  This paper is concerned with normal approximation under relaxed moment
conditions using Stein's method. We obtain the explicit rates of convergence in
the central limit theorem for (i) nonlinear statistics with finite absolute
moment of order $2+\delta\in(2,3];$ (ii) nonlinear statistics with vanishing
third moment and finite absolute moment of order $3+\delta\in(3,4].$ When
applied to specific examples, these rates are of the optimal order
$O(n^{-\frac{\delta}{2}})$ and $O(n^{-\frac{1+\delta}{2}}).$ Our proof are
based on the covariance identify formula and simple observations about the
solution of Stein's equation.
