Bootstrap of deviation probabilities with applications

We show that under different moment bounds on the underlying variables, bootstrap approximation to the large deviation probabilities of standardized sample sum, based on independent random variables, is valid for a wider zone of n , the sample size, compared to the classical normal tail probability approximation. As an application, different notions of efficiency for statistical tests are considered from Bayesian point of view. In particular, efficiency due to Pitman (1938) [11], Chernoff (1952) [1], and Bayes risk efficiency due to Rubin and Sethuraman (1965) [12] turn out to be special cases with the choice of the weight function; i.e., prior density times loss.