Accounting for Uncertainty About Variances in Small Area Estimation

where the yi are direct survey estimates of true population quantities Yi for m small areas, the ei are sampling errors (of the yi) independently distributed as N(0; vi), the ui are small area random e¤ects (model errors) distributed i:i:d: N(0; 3⁄4u), the x 0 i are 1£r row vectors of regression variables for area i, and ̄ is the corresponding vector of regression parameters. From (2), letting § = diag(3⁄4u + vi), ̄ can be estimated by generalized least squares (GLS): b̄ = (X0§¡1X)¡1X0§¡1y with Var(b̄) = (X0§¡1X)¡1, where y = (y1; : : : ; ym)0, and X is m£ r with rows x0i. Then the best linear unbiased predictors (BLUPs) of the Yi can be formed and their error variances obtained from b Yi = hiyi + (1¡ hi)xi b̄ (3) Var(Yi ¡ b Yi) = 3⁄4u(1¡ hi) + (1¡ hi)2x0iVar3b̄ ́xi (4)