Rates of Fisher information convergence in the central limit theorem for nonlinear statistics

We develop a general method to study the Fisher information distance in central limit theorem for nonlinear statistics. We first construct explicit representations for the score functions. We then use these representations to derive quantitative estimates for the Fisher information distance. To illustrate the applicability of our approach, explicit rates of Fisher information convergence for quadratic forms and the functions of sample means are provided. The case of the sums of independent random variables are discussed as well.