Nonparametric estimation of multivariate convex-transformed densities

We study estimation of multivariate densities $p$ of the form $p(x) = h(g(x))$ for $x \in R^d$ and for a fixed function $h$ and an unknown convex function $g$. The canonical example is $h(y) = e^{-y}$ for $y \in R$; in this case the resulting class of densities $\mathcal{P}(e^{-y}) = \{p = \exp(-g) : g is convex \}$ is well-known as the class of log-concave densities. Other functions $h$ allow for classes of classes of densities with heavier tails than the log-concave class. We first investigate when the MLE $\hat{p}$ 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 MLE for fairly general functions $h$, including the log-concave class $\Model(e^{-y})$ and many others. In a final section we provide asymptotic minimax lower bounds for estimation of $p$ and its vector of derivatives at a fixed point $x_0$ under natural smoothness hypotheses on $h$ and $g$. The proofs rely heavily on results from convex analysis.