Statistica Sinica 16(2006), 353-374
AN EFFECTIVE METHOD FOR HIGH-DIMENSIONAL
LOG-DENSITY ANOVA ESTIMATION, WITH APPLICATION
TO NONPARAMETRIC GRAPHICAL MODEL BUILDING
Yongho Jeon and Yi Lin
University of Wisconsin
Abstract:
The log-density functional ANOVA model provides a powerful framework for the estimation and
interpretation of high-dimensional densities. Existing methods for fitting such a model require
repeated numerical integration of high-dimensional functions, and are infeasible in problems of
dimension larger than four. We propose a new method for fitting the log-density ANOVA model
based on a penalized 
-estimation formulation with a novel loss function. Solving the penalized
-estimation problem does not require high-dimensional integration: only one-dimensional
integrals are required and they can be computed quickly by using the cumulative distribution
function of familiar one-dimensional densities. Simulations indicate that the proposed method is
statistically very efficient and computationally practical in high-dimensional problems. We apply
the new method to the construction and estimation of (undirected) nonparametric graphical models.
The graphical models use graphs to display the conditional dependence among random variables and
have become very popular, but have mostly been studied parametrically. Our method provides a
practical way to construct and estimate nonparametric graphical models.
Key words and phrases:
Density estimation, functional ANOVA model, graphical model, model selection,
penalized M-estimation.