## Nonparametric estimation of multivariate convex-transformed densities.(English)Zbl 1204.62058

Summary: 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 $\mathcal P (e^{-y}) = \{\exp (-g): g \text{ is convex }\}$ 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.

### MSC:

 62G07 Density estimation 62H12 Estimation in multivariate analysis 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference

logcondens
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### References:

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