Simple Estimators of the Mixing Proportion in a Semi-Parametric Mixture with Known Component

In this paper, we consider the problem of estimating a mixture of a background and signal component. The background distribution is fully known, while the signal distribution needs to be estimated along with the mixing proportion. We treat the special case where the support of the signal distribution is strictly included in that of the background. We show how this assumption can be accounted for in the estimation procedure to obtain a parametric rate of convergence for estimating the mixing proportion. In the case where the signal distribution admits a monotone, a monotone and convex or a log-concave density with respect to Lebesgue measure, we construct estimators that are based on the well-known shape-constrained approaches adapted for each one of these cases. Simulations are presented to illustrate the obtained theoretical results. We also showcase our methodology using prostate cancer data. Supplementary Information: The online version contains supplementary material available at 10.1007/s13171-025-00421-w.