Fully Nonparametric Efficient Estimation for Some Causal Inference Problems and Well-posedness on Mean Field Theor

In this thesis, we develop fully nonparametric efficient methods to estimate average treatment effects and natural mediation effects, which are the major concerns in causal inference framework. Besides, we also systematically study the well-posedness of the mean-field type forward-backward stochastic differential equations.;The first topic is the estimation of average treatment effects based on observational data, which is extremely important in practice and has been studied by generations of statisticians under different frameworks. Existing globally efficient estimators require non-parametric estimation of a propensity score function, an outcome regression function or both, but their performance can be poor in practical sample sizes. Without explicitly estimating either functions, we consider a wide class of calibration weights constructed to attain an exact three-way balance of the moments of observed covariates among the treated, the control, and the combined group. The wide class includes exponential tilting, empirical likelihood and generalized regression as important special cases, and extends survey calibration estimators to different statistical problems and with important distinctions. Global semiparametric efficiency for the estimation of average treatment effects is established for this general class of calibration estimators. The results show that efficiency can be achieved by solely balancing the covariate distributions without resorting to direct estimation of propensity score or outcome regression function. We also propose a consistent estimator for the efficient asymptotic variance, which does not involve additional functional estimation of either the propensity score or the outcome regression functions. The proposed variance estimator outperforms existing estimators that require a direct approximation of the efficient influence function.;The second topic is the estimation of mediation effects, which is central to understand the causal mechanism related to how the treatment works in the evaluation of a program intervention. In particular, it is often important to understand to what extent the overall treatment effects is being mediated through an intermediate variable. The overall treatment effect can be decomposed into a natural direct effect and a natural indirect effect, and the estimation of mediation effects typically involve parametric modeling of three conditional distributions. Without directly estimating these three unspecified functions, we propose a class of nonparametric calibration weights that balance certain empirical moments of the mediator and covariates. The proposed estimators are shown to be globally semiparametric efficient for the estimation of the natural direct and indirect effects, respectively. Consistent asymptotic variance estimates are also proposed.;The last topic investigates the well-posedness of mean-field type forward-backward stochastic differential equations, which plays a paramount role in mean field game and mean field control theory. Being motivated by a recent pioneer work Carmona and Delarue [11], in Chapter 9, we propose a broad class of natural monotonicity conditions under which the unique existence of the solutions to mean-field type (MFT) forward-backward stochastic differential equations (FBSDE) can be established. Our conditions provided here are consistent with those normally adopted in the traditional FBSDE (without the interference of a mean-field) frameworks, and give a generic explanation on the unique existence of solutions to common MFT-FBSDEs, such as those in the linear-quadratic setting; besides, the conditions are `optimal' in a certain sense that can elaborate on how their counter-example in Carmona and Delarue [11] just fails to ensure its well-posedness. In addition, a stability theorem is also included,...