Using $t$-distribution for Robust ‎Hierarchical Bayesian Small Area Estimation under Measurement Error in Covariates

Abstract

Small area estimation often suffers from imprecise direct estimators due to small sample sizes. One method for giving direct estimators more strength is to use models.‎ Models ‎employ area effects and ‎include supplementary information  from extra sources as covariates to increase the accuracy of direct estimators. ‎The valid covariates are the basis of ‎the ‎small ‎area ‎estimation.‎ Therefore, measurement error (ME) in covariates can produce contradictory results, i.e., even reduce the precision of direct estimators. The Gaussian distribution with known variance is generally apply as a distribution of ME. ‏‎ ‎However‎, ‎in real problem, ‎‎there might be situations in which the normality assumption fo MEs does not hold‎. In addition, the assumption of known ME variance is restricted. To address these issues and obtain a more robust model, ‎‎we propose modeling ME using a $t$-distribution with known and unknown degrees of freedom. Model parameters are estimated using a fully Bayesian framework based on MCMC methods. We validate our proposed model using simulated data and apply it to well-known crop data and the cost and income of households living in Kurdistan province of ‎Iran.‎