Identification of standardization-induced asymptotic contributions

Identify the limiting process contributed by estimated marginal Kaplan–Meier standardizations when regularity is not implied by the stated plug-in rate condition, including the joint weak limit of the marginal and directional processes and the effect of near-ties on the reduced radius and censoring indicator.

Background

The estimator can use an estimated marginal standardization, including a marginal Kaplan–Meier transformation. Under a sufficiently strong rate condition, the standardization error is asymptotically negligible and the limiting additional process M is zero. The paper explains that this need not hold in all censoring configurations, particularly when joint censoring is not of smaller order than the marginal censoring probabilities.

In the unresolved cases, the limiting contribution requires a joint weak-limit theory for the marginal standardization processes and the directional product-limit process, together with an expansion of how estimated standardization shifts both the reduced radius and its censoring mark. Near-ties couple these two effects, preventing a straightforward separation.

References

Three questions are left open for future work.

Efficiency. No efficiency claim is made here. For the full multivariate survival function under censoring the nonparametric maximum likelihood estimator is inefficient \citep{GillVanDerLaanWellner1995,vanderLaan1996}. However, the directional product limit uses, at each $q$, the one-dimensional reduction of the data. Whether the resulting asymptotic variance attains the nonparametric bound for $F\circ$ is unknown. The boundary case is settled, since without censoring, i.e., when $\Delta_{i,n}\equiv1$, the product limit KM is the empirical survival function of the reduced sample, and estimator is the empirical joint-exceedance estimator, whose optimality is classical, see \citet{DreesHuang1998}.

Identifying cases where the standardization contributes. When the regularity of Remark~\ref{rem:regular} is not implied by Assumption~\ref{a:pluginnorm}, the process $\mathbb M$ requires identification for specific standardizations. For instance, we showed that it is equal to zero for the multiplicative standardization. However, in some instances, for the Kaplan--Meier standardization, a joint weak limit of the marginal processes and the directional one is required, as well as an asymptotic expansion of the displacement of the pair $(W_{i,n}(q),\Delta_{i,n}(q))$ under the estimated standardization; the indicator and the radius move together at a near-tie, so the two cannot be separated.

— Statistics of multivariate extremes under random censoring  (2609.37904 - Bladt, 29 Sep 2026) in Section 6, Discussion and outlook, paragraph headed “Identifying cases where the standardization contributes”