We address the practical construction of asymptotic confidence intervals for smooth (i.e., pathwise differentiable), real-valued statistical
parameters by targeted learning from independent and identically
distributed data in contexts where sample size is so large that it poses
computational challenges. We observe some summary measure of all data and select a sub-sample from the complete data set by Poisson rejective sampling with unequal inclusion probabilities based on the summary measures. Targeted learning is carried out from the easier to handle sub-sample. We derive a central limit theorem for the targeted minimum loss estimator (TMLE) which enables the construction of the confidence intervals. The inclusion probabilities can be optimized to reduce the asymptotic variance of the TMLE. We illustrate the procedure with two examples where the parameters of interest are variable importance measures of an exposure (binary or continuous) on an outcome. We also conduct a simulation study and comment on its results.



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