Asymptotics of Subsampling for Generalized Linear Regression Models under Unbounded Design.

Area of Science:

• Statistics
• Computational Statistics
• Econometrics

Background:

• Generalized linear models (GLMs) are crucial for statistical analysis, but parameter estimation in massive datasets is computationally intensive.
• Existing optimal subsampling methods are limited to bounded covariates, restricting their applicability.
• Efficient inference for large-scale regression models is a significant challenge in modern data analysis.

Purpose of the Study:

• To develop and analyze optimal subsampling methodologies for generalized linear models (GLMs) with unbounded covariates.
• To extend the theoretical understanding of subsampling estimators beyond bounded covariate assumptions.
• To provide a robust statistical framework for rapid parameter estimation in massive data regression.

Main Methods:

• Derivation of the asymptotic normality for the subsampling M-estimator using the Fisher information matrix.
• Investigation of the asymptotic properties of subsampling estimators for unbounded GLMs.
• Analysis of both conditional and unconditional asymptotic properties for enhanced statistical rigor.

Main Results:

• The asymptotic normality of the subsampling M-estimator is established for generalized linear models.
• The study successfully obtains asymptotic properties for subsampling estimators involving unbounded covariates.
• Both conditional and unconditional asymptotic properties are derived, offering a comprehensive theoretical foundation.

Conclusions:

• Optimal subsampling provides an efficient statistical methodology for parameter estimation in massive data regression.
• The developed methods extend the utility of subsampling to generalized linear models with unbounded covariates and nonnatural links.
• This research offers a significant advancement in the statistical inference for large-scale datasets.