Statistical Inference for Data-adaptive Doubly Robust Estimators with Survival Outcomes

The consistency of doubly robust estimators relies on consistent estimation of at least one of two nuisance regression parameters. In moderate to large dimensions, the use of flexible data-adaptive regression estimators may aid in achieving this consistency. However, $n^{{1/2}$-consistency} of doubly robust estimators is not guaranteed if one of the nuisance estimators is inconsistent. In this paper we present a doubly robust estimator for survival analysis with the novel property that it converges to a Gaussian variable at $n^{1/2}$-rate for a large class of data-adaptive estimators of the nuisance parameters, under the only assumption that at least one of them is consistently estimated at a $n^{1/4}$-rate. This result is achieved through adaptation of recent ideas in semiparametric inference, which amount to: (i) Gaussianizing (i.e., making asymptotically linear) a drift term that arises in the asymptotic analysis of the doubly robust estimator, and (ii) using cross-fitting to avoid entropy conditions on the nuisance estimators. We present the formula of the asymptotic variance of the estimator, which allows computation of doubly robust confidence intervals and p-values. We illustrate the finite-sample properties of the estimator in simulation studies, and demonstrate its use in a phase III clinical trial for estimating the effect of a novel therapy for the treatment of HER2 positive breast cancer.