Targeted sequential design for targeted learning inference of the optimal treatment rule and its mean reward. (English) Zbl 1386.62010

Chambaz, Antoine; Zheng, Wenjing; Van der Laan, Mark J.

Ann. Stat. 45, No. 6, 2537-2564 (2017).

The authors study the targeted data-adaptive inference of an optimal treatment rule from data sampled based on a targeted sequential design. A treatment rule is an individualized treatment strategy in which treatment assigment for a patient is based on her measured baseline covariates.
The statistical model is the following: At sample size \(n\), the ordered vector \( \mathbf{O}_{n}\equiv \left( O_{1},\ldots,O_{n}\right) \), with convention \(O_{0}\equiv \emptyset \), is observed. For every \( 1\leq i \leq n \), the data structure \( O_{i} \) writes as \( O_{i}\equiv \left( W_{i},A_{i},Y_{i}\right) \), where \( W_{i} \in \mathcal{W} \) consists of the baseline covariates of the \(i\)th patient, \(A_{i}\in \mathcal{A}\equiv \left\{ 0,1 \right\} \) is the binary treatment of interest assigned to her, and \(Y_{i} \in \mathcal{Y}\) is her primary outcome of interest.
Authors’ abstract: “Our pivotal estimator, whose definition hinges on the targeted minimum loss estimation (TMLE) principle, actually infers the mean reward under the current estimate of the optimal TR. This data-adaptive statistical parameter is worthy of interest on its own. Our main result is a central limit theorem which enables the construction of confidence intervals on both mean rewards under the current estimate of the optimal TR and under the optimal TR itself. The asymptotic variance of the estimator takes the form of the variance of an efficient influence curve at a limiting distribution, allowing to discuss the efficiency of inference.
As a byproduct, we also derive confidence intervals on two cumulated pseudo-regrets, a key notion in the study of bandits problems.
A simulation study illustrates the procedure. One of the cornerstones of the theoretical study is a new maximal inequality for martingales with respect to the uniform entropy integral.”

Reviewer: Wiesław Dziubdziela (Miedziana Gora)

MSC:

62G05	Nonparametric estimation
62L05	Sequential statistical design
60F05	Central limit and other weak theorems
62G15	Nonparametric tolerance and confidence regions
62G20	Asymptotic properties of nonparametric inference
62P10	Applications of statistics to biology and medical sciences; meta analysis

Keywords:

optimal treatment rule; pseudo-regret; targeted minimum loss estimation (TMLE); central limit theorem; confidence intervals