Robust Bayesian small area estimation
Section 4. Final remarks

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The paper considers small area models for handling area level data. The new feature of this article is modeling both small area means and variances along with the use of $t-$ distribution of random effects. It is shown via both data analysis and simulation that the proposed method performs mostly better than the models of You and Chapman (2006) and Sugasawa et al. (2017) in most situations.

Acknowledgements

Ghosh’s research was partially supported in part by NSF Grant SES-1327359.

Appendix A

Proof

Theorem 2. Under the model (2.1) with modified Jeffreys’ prior (2.3), the posterior distribution (2.4) is proper, provided $\text{min}\left(n_1\mathrm{,}\dots \mathrm{,}{n}_m\right)\mathrm{<mo>></mo>}p\mathrm{.}$

Proof. Recall the posterior distribution (2.4),

$$\begin{array}{l}{\pi }_{\text{MJ}}\left(\theta \mathrm{,}\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\mathrm{,}\nu |y\mathrm{,}{s}^2\right)\hfill \\ \propto {\left({\sigma }_{\delta }^2\right)}^{-\frac{p+m}{2}-1}\text{exp}\left(-\frac{a}{2{\sigma }_{\delta }^2}\right)\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{v}_i^{-\frac{n_i}{2}-1}\right)\text{exp}\left[-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{1}{v_i}{\left(y_i-{\theta }_i\right)}^2\right]\hfill \\ \times \text{exp}\left(-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{s_i^2}{v_i}\right)\left[{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{\left\{1+\frac{{\left({\theta }_i-x_i^T\beta \right)}^2}{\nu {\sigma }_{\delta }^2}\right\}}^{-\frac{\nu +1}{2}}\right]{\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{v}{2}\right)\sqrt{\nu }}\right]}^m\hfill \\ \times {\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left[\frac{\nu g\left(v\right)}{2\left(v+3\right)}-\frac{1}{{\left(v+1\right)}^2{\left(v+3\right)}^2}\right]}^{\frac{1}{2}}\hfill \end{array}$$

where $g\left(v\right)\mathrm{<mo>=</mo>}\left\{{\Psi }^{\prime }\left(\frac{v}{2}\right)-{\Psi }^{\prime }\left(\frac{v+1}{2}\right)\right\}/4-\left(\nu +5\right)/\left\{2\nu \left(\nu +1\right)\left(\nu +3\right)\right\}.$

First of all, since $\text{log}\left[\Gamma \left\{\frac{\left(\nu +1\right)}{2}\right\}/\Gamma \left(\frac{v}{2}\right)\right]\doteq \left\{-\text{log}\left(2e\right)+\nu \text{log}\left(\nu +1\right)-\left(\nu -1\right)\text{log}\nu \right\}/2$ by Stirling’s approximation, we have

$$\frac{1}{2}\left[\Psi \left(\frac{\nu +1}{2}\right)-\Psi \left(\frac{\nu }{2}\right)\right]\doteq \frac{d}{d\nu }\text{log}\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}\mathrm{<mo>=</mo>}\frac{1}{2}\left[\text{log}\left(\nu +1\right)-\text{log}\nu \right]+\frac{1}{2\nu \left(\nu +1\right)}$$

and

$$\frac{1}{4}\left[{\Psi }^{\prime }\left(\frac{\nu +1}{2}\right)-{\Psi }^{\prime }\left(\frac{\nu }{2}\right)\right]\doteq \frac{1}{2}\left(\frac{1}{\nu +1}-\frac{1}{\nu }\right)-\frac{1}{2{\nu }^2}+\frac{1}{2{\left(\nu +1\right)}^2}\mathrm{.}$$

This leads to

$$g\left(\nu \right)\doteq \frac{5\nu +3}{2{\nu }^2{\left(\nu +1\right)}^2\left(\nu +3\right)}$$

and

$$\left[\frac{\nu g\left(v\right)}{2\left(\nu +3\right)}-\frac{1}{{\left(\nu +1\right)}^2{\left(\nu +3\right)}^2}\right]\doteq \frac{1}{4\nu {\left(\nu +1\right)}^2\left(\nu +3\right)}\mathrm{.}$$

Hence this approximation simplifies the last term in (2.4). The corresponding posterior is

$$\begin{array}{l}{\pi }_{\text{MJ}}\left(\theta \mathrm{,}\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu |y\mathrm{,}{s}^2\right)\hfill \\ \propto {\left({\sigma }_{\delta }^2\right)}^{-\frac{p+m}{2}-1}\text{exp}\left(-\frac{a}{2{\sigma }_{\delta }^2}\right)\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{v}_i^{-\frac{n_i}{2}-1}\right)\text{exp}\left[-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{1}{v_i}{\left(y_i-{\theta }_i\right)}^2\right]\hfill \\ \times \text{exp}\left(-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{s_i^2}{v_i}\right)\left[{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{\left\{1+\frac{{\left({\theta }_i-x_i^T\beta \right)}^2}{\nu {\sigma }_{\delta }^2}\right\}}^{-\frac{\nu +1}{2}}\right]{\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{v}{2}\right)\sqrt{\nu }}\right]}^m\hfill \\ \times {\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left[\frac{1}{\nu {\left(\nu +1\right)}^2\left(\nu +3\right)}\right]}^{\frac{1}{2}}\mathrm{.}(\text{A}\text{.1})\hfill \end{array}$$

First, integrating out with respect to $\beta \mathrm{.}$ By letting $w_i\mathrm{<mo>=</mo>}\left({\theta }_i-x_i^T\beta \right)/{\sigma }_{\delta }\text{},$ i.e., ${\theta }_i\mathrm{<mo>=</mo>}{x}_i^T\beta +{\sigma }_{\delta }{w}_i\text{}\mathrm{,}$ we have

$$\begin{array}{l}{\pi }_{\text{MJ}}\left(w\mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}v|y\mathrm{,}{s}^2\right)\hfill \\ \propto {\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}\text{exp}\left(-\frac{a}{2{\sigma }_{\delta }^2}\right)\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{v}_i^{-\frac{n_i}{2}-1}\right)\text{exp}\left(-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{s_i^2}{v_i}\right)\hfill \\ \times \left[{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{\left(1+\frac{w_i^2}{\nu }\right)}^{-\frac{\nu +1}{2}}\right]{\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{v}{2}\right)\sqrt{\nu }}\right]}^m{\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left[\frac{1}{\nu {\left(\nu +1\right)}^2\left(\nu +3\right)}\right]}^{\frac{1}{2}}{\left|X^T{V}^{-1}X\right|}^{-\frac{1}{2}}\hfill \\ \times \text{exp}\left[-\frac{1}{2}{\left(Y-{\sigma }_{\delta }w\right)}^T\left\{V^{-1}-V^{-1}X{\left(X^T{V}^{-1}X\right)}^{-1}{X}^T{V}^{-1}\right\}\left(Y-{\sigma }_{\delta }w\right)\right]\mathrm{.}\hfill \end{array}$$

where $w\mathrm{<mo>=</mo>}{\left(w_1\text{}\mathrm{,}\dots \mathrm{,}{w}_m\right)}^T\text{}\mathrm{.}$ Note that $V^{-1}-V^{-1}X{\left(X^T{V}^{-1}X\right)}^{-1}{X}^T{V}^{-1}$ is non-negative definite. Using $K\left(\mathrm{<mo>></mo>\; 0}\right)$ as a generic constant which depends only on the data $(y\mathrm{,}{s}^2),$ we integrate out with respect to $w$ and ${\sigma }_{\delta }^2$ respectively in order, we have

$$\begin{array}{ll}{\pi }_{\text{MJ}}\left(v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}v|y\mathrm{,}{s}^2\right)\hfill & \le K{\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}\text{exp}\left(-\frac{a}{2{\sigma }_{\delta }^2}\right)\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{v}_i^{-\frac{n_i}{2}-1}\right)\hfill \\ \hfill & \times \text{exp}\left(-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{s_i^2}{v_i}\right){\left[\frac{1}{\nu {\left(\nu +1\right)}^2\left(\nu +3\right)}\right]}^{\frac{1}{2}}{\left|X^T{V}^{-1}X\right|}^{-\frac{1}{2}}\text{},\hfill \end{array}$$

and then

$$\begin{array}{ll}{\pi }_{\text{MJ}}\left(v\mathrm{,}v|y\mathrm{,}{s}^2\right)\hfill & \le K\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{v}_i^{-\frac{n_i}{2}-1}\right)\text{exp}\left(-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{s_i^2}{v_i}\right)\hfill \\ \hfill & \times {\left[\frac{1}{\nu {\left(\nu +1\right)}^2\left(\nu +3\right)}\right]}^{\frac{1}{2}}{\left|X^T{V}^{-1}X\right|}^{-\frac{1}{2}}\text{}.\hfill \end{array}$$

Note that

$$\begin{array}{l}{\displaystyle {\int }_0^{\infty }}{\nu }^{-\frac{1}{2}}{\left(\nu +1\right)}^{-1}{\left(\nu +3\right)}^{-\frac{1}{2}}d\nu \hfill \\ \mathrm{<mo>=</mo>}{\displaystyle {\int }_0^c}{\nu }^{-\frac{1}{2}}{\left(\nu +1\right)}^{-1}{\left(\nu +3\right)}^{-\frac{1}{2}}d\nu +{\displaystyle {\int }_c^{\infty }}{\nu }^{-\frac{1}{2}}{\left(\nu +1\right)}^{-1}{\left(\nu +3\right)}^{-\frac{1}{2}}d\nu \hfill \\ \le {\displaystyle {\int }_0^c}{\nu }^{-\frac{1}{2}}d\nu +{\displaystyle {\int }_c^{\infty }}{\nu }^{-2}d\nu \mathrm{<mo>=</mo>\; 2}{c}^{\frac{1}{2}}+c^{-1}\mathrm{<mo><</mo>}\infty \text{for}\text{any}c\mathrm{<mo>></mo>\; 0.}\hfill \end{array}$$

After integrating out with respect to $\nu ,$ we have

$${\pi }_{\text{MJ}}\left(v|y\mathrm{,}{s}^2\right)\le K\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{v}_i^{-\frac{n_i}{2}-1}\right)\text{exp}\left(-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{s_i^2}{v_i}\right){\left|X^T{V}^{-1}X\right|}^{-\frac{1}{2}}\text{}\mathrm{.}$$

Finally, since ${\left|X^T{V}^{-1}X\right|}^{-\frac{1}{2}}\le {\left(v_{\text{max}}\right)}^{\frac{p}{2}}{\left|X^{T}X\right|}^{-\frac{1}{2}}\text{}\mathrm{,}$ where $v_{\text{max}}\mathrm{<mo>=</mo>}\text{max}\left(v_1\mathrm{,}\dots \mathrm{,}{v}_m\right)\mathrm{,}$ if $\text{min}\left(n_1\mathrm{,}\dots \mathrm{,}{n}_m\right)\mathrm{<mo>></mo>}p\mathrm{,}$ modified Jeffrey’s prior leads to a proper posterior.

Appendix B

Full conditional distributions

The full posterior of the parameters given the data is specified in (A.1). For the MCMC implementation, it is convenient to use the latent parameters ${\eta }_i\left(i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m\right)$ such that

$$\begin{array}{ll}{\theta }_i|{\eta }_i\hfill & \stackrel{\text{ind}}{{\displaystyle \sim }}\text{N}\left(x_i^T\beta \mathrm{,}{\eta }_i^{-1}{\sigma }_{\delta }^2\right)\hfill \\ {\eta }_i\hfill & \stackrel{\text{iid}}{{\displaystyle \sim }}\text{G}\left(\frac{\nu }{2}\mathrm{,}\frac{\nu }{2}\right)\mathrm{.}\hfill \end{array}$$

Let $\Lambda \mathrm{<mo>=</mo>}\text{Diag}({\eta }_1\mathrm{,}\dots \mathrm{,}{\eta }_m).$ The full conditional distributions are

I.      $\left[{\sigma }_{\delta }^2|\theta \mathrm{,}\Lambda \mathrm{,}\beta \mathrm{,}v\mathrm{,}\nu \mathrm{,}y\mathrm{,}{s}^2\right]\sim \text{IG}\left[\frac{p+m}{2}\mathrm{,}\frac{1}{2}\left\{a+{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m{\eta }_i{\left({\theta }_i-x_i^T\beta \right)}^2}\right\}\right]\mathrm{<mo>;</mo>}$

II.      $\left[v_i|\theta \mathrm{,}\Lambda \mathrm{,}\beta \mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \mathrm{,}y\mathrm{,}{s}^2\right]\stackrel{\text{ind}}{{\displaystyle \sim }}\text{IG}\left[\frac{n_i}{2}\mathrm{,}\frac{1}{2}\left\{{\left(y_i-{\theta }_i\right)}^2+s_i^2\right\}\right]\mathrm{<mo>;</mo>}$

III.      $\left[\beta |\theta \mathrm{,}\Lambda \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \mathrm{,}y\mathrm{,}{s}^2\right]\sim \text{N}\left[{\left(X^T\Lambda X\right)}^{-1}{X}^T\Lambda \theta \mathrm{,}{\sigma }_{\delta }^2{\left(X^T\Lambda X\right)}^{-1}\right]\mathrm{<mo>;</mo>}$

IV.      $\left[{\eta }_i|\theta \mathrm{,}\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \mathrm{,}y\mathrm{,}{s}^2\right]\stackrel{\text{ind}}{{\displaystyle \sim }}\text{G}\left[\frac{\nu +1}{2}\mathrm{,}\frac{1}{2}\left\{\nu +\frac{{\left({\theta }_i-x_i^T\beta \right)}^2}{{\sigma }_{\delta }^2}\right\}\right]\mathrm{<mo>;</mo>}$

V.      $\left[{\theta }_i|\Lambda \mathrm{,}\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \mathrm{,}y\mathrm{,}{s}^2\right]\stackrel{\text{ind}}{{\displaystyle \sim }}\text{N}\left[{\left(v_i^{-1}+{\sigma }_{\delta }^{-2}{\eta }_i\right)}^{-1}\left(v_i^{-1}{y}_i+{\sigma }_{\delta }^{-2}{\eta }_i{x}_i^T\beta \right)\mathrm{,}{\left(v_i^{-1}+{\sigma }_{\delta }^{-2}{\eta }_i\right)}^{-1}\right]\mathrm{<mo>;</mo>}$

and

VI.      $\begin{array}{ll}\left[v|\theta \mathrm{,}\Lambda \mathrm{,}\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}y\mathrm{,}{s}^2\right]\hfill & \propto {\nu }^{\frac{m-1}{2}}\text{exp}\left\{-\frac{v}{2}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m\left({\eta }_i-\text{log}{\eta }_i-1\right)}\right\}\hfill \\ \hfill & \times \left\{\frac{{\nu }^{\frac{m\nu -m}{2}}\text{exp}\left(-\frac{m\nu }{2}\right)}{2^{\frac{m\nu }{2}}{\Gamma }^m\left(\frac{v}{2}\right)}{\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left(\nu +1\right)}^{-1}{\left(\nu +3\right)}^{-\frac{1}{2}}\right\}\mathrm{<mo>.</mo>}(\text{B}\text{.1})\hfill \end{array}$

All but (VI) requires the generation of samples from standard distributions. While we use the Gibbs sampling method for (I)-(V), we use the Metropolis-Hastings algorithm for generating samples from (VI) as given in Chib and Greenberg (1995).

How to apply the result of Chib and Greenberg (1995) to (IV) in (B.1).

If the target density $\pi \left(t\right)$ can be written as $\pi \left(t\right)\propto \psi \left(t\right)h\left(t\right)\mathrm{,}$ where $h\left(t\right)$ is a density that can be sampled and $\psi \left(t\right)$ is uniformly bounded, then one can set $h\left(t\right)$ as a candidate density to draw samples and use $\psi \left(t\right)$ in $\alpha \left(x\mathrm{,}y\right)\mathrm{<mo>=</mo>}\text{min}\left\{\psi \left(y\right)/\psi \left(x\right)\mathrm{,1}\right\}$ which is the probability of move.

Recall that the full conditional distribution of $\nu$ is

$$\begin{array}{ll}\pi \left(v|\cdot \right)\hfill & \propto {\nu }^{\frac{m-1}{2}}\text{exp}\left\{-\frac{\nu }{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\left({\eta }_i-\text{log}{\eta }_i-1\right)\right\}\hfill \\ \hfill & \times \left\{\frac{{\nu }^{\frac{m\nu -m}{2}}\text{exp}\left(-\frac{m\nu }{2}\right)}{2^{\frac{m\nu }{2}}{\Gamma }^m\left(\frac{v}{2}\right)}{\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left(\nu +1\right)}^{-1}{\left(\nu +3\right)}^{-\frac{1}{2}}\right\}\mathrm{.}\hfill \end{array}$$

Since $\Gamma \left(\frac{v}{2}\right)\ge \sqrt{2\pi }\exp \left(-\frac{v}{2}\right){\left(\frac{v}{2}\right)}^{\frac{\nu -1}{2}}\text{}\mathrm{,}$ the second $\{\cdot \}$ term above is bounded by ${\left(\sqrt{2\pi }\right)}^{-m}{\left(\nu +1\right)}^{\frac{p}{2}-1}{\left(\nu +3\right)}^{-\frac{p}{2}-\frac{1}{2}}\le {\left(\sqrt{2\pi }\right)}^{-m}/\sqrt{3}\mathrm{.}$ Hence, we can apply third method in the Section 5 of Chib and Greenberg (1995) with

$$h\left(v\right)\sim \text{G}\left(\frac{m+1}{2}\mathrm{,}\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\left({\eta }_i-\text{log}{\eta }_i-1\right)\right)$$

and

$$\psi \left(v\right)\mathrm{<mo>=</mo>}\frac{{\nu }^{\frac{m\nu -m}{2}}\text{exp}\left(-\frac{m\nu }{2}\right)}{2^{\frac{m\nu }{2}}{\Gamma }^m\left(\frac{v}{2}\right)}{\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left(\nu +1\right)}^{-1}{\left(\nu +3\right)}^{-\frac{1}{2}}\text{}\mathrm{.}$$

Here $h\left(v\right)$ is a candidate-generating density, and $\psi \left(v\right)$ is uniformly bounded.

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