Asymptotic Theory for the Generalized Bootstrap of Statistical Differentiable Functionals

Let T be a statistical functional defined on a space p of probability measures (p.m.’s) on a locally compact Banach space B. Let X, X₁,…, X_n be a sequence of independent and identically distributed (i.i.d.) random variables (r.v.) with common probability P∈P, and let us define $$ {P_n}: = {n^{{ - 1}}}\Sigma_{{i = 1}}^n{\delta_{{{X_i}}}} $$, the empirical measure, where δ_Xi denotes the Dirac measure at X_i. When T is smooth in a neighborhood of P, a natural estimator of T(P) is its empirical counterpart T(P_n) (see Von Mises (1947), Huber (1981), Manski (1988)).