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We develop a general method to study the Fisher information distance incentral limit theorem for nonlinear statistics. We first construct completelynew representations for the score function. We then use these representationsto derive quantitative estimates for the Fisher information distance. Toillustrate the applicability of our approach, explicit rates of Fisherinformation convergence for quadratic forms and the functions of sample meansare provided. For the sums of independent random variables, we obtain theFisher information bounds without requiring the finiteness of Poincar\'econstant. Our method can also be used to bound the Fisher information distancein non-central limit theorems.

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