ABSTRACT
Let J be the m × 1 vector of the parameters of inferential interest (typically small area totals yi, small area means with i = 1, . . ., m) and let us assume that the direct estimator is available and design-unbiased, i.e.:
, (14.1)
where e is a vector of independent sampling errors with mean vector 0 and known diagonal variance matrix R = diag(φi), φ representing the sampling variances of the direct estimators of the area parameters of interest. Usually φi is unknown and estimated according to a variety of methods, including ‘generalised variance functions’: see Wolter (1985) and Wang and Fuller (2003) for details. The basic area level model assumes that a m × p matrix of area-specific auxiliary variables (including an intercept term), X, is linearly related to J so:
, (14.2)
where α is the p × 1vector of regression parameters, and u is the m × 1 vector of independent random area-specific effects, with zero mean and m × m covariance matrix , and with Im being the m × m identity matrix. The combined model (Fay-Herriot, 1979) can be written as:
, (14.3)
and is a special case of the linear mixed model. Under this model, the BLUP is used extensively to obtain model-based indirect estimators of small
area parameters J and associated measures of variability. This approach allows a combination of the survey data with other data sources in a synthetic regression fitted using population area level covariates. More details on the empirical version (EBLUP) of the predictor can be found in Rao (2003).
