🤖 AI Summary
This study addresses the lack of asymptotically valid confidence intervals for quantile estimation under heterogeneous data. The authors propose a novel consistent estimator that accurately accounts for the asymptotic variance reduction induced by data grouping, thereby enabling the construction of asymptotically efficient confidence intervals. This approach provides, for the first time, theoretically guaranteed asymptotically correct confidence intervals for quantiles in heterogeneous settings, with substantially reduced interval width. Rigorous theoretical analysis establishes both the consistency of the proposed estimator and the asymptotic validity of the resulting intervals. Extensive simulations demonstrate that the constructed intervals achieve coverage probabilities close to the nominal level across various heterogeneity scenarios and are markedly shorter than those derived under the conventional i.i.d. assumption.
📝 Abstract
It is well known that the asymptotic variance of sample quantiles can be reduced under heterogeneity relative to the i.i.d. setting. However, asymptotically correct confidence intervals for quantiles are not yet available. We propose a novel, consistent estimator of the reduced asymptotic variance arising when quantiles are computed from groups of observations, leading to asymptotically correct confidence intervals. Simulation studies show that our confidence intervals are substantially shorter than those in the i.i.d. case and attain nearly correct coverage across a wide range of heterogeneous settings.