A two-class mixture model, where the density of one of the components is known, is considered. We address the issue of the nonparametric adaptive estimation of the unknown probability density of the second component. We propose a randomly weighted kernel estimator with a fully data-driven bandwidth selection method, in the spirit of the Goldenshluger and Lepski method. An oracle-type inequality for the pointwise quadratic risk is derived as well as convergence rates over Hölder smoothness classes. The theoretical results are illustrated by numerical simulations.

中文翻译：

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二类混合模型中成分密度的自适应非参数估计

考虑一个二类混合模型，其中一个组件的密度是已知的。我们解决了第二个分量的未知概率密度的非参数自适应估计问题。本着 Goldenshluger 和 Lepski 方法的精神，我们提出了一个随机加权的内核估计器，它具有完全数据驱动的带宽选择方法。推导出逐点二次风险的预言式不等式以及 Hölder 平滑类上的收敛率。数值模拟说明了理论结果。