Assessing the Role of Distribution-Free Statistical Methods in Handling Data with Unknown Probability Structures

Posted: May 05, 2023

• Riley Baker
• Harper Anderson
• Noah Rodriguez

Abstract

Statistical analysis forms the backbone of empirical research across numerous disciplines, from economics and biology to computer science and social sciences. Traditional statistical methods typically rely on strong assumptions about the underlying probability distributions of the data being analyzed. Parametric approaches, such as t-tests, ANOVA, and linear regression, assume specific distributional forms (e.g., normality) that may not hold in practice. When these assumptions are violated, statistical inferences can become unreliable, leading to incorrect conclusions and potentially costly decisions. The increasing complexity of modern datasets, characterized by heterogeneity, outliers, and unknown generating processes, has exposed the limitations of conventional parametric methods. This paper addresses the critical challenge of statistical inference when the probability structure of data is unknown or poorly understood. We present a comprehensive assessment of distribution-free statistical methods, which make minimal assumptions about the underlying data distribution. Unlike traditional approaches that require specification of parametric families, distribution-free methods rely on weaker assumptions, typically concerning only the continuity or boundedness of distributions. These methods include rank-based procedures, permutation tests, bootstrap methods, and other nonparametric techniques that have gained prominence in recent decades. Our research is motivated by the observation that many real-world datasets exhibit characteristics that violate standard distributional assumptions. Financial returns often display heavy tails and volatility clustering, ecological data may show complex spatial and temporal dependencies, and social network data frequently exhibits power-law distributions. In such contexts, distribution-free methods offer a more robust alternative to parametric approaches. However, the adoption of these methods has been hindered by several factors, including computational complexity, limited power in small samples, and a lack of comprehensive comparative studies. This paper makes several original contributions to the literature on statistical methodology. First, we introduce a novel framework called Adaptive Non-