  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.
