NONPARAMETRIC ESTIMATION OF MULTIVARIATE CONVEX-TRANSFORMED DENSITIES

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Multivariate density estimation is crucial in various statistical and machine learning applications.
• Log-concave densities are a well-studied class, but their limitations necessitate broader models.
• Existing methods often struggle with complex density structures.

Purpose of the Study:

• To investigate the estimation of multivariate densities of the form p(x) = h(g(x)), where h is monotone and g is convex.
• To determine the conditions for the existence and consistency of maximum likelihood estimators for these generalized densities.
• To establish asymptotic minimax lower bounds for density and derivative estimation.

Main Methods:

• Exploration of density estimation using transformations of convex functions.
• Analysis of the existence of maximum likelihood estimators for various monotone functions h.
• Establishment of consistency for maximum likelihood estimators.
• Derivation of asymptotic minimax lower bounds using convex analysis.

Main Results:

• The study demonstrates the existence of maximum likelihood estimators for a broad class of densities, including extensions of log-convex densities.
• Consistency of the maximum likelihood estimator is established for general monotone functions h.
• Asymptotic minimax lower bounds are derived for density and derivative estimation.

Conclusions:

• The proposed framework for density estimation is robust and extends existing methodologies.
• The findings provide theoretical guarantees for the maximum likelihood estimator in this generalized setting.
• This work offers a foundation for more flexible and accurate density estimation in complex scenarios.