Confidence procedures play a fundamental role in statistical inference, providing a principled mechanism for quantifying uncertainty around parameter estimates. Since their introduction in the early 20th century, confidence intervals have become standard throughout applied statistics. However, their classical construction typically relies on asymptotic approximations, and quantitative properties—most notably coverage probability—can deteriorate in small-sample settings. While numerous refinements and bootstrap-based solutions have been proposed for specific estimation problems, general distribution-free approaches remain comparatively limited.
In the context of small area and domain estimation, the lack of reliable inferential tools is particularly acute. The demand for valid and robust methods has never been more pressing. As an initial motivation, we conducted extensive simulation studies to assess the validity of existing and tailored procedures for constructing simultaneous confidence intervals in domain estimation. The conclusion is unequivocal: statisticians require inference methods specifically designed for the small-sample, complex-estimation setting.
This thesis develops a methodology for confidence procedures in small-sample, distribution-free settings, grounded in optimal transport theory. Our novel methodology generalizes the classical quantile-based approach and adapts to the geometry of the underlying probability measure of interest. Under mild regularity assumptions, we derive explicit finite-sample bounds on the coverage probability. We also establish concentration bounds for the length of the empirical confidence intervals.
To provide a complete theoretical framework, we complement our non-asymptotic analysis with large-sample results. In particular, we show that the empirical confidence intervals converge stochastically—in the Hausdorff metric—to their corresponding population-level confidence regions.
The practical relevance of the proposed methodology is assessed through extensive Monte Carlo simulations. Our optimal transport-based intervals systematically achieve empirical coverage closer to the specified level than several established bootstrap procedures. In addition, they exhibit better stability, both in terms of variability and mean squared error of the coverage probability.
Finally, we apply the methodology to small-area estimation to construct confidence regions that simultaneously cover multiple area parameters. In this context, uniform coverage is challenging to achieve with standard resampling techniques. We adapt our optimal transport strategy and demonstrate, through simulations and comparative studies, that the resulting simultaneous confidence intervals provide improved coverage and enhanced stability, especially for a moderate number of areas.
