\text{GREG}}$ are calibrated on $x_{ij}^{*}$ at the small area level if they satisfy the following equations

$${\displaystyle \sum _{j\in s_i}{w}_{ij}^{\text{GREG}}{x}_{ij}^{*}}=X_i^{*},\text{for}i=1,\dots ,m.(3.10)$$

Equations (3.10) implies equation (3.3), however, the reverse is not true. If the weights $w_{ij}^{\text{GREG}}$ satisfy (3.10), and since $x_{ij}\subseteq x_{ij}^{*},$ it follows that the weights $w_{ij}^{\text{GREG}}$ are also calibrated on $x_{ij}$ at the small area level. In turn, this implies that ${\widehat{N}}_i^{\text{GREG}}=N_i,$ as we assume that vector $x_{ij}$ contains the constant regressor equal to 1. It follows that ${\widehat{\overline{Y}}}_i^{\text{YR}}={\widehat{\overline{Y}}}_{ib}^{\text{YR}}.$ Thus, the YR estimator ${\widehat{\overline{Y}}}_i^{\text{YR}}$ constructed with $w_{ij}^{\text{GREG}}-1$ is self-benchmarked to ${\widehat{Y}}^{\text{GREG}}$ in the special case when the GREG weights are calibrated at the small area level (see You and Rao, 2002).

3.3   Restricted EBLUP benchmarked estimator

In Section 2 we showed that the EBLUP estimators of $\left(\beta ,v\right)$ can be obtained if the function $\varphi$ defined in (2.5) is minimized with respect to $\left(\beta ,v\right).$ It therefore follows that an EBLUP estimator can be viewed as the solution to an unrestricted minimization problem. The idea of restricted EBLUP estimators is to obtain new estimators of $\left(\beta ,v\right)$ by minimizing $\varphi$ subject to the restriction given by the benchmark condition. The procedure was used by Pfeffermann and Barnard (1991) under the FH area-level model. More recently, Ugarte et al. (2009) applied the procedure under the BHF unit-level model to obtain benchmarking to a synthetic estimator. Ugarte et al. (2009) described the restricted estimator as a generalized least squares estimator subject to a restriction by noticing that the minimization can be conducted as in the econometrics theory of regression estimation under linear constraints. We now describe the procedure in Ugarte et al. (2009).

We denote by ${\widehat{\beta }}^R$ and ${\widehat{v}}^R={\left({\widehat{v}}_1^R,\dots ,{\widehat{v}}_m^R\right)}^T$ the new restricted EBLUP estimators of $\left(\beta ,v\right).$ Then, the restricted EBLUP estimator of ${\overline{Y}}_i,$ denoted as ${\widehat{\overline{Y}}}_{ib}^{\text{REBLUP}},$ is given by equation (2.4), where ${\widehat{y}}_{ij}$ are replaced by ${\widehat{y}}_{ij}^R=x_{ij}^T{\widehat{\beta }}^R+{\widehat{v}}_i^R,$ for $j\in r_i.$ We impose that the estimators ${\widehat{\overline{Y}}}_{ib}^{\text{REBLUP}},i=1,\dots ,m$ be benchmarked to ${\widehat{Y}}^{\text{GREG}},$ that is they satisfy equation (3.1) with ${\widehat{Y}}_w={\widehat{Y}}^{\text{GREG}}.$ After carrying out some algebra, it can be shown that the benchmark to ${\widehat{Y}}^{\text{GREG}}$ of estimators ${\widehat{\overline{Y}}}_{ib}^{\text{REBLUP}},i=1,\dots ,m$ is equivalent to the following linear constraint equation

$$a_1^T{\widehat{\beta }}^R+a_2^T{\widehat{v}}^R={\widehat{Y}}_r^{\text{GREG}},(3.11)$$

where $a_1={\displaystyle {\sum }_{i=1}^m{x}_{ir}},$ $a_2={\left(N_1-n_1,\dots ,N_m-n_m\right)}^T,$ $Y_r=Y-{\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j\in s_i}{y}_{ij}}}$ is the total of non-observed $y_{ij}$ values with $i=1,\dots ,m;j\in r_i,$ and ${\widehat{Y}}_r^{\text{GREG}}={\widehat{Y}}^{\text{GREG}}-{\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j\in s_i}{y}_{ij}}}$ is an estimator of $Y_r$ based on ${\widehat{Y}}^{\text{GREG}}.$ The restricted EBLUP estimators $\left({\widehat{\beta }}^R,{\widehat{v}}^R\right)$ are therefore obtained as the solution to the minimization of function $\varphi$ given by (2.5) subject to the linear constraint (3.11).

The Lagrange multiplier method can be used to solve the constrained minimization of $\varphi .$ After straightforward algebra, it can be shown that estimators $\left({\widehat{\beta }}^R,{\widehat{v}}^R\right)$ are given by

$$\left(\begin{array}{l}{\widehat{\beta }}^R\\ {\widehat{v}}^R\end{array}\right)=\left(\begin{array}{l}\widehat{\beta }\\ \widehat{v}\end{array}\right)+\frac{1}{a^T\widehat{A}a}{\widehat{A}}^{-1}a\left[{\widehat{Y}}_r^{\text{GREG}}-a^T\left(\begin{array}{l}\widehat{\beta }\\ \widehat{v}\end{array}\right)\right],(3.12)$$

where $\left(\widehat{\beta },\widehat{v}\right)$ are the (unconstrained) EBLUP estimators of $\left(\beta ,v\right),$ $\widehat{A}$ is the empirical version of matrix $A$ defined in (2.7), and $a={\left(a_1^T{a}_2^T\right)}^T.$ Then, using ${\widehat{y}}_{ij}^R$ in (2.4), the estimator ${\widehat{\overline{Y}}}_{ib}^{\text{REBLUP}}$ can be rewritten as

$${\widehat{\overline{Y}}}_{ib}^{\text{REBLUP}}=\frac{1}{N_i}\left[{\displaystyle \sum _{j\in s_i}{y}_{ij}}+x_{ir}^T{\widehat{\beta }}^R+\left(N_i-n_i\right){\widehat{v}}_i^R\right].(3.13)$$

Remark 2. The matrix $\widehat{A}$ does not exist for samples when ${\widehat{\sigma }}_v^2=0.$ In such cases, we noticed that equation (2.8) cannot be used to compute the unconstrained estimators $\left(\widehat{\beta },\widehat{v}\right).$ However $\left(\widehat{\beta },\widehat{v}\right)$ can still be computed when ${\widehat{\sigma }}_v^2=0$ because the alternative equation (2.9) can be used for $\left(\widehat{\beta },\widehat{v}\right)$ . Equation (3.12) clearly shows that the constrained $\left({\widehat{\beta }}^R,{\widehat{v}}^R\right)$ cannot be computed for samples when estimator ${\widehat{\sigma }}_v^2$ is truncated to zero, and no alternative equation exists in these cases.

It, therefore, follows that the methods of estimation for the variance components commonly used in SAE cannot be used to compute the restricted EBLUP estimator. In Section 3.4 and Appendix B we describe an alternative method that produces a strictly positive estimation of ${\sigma }_v^2$ that can be applied in conjunction with $\left({\widehat{\beta }}^R,{\widehat{v}}^R\right)$ such that a restricted benchmarked estimator of ${\overline{Y}}_i$ always exists.

3.4   Restricted You-Rao benchmarked estimator

We showed in Section 2.2 that YR estimators of $\beta$ and $v$ can be obtained as a solution to mixed model equations obtained by minimizing the sample weighted function ${\varphi }_w$ given by (2.14). That is, we showed that, by defining a function ${\varphi }_w$ with weights $\left\{w_{ij}\right\},i=1,\dots ,m;j\in s_i$ and $\left\{{\omega }_i\right\},i=1,\dots ,m,$ and then minimizing ${\varphi }_w,$ we obtain the same estimators as those given by the You and Rao’s (2002) procedure. We now minimize function ${\varphi }_w$ under the benchmark constraint given by (3.11). The result is a restricted YR estimator that is benchmarked to ${\widehat{Y}}^{\text{GREG}}.$

Minimization of ${\varphi }_w$ given the benchmark restriction (3.11) results in estimators of ${\overline{Y}}_i,i=1,\dots ,m$ that are guaranteed to be benchmarked for any weights that define the function ${\varphi }_w.$ Thus, one may choose any set of weights $w_{ij}$ in ${\varphi }_w.$ In a limited design-based simulation study, we compared three restricted YR estimators based on three options with respect to $w_{ij}\text{:}$ i. $w_{ij}=w_{ij}^{\text{GREG}}-1;$ ii. $w_{ij}=w_{ij}^{\text{GREG}}$ and iii. $w_{ij}=d_{ij}.$ We found no significant difference between these three estimators in terms of design mean squared error. Given this last point and that the unrestricted benchmarked YR estimators described in Section 3.2 were based on $w_{ij}=w_{ij}^{\text{GREG}}-1,$ we chose to define the restricted YR estimator based on these weights.

Let ${\varphi }_w$ be defined in terms of $w_{ij}=w_{ij}^{\text{GREG}}-1$ and ${\omega }_i={\displaystyle {\sum }_{j\in s_i}{w}_{ij}^2}/{\displaystyle {\sum }_{j\in s_i}{w}_{ij}}.$ Minimization of ${\varphi }_w$ with respect to $\left(\beta ,v\right)$ subject to the benchmark constraint (3.11) results in the restricted YR estimators of $\left(\beta ,v\right),$ denoted as $\left({\widehat{\beta }}^{\text{RYR}},{\widehat{v}}^{\text{RYR}}\right).$ They are given by:

$$\left(\begin{array}{l}{\widehat{\beta }}^{\text{RYR}}\\ {\widehat{v}}^{\text{RYR}}\end{array}\right)=\left(\begin{array}{l}{\widehat{\beta }}^{\text{YR}}\\ {\widehat{v}}^{\text{YR}}\end{array}\right)+\frac{1}{a^T{\widehat{A}}_{w}a}{\widehat{A}}_w^{-1}a\left[{\widehat{Y}}_r^{\text{GREG}}-a^T\left(\begin{array}{l}{\widehat{\beta }}^{\text{YR}}\\ {\widehat{v}}^{\text{YR}}\end{array}\right)\right],(3.14)$$

where estimators $\left({\widehat{\beta }}^{\text{YR}},{\widehat{v}}^{\text{YR}}\right)$ are given by (2.15), and ${\widehat{A}}_w$ is the empirical version of $A_w$ given by (2.16). Using ${\widehat{\beta }}^{\text{RYR}}$ and ${\widehat{v}}_i^{\text{RYR}}$ of ${\widehat{v}}^{\text{RYR}}={\left({\widehat{v}}_1^{\text{RYR}},\dots ,{\widehat{v}}_m^{\text{RYR}}\right)}^T,$ restricted YR estimates ${\widehat{y}}_{ij}^{\text{RYR}}=x_{ij}^T{\widehat{\beta }}^{\text{RYR}}+{\widehat{v}}_i^{\text{RYR}}$ of unobserved $y_{ij}$ for $j\in r_i$ are then used to compute a benchmarked restricted YR estimator:

$${\widehat{\overline{Y}}}_{ib}^{\text{RYR}}=\frac{1}{N_i}\left[{\displaystyle \sum _{j\in s_i}{y}_{ij}}+x_{ir}^T{\widehat{\beta }}^{\text{RYR}}+\left(N_i-n_i\right){\widehat{v}}_i^{\text{RYR}}\right].(3.15)$$

As in the case of the restricted EBLUP estimator, the estimators $\left({\widehat{\beta }}^{\text{RYR}},{\widehat{v}}^{\text{RYR}}\right)$ given by (3.14) do not exist if FC, ML or REML results in a truncated estimate for ${\sigma }_v^2.$ Consequently, ${\overline{Y}}_i$ can only be estimated by ${\widehat{\overline{Y}}}_{ib}^{\text{RYR}}$ with a method of estimation for the variance components that always leads to strictly positive estimates for ${\sigma }_v^2.$

A null estimate of ${\sigma }_v^2$ poses no problem in computing EBLUP and YR estimators. However, we noticed that the restricted EBLUP and the restricted YR estimators cannot be computed if ${\widehat{\sigma }}_v^2=0.$ In order to get around this problem, we use a method proposed by Moghtased-Azar, Tehranchi and Amiri-Simkooei (2014) that guarantees that the estimator of ${\sigma }_v^2$ will be strictly positive. This method is based on the concept of a re-parameterized restricted maximum likelihood estimation (reREML). Their idea is to use functions whose range is the set of all positive real numbers, namely positive-valued functions (PVFs), for unknown variance components in the stochastic model instead of using variance components themselves. Their numerical results showed the successful estimation of non-negativity estimation of variance components (as positive values) as well as covariance components (as negative or positive values).

We used a Fisher-scoring algorithm to obtain iteratively the reREML estimates of the variance components of the basic unit-level model given by (2.2) (see Appendix B for details). We also carried out a small simulation and found out that for area sample sizes equal to or larger than 3, the Fisher-scoring algorithm converged in less than 15 iterations. When we only considered the samples that produced a null estimate ${\widehat{\sigma }}_v^2=0,$ we observed that the algorithm converged even faster (see Figure 4.1 in Section 4).

• Previous
• Article main page
• Next