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Approximation and estimation of \(s\)-concave densities via Rényi divergences. (English) Zbl 1338.62105

Summary: In this paper, we study the approximation and estimation of \(s\)-concave densities via Rényi divergence. We first show that the approximation of a probability measure \(Q\) by an \(s\)-concave density exists and is unique via the procedure of minimizing a divergence functional proposed by R. Koenker and I. Mizera [Ann. Stat. 38, No. 5, 2998–3027 (2010; Zbl 1200.62031)] if and only if \(Q\) admits full-dimensional support and a first moment. We also show continuity of the divergence functional in \(Q\): if \(Q_{n}\to Q\) in the Wasserstein metric, then the projected densities converge in weighted \(L_{1}\) metrics and uniformly on closed subsets of the continuity set of the limit. Moreover, directional derivatives of the projected densities also enjoy local uniform convergence. This contains both on-the-model and off-the-model situations, and entails strong consistency of the divergence estimator of an \(s\)-concave density under mild conditions. One interesting and important feature for the Rényi divergence estimator of an \(s\)-concave density is that the estimator is intrinsically related with the estimation of log-concave densities via maximum likelihood methods. In fact, we show that for \(d=1\) at least, the Rényi divergence estimators for \(s\)-concave densities converge to the maximum likelihood estimator of a log-concave density as \(s\nearrow 0\). The Rényi divergence estimator shares similar characterizations as the MLE for log-concave distributions, which allows us to develop pointwise asymptotic distribution theory assuming that the underlying density is \(s\)-concave.

MSC:

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

Citations:

Zbl 1200.62031
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