HIGHER ORDER ESTIMATING EQUATIONS FOR HIGH-DIMENSIONAL MODELS.

Area of Science:

• Statistics
• Econometrics
• Biostatistics

Background:

• Semi-parametric and non-parametric models are widely used in statistical analysis.
• Estimating parameters in these models, especially with high-dimensional nuisance parameters, presents significant challenges.
• Existing methods may not achieve optimal convergence rates for all parameters of interest.

Purpose of the Study:

• To introduce a new estimation method for parameters in semi-parametric and non-parametric models.
• To develop estimators that can achieve optimal convergence rates, even in challenging model settings.
• To address the estimation of parameters with high-dimensional or low-regularity nuisance parameters.

Main Methods:

• The proposed method utilizes estimating equations based on U-statistics.
• These U-statistics are constructed using higher-order influence functions, extending traditional linear influence functions.
• The approach is applied to models with nuisance parameters and demonstrated on mean response estimation with missing data.

Main Results:

• The method provides a bias-variance trade-off for parameters where perfect representation is not feasible, leading to slower convergence rates.
• In specific examples, the resulting estimation rates are shown to be optimal.
• The technique is particularly effective for models with high-dimensional or low-regularity nuisance parameters, where standard methods fail.

Conclusions:

• The developed U-statistics-based method offers a powerful new tool for parameter estimation in complex statistical models.
• It provides a flexible framework for achieving optimal estimation rates, especially in the presence of challenging nuisance parameters.
• The method demonstrates practical utility through its application to missing data problems.