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

Area of Science:

• Statistical modeling
• Probability theory
• Biostatistics

Background:

• Estimating mixtures of known background and unknown signal distributions is a common statistical challenge.
• The signal distribution's support being within the background's support is a key simplifying assumption.
• Accurate estimation is vital for various scientific and medical applications.

Purpose of the Study:

• To develop and analyze statistical methods for estimating a mixture model with a known background and an unknown signal distribution.
• To leverage the assumption that the signal distribution's support is contained within the background's support for improved estimation.
• To construct shape-constrained estimators for signal distributions with specific properties (monotone, convex, log-concave).

Main Methods:

• Parametric rate of convergence analysis for estimating the mixing proportion.
• Development of shape-constrained estimators tailored for monotone, monotone and convex, and log-concave signal densities.
• Utilizing techniques adapted from established shape-constrained estimation approaches.
• Monte Carlo simulations to validate theoretical findings.

Main Results:

• A parametric rate of convergence is achieved for estimating the mixing proportion under the specified support condition.
• Novel estimators are proposed and analyzed for various signal density shapes.
• Simulations demonstrate the effectiveness of the proposed methods in practical scenarios.
• The methodology is successfully applied to real-world prostate cancer data.

Conclusions:

• The proposed methods provide efficient and theoretically sound approaches for mixture model estimation when signal distribution properties are partially known.
• The assumption of signal support within background support significantly aids in achieving parametric convergence rates.
• The developed shape-constrained estimators offer practical tools for analyzing complex data, as shown by the prostate cancer case study.