Berry-Esseen bounds for general nonlinear statistics, with applications to Pearson's and non-central Student's and Hotelling's

Recently Chen and Shao developed a Stein-type method to obtain bounds on the closeness of the distribution of a general nonlinear statistic to that of a linear approximation. We generalize these results so as to allow one to use lesser moment restrictions when applied to nonlinear statistics expressed as smooth enough functions of sums of independent random vectors. Our main innovation in the method is the use of a Cramer-type of tilt transform. Other techniques used to obtain improvements include exponential and Rosenthal-type inequalities for sums of random vectors established by Pinelis and Sakhanenko. As applications, Berry-Esseen type bounds are obtained for concrete nonlinear statistics such as the Pearson correlation coefficient and the non-central Student and Hotelling statistics.