Robust Bayesian small area estimation
Section 2. The model

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A typical area level model is given by $y_i=x_i^T\beta +u_i+e_i\text{}\mathrm{,}\left(i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m\right)\mathrm{,}$ where $m$ denotes the number of small areas, $x_1\mathrm{,}\dots \mathrm{,}{x}_m$ are $p\left(\mathrm{<mo><</mo>}m\right)$ dimensional covariates, and $\beta \left(p\times 1\right)$ is the vector of regression coefficients. The random effects $u_i$ and the sampling errors $e_i$ are assumed to be independently distributed with the $u_i\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(\mathrm{0,}{\sigma }_u^2\right)$ and the $e_i\stackrel{\text{ind}}{{\displaystyle \sim }}N\left(\mathrm{0,}{v}_i\right).$ That is, the classic area level model is

$$\begin{array}{ll}{y}_i|{\theta }_i\hfill & \stackrel{\text{ind}}{{\displaystyle \sim }}N\left({\theta }_i\mathrm{,}{v}_i\right)\mathrm{,}\hfill \\ {\theta }_i|\beta \mathrm{,}{\sigma }_u^2\hfill & \stackrel{\text{ind}}{{\displaystyle \sim }}N\left(x_i^T\beta \mathrm{,}{\sigma }_u^2\right)\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m.\hfill \end{array}$$

The $v_i$ are assumed to be known in order to avoid non-identifiability. The assumption of known $v_i$ almost becomes mandatory for secondary users of survey data who do not have access to any micro data for modeling the $v_i\mathrm{.}$ However, in reality they are random, based on sampled data. In situations when one has additional data to model the $v_i,$ the data can be used efficiently for estimating the $v_i.$ Moreover, in such situations, it is possible to have shrinkage estimators of the small area means ${\theta }_i$ as well as of the variances  $v_i\mathrm{.}$

We address small area estimation problems where we have additional data to model the $v_i\mathrm{.}$ Also, for robustification, we assume $t-$ distribution of the random effects instead of the normal distribution. We state our model as follows,

$$\begin{array}{ll}{y}_i|{\theta }_i\mathrm{,}{v}_i\hfill & \stackrel{\text{ind}}{{\displaystyle \sim }}N\left({\theta }_i\mathrm{,}{v}_i\right)\mathrm{,}{s}_i^2|v_i\stackrel{\text{ind}}{{\displaystyle \sim }}G\left(\frac{n_i-1}{2}\mathrm{,}\frac{1}{2v_i}\right)\hfill \\ {\theta }_i|\beta \mathrm{,}{\sigma }_{\delta }^2\mathrm{,}\nu \hfill & \stackrel{\text{ind}\text{.}}{{\displaystyle \sim }}{t}_{\nu }\left(x_i^T\beta \mathrm{,}{\sigma }_{\delta }\right)\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m\mathrm{,}(2.1)\hfill \end{array}$$

where $n_i$ is the sample size in the $i^{\text{th}}$ area, $t_{\nu }\left(\mu \mathrm{,}\sigma \right)$ denotes the Student’s $t-$ distribution with location $\mu \mathrm{,}$ scale $\sigma$ and degrees of freedom $\nu \mathrm{,}$ and $G\left(c\mathrm{,}d\right)$ denotes the gamma distribution with the kernel density $x^{c-1}\text{exp}\left(-dx\right)$ for $x\mathrm{<mo>></mo>\; 0.}$

For a full Bayesian analysis, our objective is to find the posterior distribution of $\theta \mathrm{<mo>=</mo>}{\left({\theta }_1\mathrm{,}\dots \mathrm{,}{\theta }_m\right)}^T\text{}\mathrm{,}$ given $y\mathrm{<mo>=</mo>}{\left(y_1\mathrm{,}\dots \mathrm{,}{y}_m\right)}^T$ and $s^2\mathrm{<mo>=</mo>}{\left(s_1^2\mathrm{,}\dots \mathrm{,}{s}_m^2\right)}^T\text{}\mathrm{.}$ To this end, we first need to find the prior distributions for all the hyper-parameters, $\beta \mathrm{,}v\mathrm{<mo>=</mo>}{\left(v_1\mathrm{,}\dots \mathrm{,}{v}_m\right)}^T\text{}\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}$ and $\nu \mathrm{.}$ The usual first try is Jeffreys’ prior which is proportional to the positive square root of the determinant of the Fisher information matrix. The Fisher information matrix in our case is

$$I\left(\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \right)\mathrm{<mo>=</mo>}\left[\begin{array}{cccc}\frac{\left(\nu +1\right)}{{\sigma }_{\delta }^2\left(\nu +3\right)}{X}^{T}X& 0& 0& 0\\ 0& D& 0& 0\\ 0& 0& \frac{m\nu }{2{\left({\sigma }_{\delta }^2\right)}^2\left(\nu +3\right)}& \frac{-m}{{\sigma }_{\delta }^2\left(\nu +1\right)\left(\nu +3\right)}\\ 0& 0& \frac{-m}{{\sigma }_{\delta }^2\left(\nu +1\right)\left(\nu +3\right)}& mg\left(v\right)\end{array}\right]\mathrm{,}$$

where $X={\left(x_1\mathrm{,}\dots \mathrm{,}{x}_m\right)}^T\text{}\mathrm{,}D=\text{Diag}\left(\frac{n_1}{2v_1^2}\text{}\mathrm{,}\dots \mathrm{,}\frac{n_m}{2v_m^2}\right)\mathrm{,}$ and $g\left(\nu \right)\mathrm{<mo>=</mo>}\left\{{\Psi }^{\prime }\left(\frac{v}{2}\right)-{\Psi }^{\prime }\left(\frac{v+1}{2}\right)\right\}/4-\left(\nu +5\right)/\left\{2\nu \left(\nu +1\right)\left(\nu +3\right)\right\}\mathrm{,}$ with $\Psi \left(z\right)\mathrm{<mo>=</mo>}{\Gamma }^{\prime }\left(z\right)/\Gamma \left(z\right)$ and ${\Psi }^{\prime }\left(z\right)\mathrm{<mo>=</mo>}d\Psi \left(z\right)/dz$ which are the digamma and the trigamma functions. Thus, Jeffreys’ prior is

$${\pi }_J\left(\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \right)\propto {\mathrm{<mo\; stretchy="false">(</mo>}{\sigma }_{\delta }^2\mathrm{<mo\; stretchy="false">)</mo>}}^{-\frac{p}{2}-1}{\left|X^{T}X\right|}^{\frac{1}{2}}{\left|D\right|}^{\frac{1}{2}}{\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left[\frac{\nu g\left(v\right)}{2\left(\nu +3\right)}-\frac{1}{{\left(\nu +1\right)}^2{\left(\nu +3\right)}^2}\right]}^{\frac{1}{2}}\text{}\mathrm{.}(2.2)$$

However, Jeffreys’ prior leads to an improper posterior due to the factor of ${\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}$ in (2.2).

Theorem 1. Jeffreys’ prior (2.2) leads an improper posterior.

Proof. Let ${\pi }_J\equiv {\pi }_J\left(\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu |y\mathrm{,}{s}^2\right)$ be the posterior density with Jeffreys’ prior (2.2). Considering the terms that contain ${\sigma }_{\delta }^2$ in ${\pi }_J$ and taking the transformation $w_i\mathrm{<mo>=</mo>}\left({\theta }_i-x_i^T\beta \right)/{\sigma }_{\delta }\text{}\mathrm{,}$ i.e., ${\theta }_i\mathrm{<mo>=</mo>}{x}_i^T\beta +{\sigma }_{\delta }{w}_i\text{}\mathrm{,}$ we have

$$\begin{array}{l}{\displaystyle {\int }_0^{\infty }}{\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}\text{exp}\left[-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{1}{v_i}{\left(y_i-x_i^T\beta -{\sigma }_{\delta }{w}_i\right)}^2\right]d{\sigma }_{\delta }^2\hfill \\ \ge {\displaystyle {\int }_0^{\infty }}{\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}\text{exp}\left[-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\left\{\frac{{\left(y_i-x_i^T\beta \right)}^2}{v_i}+\frac{{\sigma }_{\delta }^2{w}_i^2}{v_i}\right\}\right]d{\sigma }_{\delta }^2\hfill \\ \ge \text{exp}\left\{-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{{\left(y_i-x_i^T\beta \right)}^2}{v_i}\right\}{\displaystyle {\int }_0^k}{\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}\text{exp}\left(-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{{\sigma }_{\delta }^2{w}_i^2}{v_i}\right)d{\sigma }_{\delta }^2\text{for}\text{a}\text{constant}k\mathrm{<mo>></mo>\; 0,}\hfill \\ \ge \text{exp}\left[-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\left\{\frac{{\left(y_i-x_i^T\beta \right)}^2}{v_i}+\frac{k{w}_i^2}{v_i}\right\}\right]{\displaystyle {\int }_0^k}{\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}d{\sigma }_{\delta }^2\mathrm{<mo>=</mo>}\infty \mathrm{.}\hfill \end{array}$$

Therefore, Jeffreys’ prior leads to an improper posterior.

However, once the component ${\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}$ in (2.2) is replaced by ${\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}\text{exp}\left(-a/2{\sigma }_{\delta }^2\right)$ for some $a\mathrm{<mo>></mo>\; 0},$ this modified version of Jeffreys’ prior will lead a proper posterior under the condition, $\text{min}\left(n_1\text{}\mathrm{,}\dots \mathrm{,}{n}_m\right)\mathrm{<mo>></mo>}p\mathrm{.}$ Therefore, we suggest a modified Jeffreys’ prior for our model as follows:

$$\begin{array}{ll}{\pi }_{\text{MJ}}\left(\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \right)\hfill & \propto {\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}\text{exp}\left(-\frac{a}{2{\sigma }_{\delta }^2}\right)\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{1}{v_i}\right){\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}\hfill \\ \hfill & \times {\left[\frac{\nu g\left(v\right)}{2\left(v+3\right)}-\frac{1}{{\left(v+1\right)}^2{\left(v+3\right)}^2}\right]}^{\frac{1}{2}}\text{where}a\mathrm{<mo>></mo>\; 0.}(2.3)\hfill \end{array}$$

By combining the likelihood of (2.1) and modified Jeffreys’ prior (2.3), the full posterior of parameters given the data is

$$\begin{array}{l}{\pi }_{\text{MJ}}\left(\theta ,\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}v|y\mathrm{,}{s}^2\right)\hfill \\ \propto {\left({\sigma }_{\delta }^2\right)}^{-\frac{p+m}{2}-1}\text{exp}\left(-\frac{a}{2{\sigma }_{\delta }^2}\right)\left({\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{v}_i^{-\frac{n_i}{2}-1}\right)\text{exp}\left[-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{1}{v_i}{\left(y_i-{\theta }_i\right)}^2\right]\hfill \\ \times \text{exp}\left(-\frac{1}{2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}\frac{s_i^2}{v_i}\right)\left[{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^m}{\left\{1+\frac{{\left({\theta }_i-x_i^T\beta \right)}^2}{\nu {\sigma }_{\delta }^2}\right\}}^{-\frac{\nu +1}{2}}\right]{\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{v}{2}\right)\sqrt{\nu }}\right]}^m\hfill \\ \times {\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left[\frac{\nu g\left(v\right)}{2\left(v+3\right)}-\frac{1}{{\left(v+1\right)}^2{\left(v+3\right)}^2}\right]}^{\frac{1}{2}}\text{}\mathrm{.}(2.4)\hfill \end{array}$$

Theorem 2. Under the model (2.1), the posterior ${\pi }_{\text{MJ}}\left(\theta ,\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}v|y\mathrm{,}{s}^2\right)$  in (2.4) is proper, provided $\text{min}\left(n_1\mathrm{,}\dots \mathrm{,}{n}_m\right)\mathrm{<mo>></mo>}p\mathrm{.}$

Proof. See Appendix A.

Theorem 2 shows that modified Jeffreys’ prior (2.3) leads to a proper posterior (2.4). The key idea is that we need a prior for ${\sigma }_{\delta }^2$ such that ${\displaystyle {\int }_0^{\infty }}\pi \left({\sigma }_{\delta }^2\right){\left({\sigma }_{\delta }^2\right)}^{-\frac{p}{2}-1}d{\sigma }_{\delta }^2\mathrm{<mo><</mo>}\infty \mathrm{.}$

Remark 1. ${\pi }_{\text{MJ}}\left(\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \right)$ can be factored into four independent priors for each parameter.

$${\pi }_{\text{MJ}}\left(\beta \mathrm{,}v\mathrm{,}{\sigma }_{\delta }^2\text{}\mathrm{,}\nu \right)\propto \pi \left(\beta \right)\pi \left(v\right)\pi \left({\sigma }_{\delta }^2\right)\pi \left(v\right)$$

where

$$\pi \left(\beta \right)\propto \mathrm{1,}\pi \left(v_i\right)\propto \frac{1}{v_i}\text{for}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m\mathrm{,}$$

$$\pi \left({\sigma }_{\delta }^2\right)\sim \text{IG}\left(\frac{p}{2}\mathrm{,}\frac{a}{2}\right)$$

and

$$\pi \left(v\right)\propto {\left(\frac{\nu +1}{\nu +3}\right)}^{\frac{p}{2}}{\left[\frac{\nu g\left(v\right)}{2\left(v+3\right)}-\frac{1}{{\left(v+1\right)}^2{\left(v+3\right)}^2}\right]}^{\frac{1}{2}}\text{}\mathrm{.}$$

Here $\text{IG}\left(c\mathrm{,}d\right)$ denotes the inverse gamma distribution with the kernel density $x^{-c-1}\text{exp}\left(-d/x\right)$ for $x\mathrm{<mo>></mo>\; 0.}$

The full conditional distributions to implement the Markov chain Monte Carlo (MCMC) are given in details in Appendix B. To generate samples, we use Gibbs sampling with Metropolis-Hastings algorithm where the conditional distribution of a parameter is known only up to a multiplicative constant. We provide details on how to apply a result of Chib and Greenberg (1995) for the Metropolis-Hastings algorithm to generate samples.

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