Adaptive demixing in Poisson mixture models

• 成果类型: Article
• 署名作者: Hengartner, NW
• 署名单位: Yale University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1069362730
• 发表日期: 1997
• 页码: 917-928
• 关键词: convergence; rates

摘要

Let X-1,X-2,...,X-n be an i.i.d. sample from the Poisson mixture distribution p(x) = (1/x!)integral(o)(infinity) s(x)e(-s) f(s)ds. Rates of convergence in mean integrated squared error (MISE) of orthogonal series estimators for the mixing density f supported on [a, b] are studied. For the Holder class of densities whose rth derivative is Lipschitz alpha, the MISE converges at the rate (log n/log log n)(-2(r + alpha)). For Sobolev classes of densities whose rth derivative is square integrable, the MISE converges at the rate (log n/log log n)(-2r). The estimator is adaptive over both these classes. For the Sobolev class, a lower bound on the minimax rate of convergence is (log n/log log n)(-2r), and so the orthogonal polynomial estimator is rate optimal.