Nonparametric estimation of multivariate convex-transformed densities

Seregin, Arseni ; Wellner, Jon A.

Ann. Statist., Tome 38 (2010) no. 1, p. 3751-3781 / Harvested from Project Euclid

Résumé

We study estimation of multivariate densities p of the form p(x)=h(g(x)) for x∈ℝ^d and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y)=e^−y for y∈ℝ; in this case, the resulting class of densities ¶ \[\mathcal {P}(e^{-y})=\{p=\exp(-g)\dvtx g\mbox{ is convex}\}\] ¶ 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 p̂ 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.

Détail

Publié le : 2010-12-15
Classification:  Consistency,  log-concave density estimation,  lower bounds,  maximum likelihood,  mode estimation,  nonparametric estimation,  qualitative assumptions,  shape constraints,  strongly unimodal,  unimodal,  62G07,  62H12,  62G05,  62G20