Self-Tuned VWS within Gibbs and Small Area Estimation
Raim, Irimata, and Livsey (2025+) consider an application of the vertical weighted strips (VWS) method (Raim, Livsey, and Irimata 2024+) to small area estimation (SAE). SAE is used in official statistics to augment estimates from sample surveys (“direct estimates”) with a model. “Small areas” are cross-sections of a population - based on geography or other characteristics - where the sample size from the survey is small.
In particular, You (2021) presents an SAE model with regressions on both direct point estimates and corresponding variance estimates. A Gibbs sampler is proposed for Bayesian analysis, with one family of conditionals being an unfamiliar distribution that arises from assuming a lognormal regression in the variance model. VWS may be used to generate from this family for each small area. We find that some of the latent variances mix poorly using the independent Metropolis-Hastings step considered in You (2021), which is a routine choice for use within Gibbs samplers. In these cases, mixing is seen to be greatly improved by taking exact draws from the conditionals with rejection sampling.
It is computationally burdensome to construct new VWS proposals for each small area over the course of many Gibbs sampling iterations. To address this, we consider a rule-of-thumb to adjust the proposals as the chain evolves. In particular, knots are added using rejected draws when the probability of rejection is too large. Knots are removed when the probability of rejection is sufficiently small and their corresponding regions are found to have a low contribution. The number of adjustments is seen to diminish as the chain moves to the target posterior distribution.