APPROXIMATION AND ESTIMATION OF s-CONCAVE DENSITIES VIA RENYI DIVERGENCES

• 成果类型: Article
• 署名作者: Han, Qiyang; Wellner, Jon A.
• 署名单位: University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1408
• 发表日期: 2016
• 页码: 1332-1359
• 关键词: maximum-likelihood-estimation; limit distribution-theory; log-concave; CONVERGENCE; bounds; rates

摘要

In this paper, we study the approximation and estimation of s-concave densities via Renyi 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 [Ann. Statist. 38 (2010) 2998-3027] 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) -> 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 Renyi 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 Renyi divergence estimators for s-concave densities converge to the maximum likelihood estimator of a log-concave density as s NE arrow 0. The Renyi 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.