Bayesian spatial models for estimating means of sampled and non-sampled small areas
Section 3. Simulating posterior distributions

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In this section, we illustrate the rejection sampling steps to obtain independent posterior samples from the posterior distributions of proposed models. We assume that the components of the small area mean vector $\theta$ are arranged so that $\theta ={({\theta }_{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}^T\mathrm{,}{\theta }_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T\mathrm{<mo\; stretchy="false">)</mo>}}^T,$ where ${\theta }_{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}\in {ℝ}^{m_1}$ and ${\theta }_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\in {ℝ}^{m_2}$ are the small area mean vectors corresponding to the non-sampled and sampled areas, respectively. For notational convenience, we denote the precision matrix of the $k^{\text{th}}$ spatial model by $\Omega ={\mathrm{<mo\; stretchy="false">(</mo>}{\sigma }_v^2\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}{\Omega }_k\mathrm{<mo\; stretchy="false">(</mo>}\rho \mathrm{<mo\; stretchy="false">)</mo>}$ and the permissible range of $\rho$ by $\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{,}u\mathrm{<mo\; stretchy="false">)</mo>}$ suppressing the model index $k.$

We first derive the marginal posterior density of $\mathrm{<mo\; stretchy="false">(</mo>}{\sigma }_v^2\mathrm{,}\rho \mathrm{<mo\; stretchy="false">)</mo>}$ and provide subsequent sampling procedures. Let $0_{m_2\times m_1}$ be the $m_2\times m_1$ null matrix and $M=\mathrm{<mo\; stretchy="false">[</mo>}{0}_{m_2\times m_1}\mathrm{,}{I}_{m_2}\mathrm{<mo\; stretchy="false">]</mo>}$ such that ${\theta }_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}=M\theta .$ We also let $X_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}=MX.$ Integrating out $\theta$ from the model (2.11)-(2.12), we have $Y_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}|\beta \mathrm{,}{\sigma }_v^2\mathrm{,}\rho ~N_{m_2}\mathrm{<mo\; stretchy="false">(</mo>}{X}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\beta \mathrm{,}\Delta \mathrm{<mo\; stretchy="false">)</mo>},$ where $\Delta =D_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}+M{\Omega }^{-1}{M}^T.$ Subsequent marginalization of $\beta$ gives the marginal posterior density $p\mathrm{<mo\; stretchy="false">(</mo>}{\sigma }_v^2\mathrm{,}\rho |y_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}$ as

$$p\mathrm{<mo\; stretchy="false">(</mo>}{\sigma }_v^2\mathrm{,}\rho |y_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}\propto \frac{\exp \left[-\frac{1}{2}{y}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T{\Delta }^{-1}\left\{\Delta -X_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>}{X}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T{\Delta }^{-1}{X}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}{X}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T\right\}{\Delta }^{-1}{y}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\right]}{{\left|\Delta \right|}^{1/2}{\left|X_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T{\Delta }^{-1}{X}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\right|}^{1/2}}I\mathrm{<mo\; stretchy="false">(</mo>}l<\rho <u\mathrm{<mo\; stretchy="false">)</mo>.}(3.1)$$

Furthermore, we have conditional posterior distributions of $\beta$ and $\theta$ as

$$\beta |{\sigma }_v^2\mathrm{,}\rho \mathrm{,}y~N_p\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{,}\Gamma \mathrm{<mo\; stretchy="false">)</mo>,}(3.2)$$

$$\theta |\beta \mathrm{,}{\sigma }_v^2\mathrm{,}\rho \mathrm{,}y~N_m\mathrm{<mo\; stretchy="false">(</mo>}\mu \mathrm{,}\Psi \mathrm{<mo\; stretchy="false">)</mo>,}(3.3)$$

where $\Gamma ={\mathrm{<mo\; stretchy="false">(</mo>}{X}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T{\Delta }^{-1}{X}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}}^{-1},$ $\gamma =\Gamma X_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T{\Delta }^{-1}{y}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}},$ $\mu \mathrm{<mo>=</mo>}{y}_{*}-\Psi \Omega \mathrm{<mo\; stretchy="false">(</mo>}{y}_{*}-X\beta \mathrm{<mo\; stretchy="false">)</mo>},$ $y_{*}={\mathrm{<mo\; stretchy="false">(</mo>}{0}_{m_1}^T\mathrm{,}{y}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^T\mathrm{<mo\; stretchy="false">)</mo>}}^T,$ and ${\Psi }^{-1}=M^T{D}_{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}^{-1}M+\Omega .$ Accordingly, we can obtain a independent posterior sample via rejection sampling from (3.1) and subsequent samplings from (3.2) and (3.3). For the data with no non-sampled area, we have desired sampling procedures by setting $M=I_m$ and $y_{*}=y.$

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