Parametric Bootstrap Confidence Intervals for the Multivariate Fay–Herriot Model

Various multivariate extensions to the well-known Fay–Herriot model have been proposed in the small area estimation literature. Such multivariate models are quite effective in combining information through correlations among small area survey estimates of related variables or historical survey estimates of the same variable or both. Though the literature on small area estimation is already very rich, construction of second-order efficient confidence intervals from multivariate models has received little attention. In this article, we develop a parametric bootstrap method for constructing a second-order efficient confidence interval for a general linear combination of small area means using the multivariate Fay–Herriot normal model. The proposed parametric bootstrap method replaces difficult and tedious analytical derivations by the power of efficient algorithm and high speed computer. Moreover, the proposed method is more versatile than the analytical method because the parametric bootstrap method can be easily applied to any method of model parameter estimation and any specific structure of the variance–covariance matrix of the multivariate Fay–Herriot model avoiding all the cumbersome and time-consuming calculations required in the analytical method. We apply our proposed methodology in constructing confidence intervals for the median income of four-person families for the fifty states and the District of Columbia in the United States. Our data analysis demonstrates that the proposed parametric bootstrap method, applied to both multivariate and univariate Fay–Herriot models, generally provides much shorter confidence intervals compared to the corresponding traditional direct method. Moreover, the confidence intervals obtained from the multivariate model are generally shorter than the corresponding intervals from the univariate model indicating the potential advantage of exploiting correlations of median income of four-person families with median incomes of three- and five-person families.