A Note on Bootstrap Moment Consistency 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 note, we provide a 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 worthy pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency (obtained in Cheng and Huang (2010)) is to simply strengthen the L1 maximal inequality condition required in the latter to the Lp maximal inequality condition for p ≥ 1. The general Lp multiplier inequality developed in this note is the key technical tool, and is also of independent interest. These general conclusions hold for the bootstrap methods with exchangeable bootstrap weights, e.g., nonparametric bootstrap, and apply to a broad class of semiparametric models with root-n convergent nuisance parameters. Our general theory is illustrated in the celebrated Cox regression model.