Model-assisted optimal allocation for planned domains using composite estimation 2. Composite estimationModel-assisted optimal allocation for planned domains using composite estimation 2. Composite estimation

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Composite estimators for small areas are defined as convex combinations of direct (unbiased) and synthetic (biased) estimators. A simple example is the composition $\left(1-{\varphi }_h\right){\overline{y}}_h+{\varphi }_h\overline{y}$ of the sample mean ${\overline{y}}_h$ for the target area $h$ and the overall sample mean $\overline{y}$ of the target variable. The coefficients ${\varphi }_h$ are set with the intent to minimise its mean squared error (MSE), see for example Rao (2003, Section 4.3). The coefficients by which the MSE is minimized depend on some unknown parameters which have to be estimated.

Better results can be obtained if there are some regressors $x_i,$ for which domain population means are available, as well as sample data at either unit or domain level enabling $Y$ to be regressed on $x.$ A synthetic estimator for domain $h$ is then defined by ${\widehat{\overline{Y}}}_{h\left(\text{syn}\right)}={\widehat{\beta }}^T{\overline{X}}_h,$ where $\widehat{\beta }$ is the estimated regression coefficient, and ${\overline{X}}_h$ is the domain population mean of the regressor variables. An efficient direct estimator which is particularly suitable when domain sizes may be small is ${\overline{y}}_{hr}={\overline{y}}_h+{\widehat{\beta }}^T\left({\overline{x}}_h-{\overline{X}}_h\right)$ (Hidiroglou and Patak 2004) where ${\overline{y}}_h$ and ${\overline{x}}_h$ are the domain $h$ sample means of $Y$ and $X.$ A composite estimator can then be constructed as ${\tilde{y}}_h^{\mathcal{C}}=\left(1-{\varphi }_h\right){\overline{y}}_{hr}+{\varphi }_h{\widehat{\beta }}^T{\overline{X}}_h\mathrm{.}$

The design-based MSE of the composite estimator is given by:

$${\text{MSE}}_p\left({\tilde{y}}_h^{\mathcal{C}};{\overline{Y}}_h\right)={\left(1-{\varphi }_h\right)}^2{v}_{hr}+{\varphi }_h^2\left\{v_{h\left(\text{syn}\right)}+B_h^2\right\}+2{\varphi }_h\left(1-{\varphi }_h\right)c_h$$

where $c_h$ is the sampling covariance of ${\overline{y}}_{hr}$ and ${\widehat{\overline{Y}}}_{h\left(\text{syn}\right)},v_{hr}$ is the sampling variance of the direct estimator ${\overline{y}}_{hr},v_{h\left(\text{syn}\right)}$ is the sampling variance of the synthetic estimator ${\widehat{\overline{Y}}}_{h\left(\text{syn}\right)},$ and $B_h={\beta }_U^T{\overline{X}}_h-{\overline{Y}}_h$ is the bias of using ${\widehat{\overline{Y}}}_{h\left(\text{syn}\right)}$ to estimate ${\overline{Y}}_h,$ with ${\beta }_U$ denoting the approximate design-based expectation of $\widehat{\beta }.$ Further,

$${\text{MSE}}_p\left({\tilde{y}}_h^{\mathcal{C}};{\overline{Y}}_h\right)\approx {\left(1-{\varphi }_h\right)}^2{v}_{h\left(\text{syn}\right)}+{\varphi }_h^2{B}_h^2(2.1)$$

because $c_h\ll v_h$ and $v\ll v_h$ when the number of small areas is large, under regularity conditions.

A two-level linear model $\xi$ conditional on the values of $x$ will be assumed, with uncorrelated stratum random effects $u_h$ and unit residuals ${\epsilon }_i:$

$$\begin{array}{r}\hfill Y_i={\beta }^T{x}_i+u_h+{\epsilon }_i\\ \hfill E_{\xi }\left[u_h\right]=E_{\xi }\left[{\epsilon }_i\right]=0\\ \hfill {\text{var}}_{\xi }\left[u_h\right]={\sigma }_{uh}^2\\ \hfill {\text{var}}_{\xi }\left[{\epsilon }_j\right]={\sigma }_{eh}^2\end{array}\}(2.2)$$

for $h\in U^1$ and $i\in U_h.$ This implies that ${\text{var}}_{\xi }\left[Y_i\right]={\sigma }_{uh}^2+{\sigma }_{eh}^2={\sigma }_h^2$ for all $i\in U,$ and that the covariance ${\text{cov}}_{\xi }\left[Y_i\mathrm{,}{Y}_j\right]$ equals ${\rho }_h{\sigma }_h^2$ for units $i\ne j$ in the same strata and 0 for units from different strata, where ${\rho }_h={\sigma }_{uh}^2/\left({\sigma }_{uh}^2+{\sigma }_{eh}^2\right).$ For simplicity, it will be assumed that ${\rho }_h=\rho$ are equal for all strata.

Under model (2.1),

$$\begin{array}{ll}{E}_{\xi }\left[v_{hr}\right]\hfill & =E_{\xi }\left[n_h^{-1}{S}_{hw}^2\right]=n_h^{-1}{\sigma }_h^2\left(1-\rho \right)\hfill \end{array}$$

where $S_{hw}^2$ is the within-stratum-h sample variance of $y_i-{\beta }_U^T{x}_i;$ and

$$\begin{array}{ll}{E}_{\xi }\left[B_h^2\right]\hfill & =E_{\xi }\left[{\left({\overline{Y}}_h-{\overline{Y}}_{h\left(\text{syn}\right)}\right)}^2\right]\approx E_{\xi }\left[{\left({\overline{Y}}_h-{\beta }^T{\overline{X}}_h\right)}^2\right]\hfill \\ \hfill & ={\text{var}}_{\xi }\left[{\overline{Y}}_h\right]={\sigma }_h^2{N}_h^{-1}\left[1+\left(N_h-1\right)\rho \right]\mathrm{.}\hfill \end{array}$$

To simplify expressions, we assume that $n,N_h$ and $H$ are all large, although we do not derive rigorous asymptotic results. Assuming that $N_h$ is large, we firstly obtain $E_{\xi }\left[B_h^2\right]\approx {\sigma }_h^2\rho$ . Substituting for $E_{\xi }\left[v_{hr}\right]$ and $E_{\xi }\left[B_h^2\right]$ into (2.1) we get the anticipated MSE or approximate model assisted mean squared error, denoted ${\text{AMSE}}_h:$

$${\text{AMSE}}_h=E_{\xi }{\text{MSE}}_p\left({\tilde{y}}_h^{\mathcal{C}};{\overline{Y}}_h\right)\approx {\left(1-{\varphi }_h\right)}^2{n}_h^{-1}{\sigma }_h^2\left(1-\rho \right)+{\varphi }_h^2{\sigma }_h^2\rho .(2.3)$$

Optimizing with respect to ${\varphi }_h$ we immediately obtain the optimal weight ${\varphi }_h$ as:

$$\begin{array}{l}{\varphi }_{h\left(\text{opt}\right)}=\left(1-\rho \right){\left[1+\left(n_h-1\right)\rho \right]}^{-1}\mathrm{.}\hfill \end{array}(2.4)$$

We substitute the optimum weight (2.4) into (2.3) to obtain the approximate optimum anticipated MSE:

$$\begin{array}{ll}{\text{AMSE}}_h\hfill & =E_{\xi }{\text{MSE}}_p\left({\tilde{y}}_h^{\mathcal{C}}\left[{\varphi }_{h\left(\text{opt}\right)}\right];{\overline{Y}}_h\right)\hfill \\ \hfill & \approx {\left(n_h\rho {\left[1+\left(n_h-1\right)\rho \right]}^{-1}\right)}^2{n}_h^{-1}{\sigma }^2\left(1-\rho \right)+{\left(\left(1-\rho \right){\left[1+\left(n_h-1\right)\rho \right]}^{-1}\right)}^2{\sigma }^2\rho \hfill \\ \hfill & ={\sigma }_h^2\rho \left(1-\rho \right){\left[1+\left(n_h-1\right)\rho \right]}^{-1}\mathrm{.}\hfill \end{array}$$

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