Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M-Estimation

The bootstrap variance estimate is widely used in semiparametric inferences. However, its theoretical validity is a well known open problem. In this paper, we provide a {\em first} theoretical study on the bootstrap moment estimates in semiparametric models. Specifically, we establish the bootstrap moment consistency of the Euclidean parameter which immediately implies the consistency of $t$-type bootstrap confidence set. It is worth pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency is to simply strengthen the $L_1$ maximal inequality condition required in the latter to the $L_p$ maximal inequality condition for $p\geq 1$. The general $L_p$ multiplier inequality developed in this paper is also of independent interest. These general conclusions hold for the bootstrap methods with exchangeable bootstrap weights, e.g., nonparametric bootstrap and Bayesian bootstrap. Our general theory is illustrated in the celebrated Cox regression model.