Robust Bayesian small area estimation
Section 1. Introduction
The classic paper of Fay and Herriot (1979) has become a cornerstone of research in small area estimation for nearly four decades. The Fay-Herriot model is a random effects model with a normality assumption for both the random effects and the errors. Moreover, the error variances are assumed to be known. The latter is almost imperative due to an identifiability issue. With availability of only the area level direct small area estimates plus nonavailability of microdata, any effective modeling of the error variances is near impossible.
Some valiant remedial attempts were made by W.R. Bell and his colleagues at the US Census Bureau (Bell and Huang, 2006; Bell, 2008) for handling some census data, but questions remain regarding the universal application of their approach. Additionally, nonavailability of microdata for secondary survey users is primarily due to confidentiality reasons, especially from the Federal Agencies. If microdata becomes available, unit level models are more appropriate than area level models. A classic example is the well-cited article of Battese, Harter and Fuller (1988). However, area level models are widely used due to their simplicity of implementation in a complex survey setting when compared to unit level models.
As the field developed and more data started getting analyzed, researchers found the inappropriateness of the assumption of normality as well as that of known error variances. As mentioned in the previous paragraph, the latter is hard to rectify without any extra information. One of the first attempts in this regard is due to Lahiri and Rao (1995) who replaced the normality assumption of random effects by the finiteness of their eighth moments. Datta and Lahiri (1995) considered a general mixture of normal distributions for random effects that includes the distribution. There are papers, dispensing fully with the normality, but maintaining linearity of the model, and using ANOVA estimators of the variances. One may refer to Butar and Lahiri (2002) and Jiang, Lahiri and Wan (2002) who calculated the corresponding uncertainty measures either via jackknife or bootstrap. Bell and Huang (2006) used distributions for random effects or sampling errors to diminish the effects of outliers.
The objective of the present article is to address these two important issues in the context of small area estimation. First, we consider small area modeling of both the population means and population variances. This is possible due to the availability of additional data purported to estimate the error variances. Second, in order to induce some robustness of our procedure, we consider priors for the random effects.
The data set considered in this paper came from a test demographic census carried out in one municipality in Brazil consisting of 140 enumeration districts, hereafter referred to as small areas. The response variable was the average income of the heads of households for each small area, and the goal was to make predictions for the 140 population means of the heads of household’s income. The auxiliary variables were the respective small area population means of the educational attainment of the heads of households, and the respective population means of the number of rooms in the households for each small area. Only area level data was provided to us.
We propose a full non-subjective Bayesian analysis for the general small area problem, where we model both the population means and variances. The initial idea was to use Jeffreys’ general rule prior, treating all the parameters including the degrees of freedom of the Student’s distribution as unknown. However, the resultant prior yielded an improper posterior, which led to a modified Jeffreys’ prior resulting in a proper posterior.
The outline of the remaining sections is as follows. Section 2 introduces the model, the Fisher information matrix, Jeffreys’ prior and its modification. The impropriety of the posterior under the former, and its propriety under the latter are also included in this section. Section 3 contains a real data analysis as well as a simulation study. Some final remarks are made in Section 4.
The fact that error variances are really random has been recognized for a long time. The work of Otto and Bell (1995), Arora and Lahiri (1997), Wang and Fuller (2003), Rivest and Vandal (2003) and others have tried to account for this in different ways. Slud and Maiti (2006), Dass, Maiti, Ren and Sinha (2012) and Maiti, Ren and Sinha (2014) used an empirical Bayes approach towards this end by estimating the hyper-parameters. Full Bayesian analysis using hierarchical Bayesian methods with normality of area level effects has been considered in You and Chapman (2006) and Sugasawa, Tamae and Kubokawa (2017). We will demonstrate that priors for random effects often perform better than the methods of the last two papers via data analysis and simulations.
The use of priors for the errors in the standard normal regression models, but not in mixed effects models, was proposed in Lange, Little and Taylor (1989), Fernandez and Steel (1998), Vrontos, Dellaportas and Politis (2000), Jacquier, Polson and Rossi (2004), and Fonseca, Ferreira and Migon (2008) primarily for protection against outliers. However, there are situations where normality of errors is a reasonable assumption, mainly because of the central limit theorem. Also, there are model diagnostic techniques to check this. The normality assumption of random effects, however, does not always work well. For the Brazilian data that we have on hand, joint modeling of both sample means and variances along with priors for random effects yields better performance than some of the other area level models. As suggested by referee, we used the data to compute the residuals fitting a regression with the true area means and covariates to investigate the distribution of the random effects for this application. See Section 3.
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