Flexible estimation of a semiparametric two-component mixture model with one parametric component

We study a two-component semiparametric mixture model where one component distribution belongs to a parametric class, while the other is symmetric but otherwise arbitrary. This semiparametric model has wide applications in many areas such as large-scale simultaneous testing/multiple testing, sequential clustering, and robust modeling. We develop a class of estimators that are surprisingly simple and are unique in terms of their construction. A unique feature of these methods is that they do not rely on the estimation of the nonparametric component of the model. Instead, the methods only require a working model of the unspecified distribution, which may or may not reflect the true distribution. In addition, we establish connections between the existing estimator and the new methods and further derive a semiparametric efficient estimator. We compare our estimators with the existing method and investigate the advantages and cost of the relatively simple estimation procedure.