Statistica Sinica

M. T. Chao and S. H. Lo

Abstract:Given a general statistic T_{n}(X,θ)=T, 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_{n}(X_{1},...,X_{n},θ)Fand the statistic_{θ(•)}Tonly, and they appear in the form of an inner product and behave like a derivative of_{n}Twith 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)._{n}

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