Bayesian benchmarking of the Fay-Herriot model using random deletion
Section 3. Random deletion methodology

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As remarked earlier, the random deletion methodology is obtained by introducing a new variable which takes the values $\mathrm{1,}\dots \mathrm{,}l$ with equal, possibly different, probabilities (weights). In Section 3.1, we show how to construct the joint posterior density of the parameters of the BFH model under random deletion, and in Section 3.2, we show how to sample from the joint posterior density.

3.1  Construction of the joint posterior density

The Basic Benchmarking Theorem is motivated by the work reviewed in Section 1. The next goal is to construct the joint prior density of ${\theta }_1\mathrm{,}\dots \mathrm{,}{\theta }_l$ subject to the benchmarking constraint that ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^l{\theta }_i\mathrm{<mo>=</mo>}a},$ where $a$ is a known external target. The joint prior density of ${\theta }_1\mathrm{,}\dots \mathrm{,}{\theta }_l$ will be used to complete the constrained Fay-Herriot model for the last one deletion benchmarking (see Section 1).

Theorem 2

Let ${\theta }_i\stackrel{\text{ind}}{{\displaystyle \sim }}\text{Normal}\left(u_i^{\prime }\beta ,{\delta }^2\right),i=1,\dots ,l.$ Then, under the constraint, ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^l{\theta }_i}\mathrm{<mo>=</mo>}a,$ where $a$ is constant, letting ${\theta }_{\left(l\right)}$ be the vector of all the ${\theta }_i$ except the last one, the joint density of ${\theta }_i\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ is

$${\theta }_{\left(l\right)}\sim \text{Normal}\left\{\left(I-\frac{1}{l}J\right)c\mathrm{,}{\delta }^2\left(I-\frac{1}{l}J\right)\right\}\mathrm{,}{\theta }_l\mathrm{<mo>=</mo>}a-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{l-1}{\theta }_i}\mathrm{,}(3.1)$$

$c^{\prime }=a{j}^{\prime }+\left({\left(u_1-u_l\right)}^{\prime }\beta ,\dots ,{\left(u_{l-1}-u_l\right)}^{\prime }\beta \right),J$ and $j$ are respectively a $\left(l-1\right)\times \left(l-1\right)$ matrix and a $\left(l-1\right)$ vector of ones.

Proof of Theorem 2

See Appendix C.

The proof of Theorem 2 uses the multivariate normal distribution, and it will be used to prove the more general theorem when the prior is adjusted to delete any area.

The constrained BFH model is

$${\widehat{\theta }}_i|{\theta }_i\stackrel{\text{ind}}{{\displaystyle \sim }}\text{Normal}\left({\theta }_i\mathrm{,}{s}_i^2\right)\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l\mathrm{,}(3.2)$$

$${\theta }_i|\beta \mathrm{,}{\sigma }^2\stackrel{\text{ind}}{{\displaystyle \sim }}\text{Normal}\left(x_i^{\prime }\beta \mathrm{,}{\sigma }^2\right)\mathrm{,}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^l{\theta }_i}\mathrm{<mo>=</mo>}a\mathrm{,}$$

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

The constraint on the prior on ${\theta }_1\mathrm{,}\dots \mathrm{,}{\theta }_l$ essentially adjusts the joint prior density. However, to incorporate the constraint, we will use Theorem 2 to adjust the posterior density under the unconstrained model.

For $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ let ${\lambda }_i\mathrm{<mo>=</mo>}\frac{{\sigma }^2}{{\sigma }^2+s_i^2}$ and ${\widehat{\widehat{\theta }}}_i\mathrm{<mo>=</mo>}{\lambda }_i{\widehat{\theta }}_i+\left(1-{\lambda }_i\right)x_i^{\prime }\beta .$ Also, let ${\widehat{\sigma }}_i^2\mathrm{<mo>=</mo>}{\sigma }^2\left(1-{\lambda }_i\right)\mathrm{,}i\mathrm{<mo>=</mo>1,}\dots \mathrm{,}l\mathrm{.}$ Then, under the unconstrained model the joint posterior density is

$${\pi }_{nb}\left(\theta ,\beta \mathrm{,}{\sigma }^2|\widehat{\theta }\right)\propto \pi \left(\beta ,{\sigma }^2\right){\displaystyle \prod _{i=1}^l\left[{\text{Normal}}_{{\widehat{\theta }}_i}\left(x_i^{\prime }\beta ,\frac{{\sigma }^2}{{\lambda }_i}\right){\text{Normal}}_{{\theta }_i}\left({\widehat{\widehat{\theta }}}_i,{\widehat{\sigma }}_i^2\right)\right]}\mathrm{.}$$

We now extend the result of Theorem 2, to reflect our interest in the constrained BFH model. Theorem 3, below, is used to construct the random deletion benchmarking method.

Theorem 3

Using a general notation, let $y_i\stackrel{\text{ind}}{{\displaystyle \sim }}\text{Normal}\left({\mu }_i\mathrm{,}{\sigma }_i^2\right)\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l\mathrm{.}$ Let $y_{\left(i\right)}$ denote the vector of all the $y_i$ except the $i^{\text{th}}$ one. Also, let $v_i={\sigma }_i^2/\sqrt{{\displaystyle {\sum }_{j=1}^l{\sigma }_j^2}},i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ and $v_i^{*}={\sigma }_i^2/{\displaystyle {\sum }_{j=1}^l{\sigma }_j^2,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l.$ Under the constraint ${\displaystyle {\sum }_{i=1}^l{y}_i=a,}$

$$y_{\left(i\right)}\sim \text{Normal}\left\{u_{\left(i\right)}-\left({\displaystyle \sum _{j=1}^l{u}_j-a}\right)v_{\left(i\right)}^{*},\text{diagonal}\left({\sigma }_{\left(i\right)}^2\right)-v_{\left(i\right)}{v}_{\left(i\right)}^{\prime }\right\}(3.3)$$

with $y_i\mathrm{<mo>=</mo>}a-{\displaystyle {\sum }_{j\ne i}{y}_j}.$ Then,

$$p\left(y\mathrm{,}z\mathrm{<mo>=</mo>}r|\varphi \mathrm{<mo>=</mo>}0\right)\mathrm{<mo>=</mo>}{\delta }_{y_r}\left(a-{\displaystyle \sum _{j\ne r}{y}_j}\right){\text{Normal}}_{y_{\left(r\right)}}\left\{{\mu }_{\left(r\right)}-\left({\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l{\mu }_j}-a\right)v_{\left(r\right)}^{*}\mathrm{,}\text{diagonal}\left({\sigma }_{\left(r\right)}^2\right)-v_{\left(r\right)}{v}_{\left(r\right)}^{\prime }\right\}\mathrm{,}(3.4)$$

for $r\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l$ and $\psi \mathrm{<mo>=</mo>}a-{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^l{y}_i}.$

Proof of Theorem 3

The proof of Theorem 3 is similar to Theorem 2. See Appendix C.

In what follows, one of the $l$ area parameters will be deleted randomly (i.e., with probability $1/l).$ Let $z\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l$ represent the county that is deleted. That is, $P\left(z\mathrm{<mo>=</mo>}r\right)\mathrm{<mo>=</mo>}1/l\mathrm{,}r\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l.$ Then, under the constrained BFH model, using Theorem 3, the joint posterior density is

$$\begin{array}{ll}{\pi }_b\left(\theta \mathrm{,}z\mathrm{<mo>=</mo>}r\mathrm{,}\beta \mathrm{,}{\sigma }^2|\widehat{\theta }\right)\hfill & \propto \pi \left(\beta \mathrm{,}{\sigma }^2\right){\displaystyle \prod _{i\mathrm{<mo>=</mo>1}}^l\left[{\text{Normal}}_{{\widehat{\theta }}_i}\left(x_i^{\prime }\beta \mathrm{,}\frac{{\sigma }^2}{{\lambda }_i}\right)\right]}\times {\delta }_{{\theta }_r}\left(a-{\displaystyle \sum _{j\ne r}{\theta }_j}\right)\hfill \\ \hfill & \times {\text{Normal}}_{{\theta }_{\left(r\right)}}\left\{{\widehat{\widehat{\theta }}}_{\left(r\right)}-\left({\displaystyle \sum _{j=1}^l{\widehat{\widehat{\theta }}}_j-a}\right)v_{\left(r\right)}^{*},\text{diagonal}\left({\widehat{\sigma }}_{\left(r\right)}^2\right)-v_{\left(r\right)}{v}_{\left(r\right)}^{\prime }\right\}\mathrm{,}(3.5)\hfill \end{array}$$

where $r\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ and for $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ $v_i^{*}\mathrm{<mo>=</mo>}{\widehat{\sigma }}_i^2/{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^l{\widehat{\sigma }}_j^2}$ and $v_i\mathrm{<mo>=</mo>}{\widehat{\sigma }}_i^2/\sqrt{{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^l{\widehat{\sigma }}_j^2}}.$

3.2  Sampling the joint posterior density

Unlike the BFH model, the constrained model in (3.2) cannot be fit using random draws; we use a Gibbs sampler. The joint conditional posterior density (cpd) of $\left(\theta ,z\right)$ is

$$\begin{array}{ll}\pi \left(\theta \mathrm{,}z\mathrm{<mo>=</mo>}r|\beta \mathrm{,}{\sigma }^2\mathrm{,}\widehat{\theta }\right)\hfill & \propto {\delta }_{{\theta }_r}\left(a-{\displaystyle \sum _{j\ne r}{\theta }_j}\right)\hfill \\ \hfill & \times {\text{Normal}}_{{\theta }_{\left(r\right)}}\left\{{\widehat{\widehat{\theta }}}_{\left(r\right)}-\left({\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l{\widehat{\widehat{\theta }}}_j-a}\right)v_{\left(r\right)}^{*},\text{diagonal}\left({\widehat{\sigma }}_{\left(r\right)}^{\text{2}}\right)-v_{\left(r\right)}{v}_{\left(r\right)}^{\prime }\right\}\mathrm{,}(3.6)\hfill \end{array}$$

where $r\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,{\theta }_{\left(r\right)}$ denotes the vector of ${\theta }_i$ with the $r^{\text{th}}$ component deleted, and $v$ and $v^{*}$ are defined in Theorem 3. Then, the joint conditional posterior density of $\left(\beta ,{\sigma }^2\right)$ is

$$\begin{array}{ll}\pi \left(\beta \mathrm{,}{\sigma }^2|\theta \mathrm{,}\widehat{\theta }\mathrm{,}z\mathrm{<mo>=</mo>}r\right)\hfill & \propto \pi \left(\beta \mathrm{,}{\sigma }^2\right){\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^l{\text{Normal}}_{{\widehat{\theta }}_i}}\left(x_i^{\prime }\beta \mathrm{,}\frac{{\sigma }^2}{{\lambda }_i}\right)\hfill \\ \hfill & \times {\text{Normal}}_{{\theta }_{\left(r\right)}}\left\{{\widehat{\widehat{\theta }}}_{\left(r\right)}-\left({\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^l{\widehat{\widehat{\theta }}}_j}-a\right)v_{\left(r\right)}^{*}\mathrm{,}\text{diagonal}\left({\widehat{\sigma }}_{\left(r\right)}^{\text{2}}\right)-v_{\left(r\right)}{v}_{\left(r\right)}^{\prime }\right\}\mathrm{.}(3.7)\hfill \end{array}$$

It is straight forward to sample the cpd’s of $\theta$ and $z.$ However, it is not so simple to sample the cpd’s of $\beta$ and ${\sigma }^2$ that we will next discuss.

First, to obtain the cpd’s of $\beta$ and ${\sigma }^2,$ we define

$$g_1=\frac{1}{{\sigma }^2}{\displaystyle \sum _{i=1}^l{\lambda }_i{\widehat{\theta }}_i{x}_i\text{and}{A}_1=\frac{1}{{\sigma }^2}{\displaystyle \sum _{i=1}^l{\lambda }_i{x}_i{x}_i^{\prime }}}.$$

Second, for $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}l,$ let $d_i={\lambda }_i{\widehat{\theta }}_i-\left({\displaystyle {\sum }_{j=1}^l{\lambda }_j{\widehat{\theta }}_j-a}\right)v_i^{*}$ and $x_i=\left(1-{\lambda }_i\right)x_i-v_i^{*}{\displaystyle {\sum }_{j=1}^l\left(1-{\lambda }_j\right)x_j}.$ Let $d_{\left(r\right)}$ and ${\tilde{X}}_{\left(r\right)}$ respectively denote the vector with entries $d_i$ excluding the $r^{\text{th}}$ component and the matrix with columns $x_i$ excluding $x_r.$ Now, let

$$g_2={\left({\theta }_{\left(r\right)}-d_{\left(r\right)}\right)}^{\prime }{\Sigma }_{\left(r\right)}{\tilde{X}}_{\left(r\right)}^{\prime }\text{and}{A}_2={\tilde{X}}_{\left(r\right)}{\Sigma }_{\left(r\right)}{\tilde{X}}_{\left(r\right)}^{\prime },$$

where ${\Sigma }_{\left(r\right)}$ is the covariance matrix without the $r^{\text{th}}$ row and column. Third, with a multivariate normal prior on $\beta$ of the form $\beta \sim \text{Normal}\left({\beta }_0,{\Sigma }_0\right),$ where ${\beta }_0$ and ${\Sigma }_0$ are specified, we let

$$g_0={\beta }_0{A}_0\text{and}{A}_0\mathrm{<mo>=</mo>}{\Sigma }_0^{-1}\mathrm{<mo>;</mo>}$$

thereby offering some protection against posterior impropriety. It follows that

$$\beta |\theta \mathrm{,}z\mathrm{<mo>=</mo>}r\mathrm{,}{\sigma }^2\mathrm{,}\widehat{\theta }\sim \text{Normal}\left\{{\left({\displaystyle \sum _{s\mathrm{<mo>=</mo>\; 0}}^2{A}_s}\right)}^{-1}{\displaystyle \sum _{s\mathrm{<mo>=</mo>\; 0}}^2{A}_s}{g}_s,\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{\left({\displaystyle \sum _{s\mathrm{<mo>=</mo>\; 0}}^2{A}_s}\right)}^{-1}\right\}\mathrm{.}$$

We can eliminate the prior of $\beta$ by letting ${\Sigma }_0\to \infty$ (i.e., noninformative prior) to get $\pi \left(\beta \right)=1$ as in the BFH model.

Finally, we consider the cpd of ${\sigma }^2.$ Let

$$U_i={\lambda }_i{\widehat{\theta }}_i+\left(1-{\lambda }_i\right)x_i^{\prime }\beta -\left\{{\displaystyle \sum _{j=1}^l\left\{{\lambda }_j{\widehat{\theta }}_j+\left(1-{\lambda }_j\right)x_j^{\prime }\beta \right\}-a}\right\}{v}_i^{*},i=1,\dots ,l,$$

and

$$\Sigma ={\sigma }^2\left\{\text{diagonal}\left(1-{\lambda }_1,\dots ,1-{\lambda }_l\right)-\left[\left(1-{\lambda }_i\right)\left(1-{\lambda }_{i^{\prime }}\right)\right]/{\displaystyle \sum _{j=1}^l\left(1-{\lambda }_j\right)}\right\}.$$

Then, the cpd of ${\sigma }^2$ is

$$\pi \left({\sigma }^2|\theta ,z=r,\beta ,\widehat{\theta }\right)\propto \pi \left({\sigma }^2\right)\left[{\displaystyle \prod _{i=1}^l{\text{Normal}}_{{\widehat{\theta }}_i}}\left\{x_i^{\prime }\beta ,{\sigma }^2+s_i^2\right\}\right]{\text{Normal}}_{{\theta }_{\left(r\right)}}\left\{U_{\left(r\right)},{\Sigma }_{\left(r\right)}\right\},$$

where $U_{\left(r\right)}$ denotes the vector of the $U_i$ excluding the $r^{\text{th}}$ component and $\pi \left({\sigma }^2\right)$ is a prior on ${\sigma }^2.$ As in the BFH model (see Section 2), we assign the prior density $\pi \left({\sigma }^2\right)={1/\left(1+{\sigma }^2\right)}^2\text{},{\sigma }^2>0$ to ${\sigma }^2.$ Because the baseline BFH posterior density is proper, the constraint BFH posterior density will also be proper.

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