The Fay and Herriot Model and its Distributional Assumptions

The Fay and Herriot (1979) small-area estimator fundamentally relies on assumptions regarding the probability distributions of the random errors.1 The model is built upon a hierarchical structure that specifies distributional assumptions at two levels.2

At its core, the Fay-Herriot model is a linear mixed model that combines direct survey estimates with predictions from a regression model based on auxiliary data. This combination, often referred to as "borrowing strength" from related areas, is guided by the assumed distributions of the two error terms in the model.
The Fay-Herriot model is typically specified in two stages:3

Level 1: The Sampling Model
This level describes the relationship between the direct survey estimate for a small area and the true, unknown value for that area. It accounts for the sampling error inherent in the direct estimate.
The model is expressed as:
yi​=θi​+ei​
where:
●	yi​ is the direct survey estimate for small area i.
●	θi​ is the true underlying value for small area i.
●	ei​ is the sampling error for area i.
A key assumption at this level is that the sampling errors, ei​, are independent and follow a specific probability distribution. Typically, it is assumed that the sampling errors are normally distributed with a mean of 0 and a known variance ψi​:
ei​∼N(0,ψi​)
The assumption that the sampling variances, ψi​, are known is a strong one. In practice, these are often estimated from the survey data.
Level 2: The Linking Model
This level connects the true small-area value, θi​, to a set of area-level auxiliary variables (covariates), xi​. This is where the model "borrows strength" from other areas.4

The model is specified as:
θi​=xi′​β+vi​
where:
●	xi′​ is a vector of auxiliary variables for area i.
●	β is a vector of unknown regression coefficients.
●	vi​ is the area-specific random effect, which accounts for the variability in θi​ not explained by the covariates.
The second critical distributional assumption lies here. The area-specific random effects, 5vi​, are assumed to be independent and identically distributed, following a normal distribution with a mean of 0 and an unknown variance 6σv2​:7

vi​∼N(0,σv2​)

Why These Assumptions are Crucial

These normality assumptions are not merely technical details; they are central to the estimation process for several reasons:
●	Best Linear Unbiased Prediction (BLUP): The normality assumptions allow for the derivation of the Best Linear Unbiased Predictor (BLUP) of the small area means, θi​. The BLUP provides an optimal combination of the direct estimate and the regression-based synthetic estimate.
●	Estimation of the Random Effects Variance: The normality assumption for the random effects is essential for estimating the variance component, σv2​, using methods like maximum likelihood (ML) or restricted maximum likelihood (REML).
●	Measures of Uncertainty: These assumptions are also used to derive measures of uncertainty for the small-area estimates, such as the mean squared error (MSE).
In summary, the Fay and Herriot (1979) small-area estimator is not distribution-free. It explicitly assumes that both the sampling errors and the area-specific random effects are normally distributed. While subsequent research has explored relaxations and robust versions of the model, the foundational framework is firmly based on these distributional assumptions.