Impact of existence and nonexistence of pivot on the coverage of empirical best linear prediction intervals for small areas

We advance the theory of parametric bootstrap in constructing highly efficient empirical best (EB) prediction intervals of small area means. The coverage error of such a prediction interval is of the order $O(m^{-3/2})$, where $m$ is the number of small areas to be pooled using a linear mixed normal model. In the context of an area level model where the random effects follow a non-normal known distribution except possibly for unknown hyperparameters, we analytically show that the order of coverage error of empirical best linear (EBL) prediction interval remains the same even if we relax the normality of the random effects by the existence of pivot for a suitably standardized random effects when hyperpameters are known. Recognizing the challenge of showing existence of a pivot, we develop a simple moment-based method to claim non-existence of pivot. We show that existing parametric bootstrap EBL prediction interval fails to achieve the desired order of the coverage error, i.e. $O(m^{-3/2})$, in absence of a pivot. We obtain a surprising result that the order $O(m^{-1})$ term is always positive under certain conditions indicating possible overcoverage of the existing parametric bootstrap EBL prediction interval. In general, we analytically show for the first time that the coverage problem can be corrected by adopting a suitably devised double parametric bootstrap. Our Monte Carlo simulations show that our proposed single bootstrap method performs reasonably well when compared to rival methods.

💡 Research Summary

The paper addresses a fundamental problem in small‑area estimation: constructing prediction intervals for area‑level means that achieve high coverage accuracy when the random effects at level 2 follow a known but possibly non‑normal distribution. Classical approaches (Cox 1975, Yoshi­mori & Lahiri 2014) rely on normality at both levels and attain coverage errors of order (O(m^{-1})) or, with refined estimators of the variance component, (O(m^{-3/2})). However, these results do not extend to models where the level‑2 random effects are non‑normal.

The authors introduce the concept of a pivot—a statistic whose distribution does not depend on unknown parameters—to analyze how the existence or non‑existence of such a pivot influences the performance of empirical best linear (EBL) prediction intervals. When a pivot exists for the standardized random effect, the single‑parametric bootstrap procedure of Chatterjee et al. (2008) can be applied unchanged, and the resulting interval (\hat I_i^{(s)}) enjoys a coverage error of order (O(m^{-3/2})), exactly as in the normal‑normal case. The key technical condition is that
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