The strata-specific treatment effect or so-called blip for a randomly drawn strata of confounders defines a random variable and a corresponding cumulative distribution function. However, the CDF is not pathwise differentiable, necessitating a kernel smoothing approach to estimate it at a given point or perhaps many points. Assuming the CDF is continuous, we derive the efficient influence curve of the kernel smoothed version of the blip CDF and a CV-TMLE estimator. The estimator is asymptotically efficient under two conditions, one of which involves a second order remainder term which, in this case, shows us that knowledge of the treatment mechanism does not guarantee a consistent estimate. The remainder term also teaches us exactly how well we need to estimate the nuisance parameters (outcome model and treatment mechanism) to guarantee asymptotic efficiency. Through simulations we verify theoretical properties of the estimator and show the importance of machine learning over conventional regression approaches to fitting the nuisance parameters. We also derive the bias and variance of the estimator, the orders of which are analogous to a kernel density estimator. This estimator opens up the possibility of developing methodology for optimal choice of the kernel and bandwidth to form confidence bounds for the CDF itself.
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