Semiparametric Minimax Rates.

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Semiparametric models are widely used in statistical inference.
• Existing methods struggle to determine minimax rates for certain non-linear functionals, especially with multiple infinite-dimensional parameters.
• Accurate estimation of these functionals is crucial for various applications.

Purpose of the Study:

• To develop a novel statistical approach for determining the minimax rate of testing and estimation for non-linear functionals in semiparametric models.
• To extend existing methodologies to handle semiparametric models with multiple infinite-dimensional parameters.
• To provide a robust framework for lower bound determination in complex statistical settings.

Main Methods:

• The study extends a previously established method based on comparing distributions.
• The new approach involves comparing two convex mixtures of distributions, perturbing one or two parameters.
• This technique is applied to derive minimax rates for specific semiparametric functionals.

Main Results:

• A new method is presented for calculating the minimax rate of estimation in semiparametric models.
• The approach successfully addresses models with multiple infinite-dimensional parameters, a limitation of prior methods.
• The efficacy of the method is demonstrated through applications to missing data and covariance estimation.

Conclusions:

• The developed method provides a powerful tool for analyzing the statistical efficiency of estimators in complex semiparametric models.
• This research advances the theoretical understanding of estimation bounds in high-dimensional statistical settings.
• The findings have implications for improving statistical inference in areas like missing data analysis and financial modeling.