Wasserstein regression with empirical measures and density estimation for sparse data.

Area of Science:

• Statistics
• Biostatistics
• Environmental Health

Background:

• Modeling relationships between distributions and explanatory variables is challenging, particularly with limited data for some distributions.
• Existing methods struggle with density estimation when individual response distributions have few data points.
• Tuning parameter selection and bias are common issues in traditional density estimation approaches.

Purpose of the Study:

• To develop a novel distribution-response regression model that overcomes limitations of existing methods, especially with sparse data.
• To enable consistent distribution estimation even for distributions with few available data points.
• To address challenges in density estimation, including parameter selection and bias.

Main Methods:

• A new distribution-response regression model based on empirical measures is proposed.
• The approach avoids pre-estimating individual response distributions, accommodating varying sample sizes.
• Leverages data across multiple distributions to enhance estimates for sparsely sampled ones.

Main Results:

• The proposed model successfully handles situations with small sample sizes for some distributions.
• It provides consistent distribution estimates where traditional methods fail due to sparse data.
• Outperforms existing approaches in simulations and on Environmental Influences on Child Health Outcomes data.

Conclusions:

• The novel empirical measure-based regression model offers a robust solution for distribution-response modeling with heterogeneous sample sizes.
• This method significantly improves density estimation accuracy in challenging data scenarios.
• Applicable to various fields, including environmental health, where data sparsity is common.