Statistical Inference for High-Dimensional Heteroscedastic Partially Single-Index Models

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• High-dimensional statistical models present challenges in parameter estimation and variable selection.
• Heteroscedasticity and partially linear structures complicate standard modeling approaches.
• Existing methods may struggle with simultaneously addressing estimation and selection in complex settings.

Purpose of the Study:

• To develop a novel penalized empirical likelihood approach for heteroscedastic partially linear single-index models.
• To simultaneously perform parameter estimation and variable selection.
• To address models with a diverging number of parameters in high-dimensional settings.

Main Methods:

• Proposed a penalized empirical likelihood method.
• Rigorously proved the oracle property for the estimators.
• Established the asymptotic distribution of the penalized empirical log-likelihood ratio statistic.

Main Results:

• The proposed method achieves consistent estimation of zero components and asymptotic efficiency for non-zero coefficients.
• The penalized empirical log-likelihood ratio statistic follows a chi-squared distribution under the null hypothesis.
• Demonstrated applicability to pure partially linear and single-index models in high-dimensional scenarios.

Conclusions:

• The novel penalized empirical likelihood approach offers a robust solution for parameter estimation and variable selection.
• The method exhibits desirable theoretical properties, including the oracle property and asymptotic efficiency.
• Validated through simulation studies and real-world data analysis, confirming its practical utility.