Efficiency of the directional product-limit estimator

Determine whether the asymptotic variance of the directional Kaplan–Meier product-limit estimator for the normalized joint-tail limit F° attains the nonparametric efficiency bound under multivariate random censoring.

Background

The paper estimates multivariate tail dependence under componentwise random right censoring by reducing each direction to a one-dimensional randomly censored observation and applying a Kaplan–Meier product limit. This yields an asymptotic Gaussian limit with an explicitly characterized variance, but the paper does not establish an efficiency result.

The unresolved issue is whether exploiting only the one-dimensional directional reductions loses information relative to the full censored multivariate observation. The paper notes that the nonparametric maximum likelihood estimator for the full multivariate survival function is known to be inefficient, while the uncensored boundary case is already classically optimal.

References

Whether the resulting asymptotic variance attains the nonparametric bound for $F\circ$ is unknown.

— Statistics of multivariate extremes under random censoring  (2609.37904 - Bladt, 29 Sep 2026) in Section 6, Discussion and outlook, paragraph headed “Efficiency”