UC San Diego

The first two chapters consider the estimation and inference of the dynamic treatment effect when the confounders are possibly high dimensional. Chapter 1 proposes a sequential doubly robust Lasso (S-DRL) estimator using $\ell_1$-regularized nuisance estimates with DR-type imputations. The proposed method achieves consistency as long as at least one nuisance function is appropriately parametrized for each exposure time and treatment path. The key to achieving these results is the usage of DR representations for intermediate conditional outcome models, which offer superior inferential performance while requiring weaker assumptions. We establish root-n inference based on the S-DRL estimator is guaranteed when two product-sparsity conditions are satisfied. Chapter 2 further provides root-$n$ inference for the dynamic treatment effect even when model misspecification occurs. We provide valid inference based on a ``sequential model double robust'' solution as long as one of the nuisance models is correctly specified at each time spot. Chapter 3 proposes a novel construction for random forests, incorporating cyclic modification of the selection of splitting directions with the goal of achieving a faster consistency rate for the integrated mean squared error (IMSE). Setting $\alpha=0.5$ leads to the proposed cyclic forest degenerating into cyclic median forests, obtaining a minimax optimal rate for IMSE within the Lipschitz class. We further extend our exploration to local polynomial regression within each leaf, formulating cyclic local polynomial forests as generalizations of the cyclic forests. When $\alpha=0.5$, our cyclic local polynomial forests attain a minimax rate for IMSE, marking the first instance of achieving minimax optimal rates for random forests within the H\" older class. Furthermore, we establish minimax optimal rates for the uniform convergence rate.