The Annals of Statistics

Nonparametric estimation of multivariate convex-transformed densities

Arseni Seregin and Jon A. Wellner

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We study estimation of multivariate densities p of the form p(x)=h(g(x)) for x∈ℝd and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y)=ey for y∈ℝ;; in this case, the resulting class of densities $$\mathcal {P}(e^{-y})=\{p=\exp(-g): g\mbox{ is convex}\} $$ is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails than the log-concave class.

We first investigate when the maximum likelihood estimator exists for the class $\mathcal {P}(h)$ for various choices of monotone transformations h, including decreasing and increasing functions h. The resulting models for increasing transformations h extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to h(y)=exp(y).

We then establish consistency of the maximum likelihood estimator for fairly general functions h, including the log-concave class $\mathcal {P}(e^{-y})$ and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of p and its vector of derivatives at a fixed point x0 under natural smoothness hypotheses on h and g. The proofs rely heavily on results from convex analysis.

Article information

Ann. Statist., Volume 38, Number 6 (2010), 3751-3781.

First available in Project Euclid: 30 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62H12: Estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Consistency log-concave density estimation lower bounds maximum likelihood mode estimation nonparametric estimation qualitative assumptions shape constraints strongly unimodal unimodal


Seregin, Arseni; Wellner, Jon A. Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist. 38 (2010), no. 6, 3751--3781. doi:10.1214/10-AOS840.

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Supplemental materials

  • Supplementary material: Omitted Proofs and Some Facts from Convex Analysis. In the supplement, we provide omitted proofs and some basic facts from convex analysis used in this paper.