Statistica Sinica 5(1995), 129-139

SOME STATISTICAL MEAN VALUE THEOREMS

RELATED TO THE BOOTSTRAP

M. T. Chao and S. H. Lo

Academia Sinica and Columbia University

Abstract: Given a general statistic T_n(X,θ)=T_n(X₁,...,X_n,θ) , a representation is given for the difference between the bootstrapped statistic and a replica of its own image . Except for a high order error term, the difference, which explains the validity of the bootstrap method, consists of 3 components. The first component is the difference , where is used in as the bootstrap resampling base. The other two components depend on the model F_θ(•) and the statistic T_n only, and they appear in the form of an inner product and behave like a derivative of T_n with respect to θ. This representation is an application of the classical mean value theorem and it supports the superiority of the maximum likelihood summary as explored by Efron (1982b).

Key words and phrases: Bootstrap, bootstrap representation, estimation equation, maximum likelihood summary, mean value theorem, parametric bootstrap, root statistic.