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

2026-07-06

AI Summary

This study addresses the challenge of modeling uncertain data arising from frequency instability and subjective judgment in complex environments by introducing uncertainty theory into the single-index model framework for the first time, thereby providing a unified treatment of both precise and fuzzy explanatory variables. Two classes of uncertain single-index models are proposed, each integrated with a semiparametric least squares approach: one employs Nadaraya–Watson kernel estimation and the other utilizes B-spline approximation to estimate the unknown link function. Corresponding parameter estimation procedures and hypothesis testing frameworks are developed. The effectiveness and practical utility of the proposed methods are validated through residual analysis, simulation studies, and real-data applications, demonstrating significantly enhanced fitting capability for mixed-type covariates while preserving model flexibility.

Abstract

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.

Problem

• uncertain data
• single-index models
• parameter estimation
• imprecise explanatory variables
• model fitting

Innovation

• uncertain single-index models
• semiparametric least-squares
• Nadaraya-Watson kernel
• B-spline estimation
• hypothesis testing under uncertainty