Confidence intervals and bands for deconvolution estimators

Develop confidence intervals and confidence bands for the proposed estimators of the latent distribution function at continuity points, latent interval probabilities, and jump sizes under additive measurement error.

Background

The paper develops direct deconvolution estimators for three functionals of an unknown latent distribution under the classical additive measurement-error model: pointwise distribution-function values at continuity points, interval probabilities, and the sizes of prespecified jumps. It establishes non-asymptotic bias, variance, and mean squared error bounds, together with asymptotic unbiasedness and consistency, under weak assumptions that do not require a latent density, a mixture representation, or global Sobolev smoothness.

Although these results provide point-estimation and error-bound theory, the paper does not develop inferential procedures for quantifying uncertainty around the proposed estimators. In particular, constructing valid confidence intervals and confidence bands remains unresolved, potentially requiring treatment of dependence across evaluation points, measurement-error smoothness, bandwidth selection, and latent distributions with discontinuities.

References

Finally, the development of confidence intervals and confidence bands for the proposed estimators remains an open problem.

— Estimation of distribution functions, their jumps and interval probabilities under measurement error  (2608.13152 - Mynbaev et al., 13 Aug 2026) in Section 7, Conclusion

Although we are convinced that this result is a characteristic of our method, understanding the underlying reason is left for future work.

— Unfolding the low-energy reactor neutrino flux from CE$ν$NS data with a finite Dirac sum  (2608.16107 - Chen et al., 17 Aug 2026) in Section 4, Results