Bayesian benchmarking of the Fay-Herriot model using random deletion
Section 2. Bayesian Fay-Herriot model

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Assume that the observed data are $\left({\widehat{\theta }}_i\mathrm{,}{s}_i\right)\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ where ${\widehat{\theta }}_i$ and $s_i$ are respectively an estimate and its standard error (for simplicity, assumed known) of a quantity under study, e.g., the $i^{\text{th}}$ area total ${\theta }_i.$ The BFH model is

$${\widehat{\theta }}_i|{\theta }_i\stackrel{\text{ind}}{{\displaystyle \sim }}\text{Normal}\left({\theta }_i\mathrm{,}{s}_i^2\right)\mathrm{,}$$

$${\theta }_i|\beta \mathrm{,}{\sigma }^2\stackrel{\text{ind}}{{\displaystyle \sim }}\text{Normal}\left(x_i^{\prime }\beta ,{\sigma }^2\right)\mathrm{,}$$

$$\pi \left(\beta ,{\sigma }^2\right)\mathrm{,}(2.1)$$

where $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ $x_i$ are a set of covariates with $p$ components (including intercept) and $\pi \left(\beta ,{\sigma }^2\right)$ is the joint prior distribution for $\left(\beta ,{\delta }^2\right).$ A priori it is assumed that $\pi \left(\beta ,{\sigma }^2\right)=\pi \left(\beta \right)\pi \left({\sigma }^2\right),$ i.e., $\beta$ and ${\sigma }^2$ are independent, with

$$\pi \left(\beta \right)\propto 1\text{and}\pi \left({\sigma }^2\right)\mathrm{<mo>=</mo>}1/{\left(1+{\sigma }^2\right)}^2.(2.2)$$

By Bayes’ theorem, the joint posterior density of $l\mathrm{<mo>+</mo>}p+1$ model parameters is

$$\pi \left(\theta \mathrm{,}\beta \mathrm{,}{\sigma }^2|\widehat{\theta }\right)\propto \frac{1}{{\left(1+{\sigma }^2\right)}^2}{\left(\frac{1}{{\sigma }^2}\right)}^{l/2}{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^l}\left\{\exp \left[-\frac{1}{2}\left\{\frac{1}{s_i^2}{\left({\widehat{\theta }}_i-{\theta }_i\right)}^2+\frac{1}{{\sigma }^2}{\left({\theta }_i-x_i^{\prime }\beta \right)}^2\right\}\right]\right\}\mathrm{.}(2.3)$$

In (2.2), $\pi \left({\sigma }^2\right)$ is a proper prior distribution, flatter than $\pi \left({\sigma }^2\right)\propto 1/{\sigma }^2$ near zero, with no moments. In fact, any proper prior for ${\sigma }^2$ is fine; an improper prior on ${\sigma }^2$ may lead to improper posterior density. Because $\pi \left(\beta \right)$ is improper, the product $\pi \left(\beta ,{\sigma }^2\right)=\pi \left(\beta \right)\pi \left({\sigma }^2\right)$ is improper, and this could cause the joint posterior density of $\left(\theta ,\beta ,{\sigma }^2\right)$ to be improper, an undesirable scenario. Theorem 1 below establishes the propriety of the joint posterior density (2.3).

More details about the BFH model are presented in Appendix B. Specifically, letting ${\lambda }_i\mathrm{<mo>=</mo>}\frac{{\sigma }^2}{s_i^2+{\sigma }^2}\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ we have shown that

$${\theta }_i|\beta \mathrm{,}{\sigma }^2\mathrm{,}\widehat{\theta }\stackrel{\text{ind}}{{\displaystyle \sim }}\text{Normal}\left\{{\lambda }_i{\widehat{\theta }}_i+\left(1-{\lambda }_i\right)x_i^{\prime }\beta \mathrm{,}\left(1-{\lambda }_i\right){\sigma }^2\right\}\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l\mathrm{,}(2.4)$$

$$\beta |{\sigma }^2\mathrm{,}\widehat{\theta }\sim \text{Normal}\left(\widehat{\beta },\widehat{\Sigma }\right)\mathrm{,}(2.5)$$

$${\pi }_3\left({\sigma }^2|\widehat{\theta }\right)\propto Q\left({\sigma }^2\right)\frac{1}{{\left(1+{\sigma }^2\right)}^2}\mathrm{,}(2.6)$$

where

$$\widehat{\beta }\mathrm{<mo>=</mo>}\widehat{\Sigma }{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^l}\frac{{\widehat{\theta }}_i{x}_i}{s_i^2+{\sigma }^2}\mathrm{,}{\widehat{\Sigma }}^{-1}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^l}\frac{x_i{x}_i^{\prime }}{s_i^2+{\sigma }^2}\mathrm{,}$$

$$Q\left({\sigma }^2\right)\mathrm{<mo>=</mo>}{\left|\widehat{\Sigma }\right|}^{\mathrm{1\; <mo>/</mo>\; 2}}{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^l}\frac{1}{{\left(s_i^2+{\sigma }^2\right)}^{1/2}}\exp \left\{-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^l}\frac{1}{s_i^2+{\sigma }^2}{\left({\widehat{\theta }}_i-x_i^{\prime }\widehat{\beta }\right)}^2\right\}\mathrm{.}$$

Theorem 1

The joint posterior density (2.3) is proper provided the design matrix is full rank.

Proof of Theorem 1

Because the design matrix is full rank, $\widehat{\beta }$ and $\widehat{\Sigma }$ are well defined for all ${\sigma }^2.$ This implies that $Q\left({\sigma }^2\right)$ is bounded in ${\sigma }^2.$ Therefore, as $\pi \left({\sigma }^2\right)=\frac{1}{{\left(1+{\sigma }^2\right)}^2}$ is proper, the posterior density ${\pi }_3\left({\sigma }^2|\widehat{\theta }\right)$ is proper. Then, by applying the multiplication rule of probability, it follows that the joint posterior density, $\pi \left(\theta ,\beta ,{\sigma }^2|\widehat{\theta }\right),$ is proper.

With propriety assured, sampling from the joint posterior density and inference about ${\theta }_i$ can be achieved through a simple Monte Carlo procedure. Using the multiplication rule and drawing samples from (2.6), (2.5) and (2.4), the procedure follows. First, draw a sample from ${\pi }_3\left({\sigma }^2|\widehat{\theta }\right)$ in (2.6). Draws from this distribution can be made using a grid method; see Appendix B. It is then easy to draw samples from the conditional posterior density of $\beta$ in (2.5). Finally, samples of the ${\theta }_i$ can be drawn independently from (2.4). Sampling in this manner from the joint posterior density under the unconstrained BFH model does not need monitoring (unlike Markov chain Monte Carlo methods). The unconstrained BFH model will provide a basis for comparison of the estimates and related measures of uncertainty obtained under the proposed random deletion benchmarking method.

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