🤖 AI Summary
Classical empirical processes fail under heavy-tailed or skewed distributions where moments (e.g., mean or variance) do not exist. To address this, we propose the trimmed functional empirical process (TFEP) framework, which relaxes the conventional finite-variance requirement. Under mild regularity conditions, TFEP converges weakly to a Gaussian process, enabling novel asymptotic theory for one-sample and two-sample hypothesis tests as well as confidence intervals. Our methodology integrates extreme-order-statistic trimming, functional empirical process analysis, and Monte Carlo simulation, supporting robust inference for trimmed means, variances, and their ratios. Empirical evaluations demonstrate that TFEP substantially outperforms classical functional empirical processes and normal approximation methods under Pareto and Cauchy distributions—yielding more accurate confidence interval coverage. We further validate its practical utility through application to real-world income data analysis.
📝 Abstract
This paper introduces the Trimmed Functional Empirical Process (TFEP) as a robust framework for statistical inference when dealing with heavy-tailed or skewed distributions, where classical moments such as the mean or variance may be infinite or undefined. Standard approaches including the classical Functional Empirical Process (FEP), break down under such conditions, especially for distributions like Pareto, Cauchy, low degree of freedom Student-t, due to their reliance on finite-variance assumptions to guarantee asymptotic convergence. The TFEP approach addresses these limitations by trimming a controlled proportion of extreme order statistics, thereby stabilizing the empirical process and restoring asymptotic Gaussian behavior. We establish the weak convergence of the TFEP under mild regularity conditions and derive new asymptotic distributions for one-sample and twosample problems. These theoretical developments lead to robust confidence intervals for truncated means, variances, and their differences or ratios. The efficiency and reliability of the TFEP are supported by extensive Monte Carlo experiments and an empirical application to Senegalese income data. In all scenarios, the TFEP provides accurate inference where both Gaussian-based methods and the classical FEP break down. The methodology thus offers a powerful and flexible tool for statistical analysis in heavy-tailed and non-standard environments.