Inference and model selection for fuzzy regression methods

This thesis explores how fuzzy logic can be integrated into statistical modeling to address uncertainty arising from imprecise, vague, or incomplete data—situations where classical probabilistic methods are often inadequate. Building on Zadeh’s fuzzy set theory, the work develops a comprehensive framework for fuzzy statistical analysis.The thesis begins by formalizing fundamental concepts such as fuzzy sets and fuzzy numbers, along with their arithmetic and logical operations derived from Zadeh’s extension principle. These tools provide a rigorous foundation for handling fuzzy-valued data. The study then advances fuzzy linear regression by proposing new models that allow both explanatory and response variables to be fuzzy. A key theoretical contribution is the use of fuzzy orthogonality, which shows that regression parameters can be estimated independently without sacrificing generality.To support statistical inference in fuzzy environments, the thesis introduces fuzzy random variables and empirical fuzzy distributions. These concepts are used to construct fuzzy counterparts of confidence intervals, hypothesis tests, and p-values, enabling meaningful inference under epistemic uncertainty.Recognizing the importance of comparing fuzzy quantities, the thesis proposes a novel ranking approach—the Polygon Ranking Method (PRM). This geometrically motivated method captures directional asymmetries and applies to a wide range of fuzzy number shapes. Finally, the thesis extends classical model selection criteria, such as the Akaike Information Criterion and empirical likelihood methods, to fuzzy settings using resampling and PRM-based ranking.Overall, the work presents a unified framework for fuzzy statistical modeling, inference, and model selection, combining theoretical innovation with practical applicability.