Asymptotic Normality of Quadratic Estimators

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Quadratic U-statistics are fundamental in statistical inference.
• Their application is often limited by complex kernel structures and high-dimensional data.
• Existing methods struggle with degenerate second-order parts and changing kernels.

Purpose of the Study:

• To prove conditional asymptotic normality for a specific class of quadratic U-statistics.
• To extend the applicability of U-statistics in high-dimensional statistical modeling.
• To provide theoretical foundations for nonparametric confidence sets and estimators.

Main Methods:

• Analysis of quadratic U-statistics with degenerate second-order parts.
• Development of techniques for kernels that vary with the number of observations.
• Conditional asymptotic normality proofs under non-standard convergence rates.

Main Results:

• Conditional asymptotic normality is established for the studied class of U-statistics.
• The results hold even when the convergence rate is slower than the square root of the sample size.
• Demonstrated applicability in estimating integrals of squared functions and handling missing data.

Conclusions:

• The findings broaden the utility of U-statistics in advanced statistical applications.
• Enables robust estimation and confidence set construction in high-dimensional settings.
• Provides a theoretical basis for developing new statistical methods for complex data.