Shape-Constrained Univariate Density Estimation

TL;DR

This paper presents a geometric method for estimating univariate probability density functions with shape constraints, specifically a fixed number of modes, using diffeomorphisms and maximum likelihood optimization.

Contribution

It introduces a novel, efficient geometric framework for shape-constrained density estimation that leverages diffeomorphisms and tangent space representations.

Findings

The method achieves consistent density estimates with known asymptotic rates.

Application to traffic speed data demonstrates practical effectiveness.

Framework extends to conditional density estimation.

Abstract

While the problem of estimating a probability density function (pdf) from its observations is classical, the estimation under additional shape constraints is both important and challenging. We introduce an efficient, geometric approach for estimating pdfs given the number of its modes. This approach explores the space of constrained pdf's using an action of the diffeomorphism group that preserves their shapes. It starts with an initial template, with the desired number of modes and arbitrarily chosen heights at the critical points, and transforms it via: (1) composition by diffeomorphisms and (2) normalization to obtain the final density estimate. The search for optimal diffeomorphism is performed under the maximum-likelihood criterion and is accomplished by mapping diffeomorphisms to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. This framework is first applied to shape-constrained univariate, unconditional pdf estimation and then extended to conditional pdf estimation. We derive asymptotic convergence rates of the estimator and demonstrate this approach using a synthetic dataset involving speed distribution for different traffic flow on Californian driveways.