Analyze MiNCE with an estimated input distribution

Analyze the consistency of the Minimum-Norm Confidence Envelope (MiNCE) framework when the input sampling distribution is estimated rather than known a priori.

Background

The theoretical development assumes that the input distribution, including its density h_*, is known. This assumption is used in the importance-sampling-based upper bounds for the Paley–Wiener function norm and therefore enters the construction and consistency analysis of the MiNCE confidence bands.

The paper notes that an estimated input distribution could be used as a plug-in in practice, but does not analyze the resulting statistical and consistency properties. Establishing such an analysis would extend MiNCE to settings in which the sampling mechanism is unknown.

References

We leave the analysis of estimated input distributions to future work.

— MiNCE: Nonparametric, Strongly Consistent Confidence Envelopes for Band-Limited Functions and their Smoothed Spectra  (2609.09436 - Csáji et al., 8 Sep 2026) in Section 3, subsection “Main Assumptions” (paragraph following Assumption A2)

A natural direction for future work is to relax the assumption that the sampling distribution of the inputs is known, and to complement consistency with convergence rates; both are directions we are actively pursuing.

— MiNCE: Nonparametric, Strongly Consistent Confidence Envelopes for Band-Limited Functions and their Smoothed Spectra  (2609.09436 - Csáji et al., 8 Sep 2026) in Section 8, Conclusions

The analysis assumes that the operational distribution $\mathcal D$ is known exactly. When $\mathcal D$ is estimated from data, uncertainty in the distribution must also be incorporated into the reliability guarantee; extending the framework to distributionally robust certification is left for future work.

— How Often Does Your Program Fail?  (2609.20037 - Ray et al., 17 Sep 2026) in Section 6.4, subsection “A Certification Threshold: Sample Budget”