Consistency of naive PRx without separation

Establish that naive weight-localized predictive recursion (PRx), which estimates the local false discovery rate using the estimated point mass at zero without imposing a separation condition on the alternative mixing density, consistently recovers the oracle covariate-localized false discovery rate.

Background

The paper proves consistency for hole-adjusted PRx under assumption (A3), which requires the continuous alternative mixing density to vanish on a neighborhood of zero. This separation condition allows weak convergence and the Portmanteau theorem to establish consistency of the estimated point mass at zero, and hence of the local false discovery rate.

Naive PRx does not impose a hole in the initialized alternative density. Although Portmanteau arguments provide an upper bound on the limiting estimated null proportion, proving the corresponding lower bound requires showing that the estimated continuous mixing density places asymptotically vanishing mass in every shrinking neighborhood of zero. The paper states that this final step has not been proved, leaving full consistency of naive PRx unresolved without the separation assumption.

References

A more ambitious goal than the above result would be full consistency: can we show that naive PRx leads to consistent recovery of the local false discovery rate? The primary difficulty in this latter approach comes in showing consistency of the estimated term $\hat{\Psi}n({0}|x)$ towards its oracle target, $\pi_0(x)$. Towards this end, Portmanteau may be leveraged to show that $\limsup{n} \hat{\Psi}n({0}|x)\leq \pi_0(x)$. To bound the $\liminf$ term, it suffices to show that $\lim{\epsilon\rightarrow0}\limsup_{n\rightarrow\infty}\int_{-\epsilon}\epsilon\hat{\psi}_n(u|x)du=0$ --- but a proof of this final statement has so far eluded us.

— Covariate-localized False Discovery Rates  (2609.09471 - Lin et al., 8 Sep 2026) in Section 5, Theoretical Results, immediately following the proof of Theorem 5.1 (Consistency of PRx)