Inference with Unknown Reference-Standard Purity Parameters

Develop methods for incorporating uncertainty in the mixture-purity parameters \(\pi_0=\Pr(D=0\mid R=0)\) and \(\pi_1=\Pr(D=1\mid R=1)\), possibly by introducing additional information or assumptions that ensure identifiability when both parameters are unknown.

Background

The proposed Box–Cox density ratio model assumes that π0\pi_0 and π1\pi_1, the proportions of truly healthy individuals in the nominally healthy group and truly diseased individuals in the nominally diseased group, respectively, are known. These quantities are essential for identifying the underlying healthy and diseased biomarker distributions from the two contaminated samples.

The discussion identifies extending the framework to unknown π0\pi_0 and π1\pi_1 as unresolved because allowing both parameters to be unknown may compromise identifiability. The stated research problem is therefore to develop estimation and inference methods that account for uncertainty in these quantities while restoring identifiability through additional data or assumptions.

References

This assumption plays an important role in the identifiability of the model, and allowing both quantities to be unknown may lead to an identifiability issue under the current framework. Developing methods that incorporate uncertainty in \pi_0 and \pi_1, possibly by introducing additional information or assumptions to ensure identifiability, is an important direction for future research.

— Semiparametric Receiver Operating Characteristic Analysis in the Presence of an Imperfect Reference Standard via a Box-Cox Density Ratio Model  (2609.09401 - Chang et al., 8 Sep 2026) in Section 7, Discussion