Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas

TL;DR

This paper introduces a vine copula-based nonparametric density estimator that overcomes the curse of dimensionality, showing improved convergence rates and practical performance in high-dimensional settings.

Contribution

It proposes a novel nonparametric estimator using simplified vine copulas, demonstrating dimension-independent convergence and validating its effectiveness through simulations and an astrophysics application.

Findings

Convergence rate is independent of dimension under certain assumptions.

Simulation shows significant finite sample improvements.

Method remains advantageous even when simplifying assumptions are violated.

Abstract

Practical applications of nonparametric density estimators in more than three dimensions suffer a great deal from the well-known curse of dimensionality: convergence slows down as dimension increases. We show that one can evade the curse of dimensionality by assuming a simplified vine copula model for the dependence between variables. We formulate a general nonparametric estimator for such a model and show under high-level assumptions that the speed of convergence is independent of dimension. We further discuss a particular implementation for which we validate the high-level assumptions and establish its asymptotic normality. Simulation experiments illustrate a large gain in finite sample performance when the simplifying assumption is at least approximately true. But even when it is severely violated, the vine copula based approach proves advantageous as soon as more than a few variables are involved. Lastly, we give an application of the estimator to a classification problem from astrophysics.