Bootstrap of Deviation Probabilities with Applications II

Bootstrap of deviation probabilities is considered for an extended zone than that obtained in Dasgupta (J Multivariate Anal 101(9):2137–2148, 2010), where it is shown that under different moment bounds on the underlying variables, bootstrap approximation for deviation probabilities of standardized sample sum, based on independent random variables, is valid for a wider zone compared to the classical normal tail probability $$\Phi (-t), t\rightarrow \infty $$ approximation. Here, we show that the bootstrap approximation zone may further be extended compared to the earlier results of $$t=o(n^{3/8}),$$ where n is the sample size. When skewness and kurtosis of the random variable X are zero, then by bootstrapping from a modified constructed sample where skewness and kurtosis are near to zero, the bootstrap approximation zone may be extended to $$t=o(n^{2/5})$$ under the assumption that $$Ee^{s|X|^{8/9}} 0.$$ Zero skewness may be achieved by bootstrapping from a symmetrized sample. Reduction in the magnitude of the fourth semi-invariant $$\hat{\psi }_4$$ in modified sample such that $$|\hat{\psi }_4|=o(n^{-3/5})$$ suffices for bootstrap approximation to hold in an enlarged zone of $$t=o(n^{2/5})$$ . The results are extended to a triangular array of independent random variables. Construction of modified sample with nearly nil skewness and kurtosis from original sample is explained. Applications of the results include the distinction of growth processes with different levels of contamination mixed with normal process.