Efficient estimation of pathwise differentiable target parameters with the undersmoothed highly adaptive lasso.

Area of Science:

• Statistics
• Statistical Inference
• Machine Learning

Background:

• Functional parameter estimation is crucial for understanding complex data distributions.
• Existing methods may face challenges with high-dimensional or complex functional parameters.
• The highly adaptive lasso (HAL) estimator offers a novel approach to functional parameter estimation.

Purpose of the Study:

• To establish the asymptotic efficiency of the HAL estimator for functional parameters.
• To investigate the role and impact of undersmoothing in HAL estimation.
• To demonstrate the practical utility and performance of the HAL estimator.

Main Methods:

• Defining the HAL estimator as a minimizer of empirical risk over a specific function class.
• Analyzing the estimator's behavior under a global undersmoothing condition.
• Formally showing the HAL estimator's ability to solve score equations without enforcing L1-restriction.

Main Results:

• The HAL estimator achieves asymptotic efficiency for smooth features of functional parameters.
• Undersmoothing is identified as a strategy to accommodate potentially less complex true functions.
• Practical verification shows the proposed targeted undersmoothing method performs well.

Conclusions:

• The HAL estimator is a theoretically sound and practically effective tool for functional parameter estimation.
• The study confirms the benefits of targeted undersmoothing in specific scenarios.
• Demonstrated applications include estimating treatment-specific means and integrated square densities.