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Chemical equations represent the identities and relative quantities of substances involved in a chemical reaction. The substances undergoing reaction are called reactants, and their formulas are placed on the left side of the equation. The substances generated by the reaction are called products, and their formulas are placed on the right side of the equation. Plus signs (+) separate individual reactant and product formulas, and an arrow (→) separates the reactant and product (left and right)...
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James Robins1, Lingling Li1, Rajarshi Mukherjee1

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Summary
This summary is machine-generated.

This study introduces a novel U-statistics-based estimation method for semi-parametric and non-parametric models. It offers optimal estimation rates, particularly for complex models with high-dimensional nuisance parameters.

Keywords:
62F2562G20Nonlinear functionalPrimary 62G05U-statisticinfluence functionnonparametric estimationtangent space

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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.