AbstractsMathematics

Suppose we observe baseline covariates, a binary indicator of treatment, and an outcome occuring after treatment. An individualized treatment rule (ITR) is a treatment rule which assigns treatments to individuals based on their measured covariates. An optimal ITR is the ITR which maximizes the population mean outcome. The mean outcome of the optimal ITR is referred to as the optimal value. This dissertation considers three inferential challenges related to these parameters in the large semiparametric model that at most places restrictions on the probability of receiving treatment given covariates.The first is to develop confidence intervals for the optimal value. Constructing valid confidence intervals for this quantity is surprisingly difficult when the stratum specific treatment effect, also called the blip function, is null with positive probability. This null treatment effect seems possible in many studies. While it has been claimed in the literature that no regular and asymptotically linear (RAL) estimator exists in this case, we prove that RAL estimators of the optimal value can exist in a slightly more general setting. We then describe an approach to obtain root-n rate confidence intervals for the optimal value even when regular estimation is not possible. We also provide sufficient conditions under which our estimator is RAL and asymptotically efficient  – a necessary condition is of course that regular estimation is possible under the data generating distribution.We have thus far assumed that treatment is an unlimited resource so that the entire population can be treated if this strategy maximizes the population mean outcome. In the second part of this dissertation, we consider optimal ITRs in settings where the treatment resource is limited so that there is a maximum proportion of the population that can be treated. We give a general closed-form expression for an optimal stochastic ITR in this resource-limited setting, and a closed-form expression for the optimal deterministic ITR under an additional assumption. We also present an estimator of the mean outcome under the optimal stochastic ITR and give conditions under which our estimator is efficient among all RAL estimators.Both of the first two inferential challenges considered give parametric-rate confidence intervals for finite-dimensional parameters in our large semiparametric model. In the third part of this dissertation we focus on developing hypothesis tests and confidence sets for infinite-dimensional parameters that one typically estimates using data adaptive techniques. Parametric-rate inference is not typically expected in this setting. Our primary motivating example concerns the blip function, which is closely related to the optimal ITRs in both the resource-unconstrained and constrained settings. For any fixed function, we give valid hypothesis tests that the blip function is equal to this fixed function. These tests can then be inverted to develop a confidence set for the blip function. Surprisingly, the hypothesis test achieves a parametric…