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

## TL;DR

This paper introduces new methods to estimate the proportion of signal in a mixture model where the background is known and the signal has a constrained shape.

## Contribution

The paper proposes novel estimators for the mixing proportion under shape constraints on the signal distribution.

## Key findings

- The proposed estimators achieve a parametric rate of convergence for the mixing proportion.
- Simulation results confirm the theoretical findings and demonstrate the method's effectiveness.
- The methodology is applied to real prostate cancer data, showcasing its practical utility.

## Abstract

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.

The online version contains supplementary material available at 10.1007/s13171-025-00421-w.