Patra and Sen (2016) consider a two-component mixture model, where one component plays the role of background while the other plays the role of signal, and propose to estimate the background component by simply "maximizing" its weight. While in their work the background component is a completely known distribution, we extend their approach here to three emblematic settings: when the background distribution is symmetric; when it is monotonic; and when it is log-concave. In each setting, we derive estimators for the background component, establish consistency, and provide a confidence band. While the estimation of a background component is straightforward when it is taken to be symmetric or monotonic, when it is log-concave its estimation requires the computation of a largest concave minorant, which we implement using sequential quadratic programming. Compared to existing methods, our method has the advantage of requiring much less prior knowledge on the background component, and is thus less prone to model misspecification. We illustrate this methodology on a number of synthetic and real datasets.
翻译:Patra和Sen(Sen)(Patra和Sen(Sen))考虑一种双重成分混合模型,其中一种成分发挥背景作用,而另一种成分则发挥信号的作用,并提议仅用“最大化”其重量来估计背景成分。虽然在其工作中,背景成分的分布是完全已知的,但我们将其方法扩大到三个符号性环境:当背景分布是对称的;当它是单调的;当它是单调的;当它是对称的;当它是对称或单调时,我们得出背景成分的估测符,建立一致性并提供一个信任带。虽然当一个背景成分的估算被视为对称或单调时,其估算是直截的,但当它是一个对称的或单调的时,其估算需要计算一个最大的二次的微小,我们用顺序的四方形编程来进行计算。与现有方法相比,我们的方法的优势是,对背景成分的先前知识要求要少得多,因此不易于模型的辨别。我们用一些合成和真实的数据集来说明这一方法。