  We study estimation of multivariate densities $p$ of the form $p(x)=h(g(x))$
for $x\in \mathbb {R}^d$ and for a fixed monotone function $h$ and an unknown
convex function $g$. The canonical example is $h(y)=e^{-y}$ for $y\in \mathbb
{R}$; 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 densities with heavier
tails than the log-concave class. We first investigate when the maximum
likelihood estimator $\hat{p}$ 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 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 $x_0$ under natural
smoothness hypotheses on $h$ and $g$. The proofs rely heavily on results from
convex analysis.
