Parameter estimation and application in two types of uncertain single-index models

Uncertain data often arises in complex environments because of frequency instability and subjective judgment. This paper establishes two types of uncertain single-index models to capture the inherent properties of such data. Based on the semiparametric least-squares principle, the Nadaraya-Watson kernel and B-spline methods are used to estimate the unknown coefficients in various scenarios with both crisp and imprecise explanatory variables. Residual analysis and hypothesis testing under uncertainty assess the fit of the proposed models. Furthermore, simulation studies verify the models' validity, and a real-data application demonstrates their effectiveness in practical settings.