A Note on Bootstrap Moment Consistency for Semiparametric M-Estimation

Cheng and Huang (2010) have recently proven that the bootstrap is asymptotically consistent in estimating the distribution of the M-estimate of Euclidean parameter. 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 beyond the distribution consistency is to strengthen the $L_1$ maximal inequality condition required in the latter to the $L_p$ maximal inequality condition for $p\geq 1$. The key technical tool in deriving the above results is the general $L_p$ multiplier inequality developed in this note. These general conclusions hold when the infinite dimensional nuisance parameter is root-n consistent, and apply to a broad class of bootstrap methods with exchangeable bootstrap weights. Our general theory is illustrated in the celebrated Cox regression model.