  In a recent paper, Gaunt 2020 extended Stein's method to limit distributions
that can be represented as a function $g:\mathbb{R}^d\rightarrow\mathbb{R}$ of
a centered multivariate normal random vector $\Sigma^{1/2}\mathbf{Z}$ with
$\mathbf{Z}$ a standard $d$-dimensional multivariate normal random vector and
$\Sigma$ a non-negative definite covariance matrix. In this paper, we obtain
improved bounds, in the sense of weaker moment conditions, smaller constants
and simpler forms, for the case that $g$ has derivatives with polynomial
growth. We obtain new non-uniform bounds for the derivatives of the solution of
the Stein equation and use these inequalities to obtain general bounds on the
distance, measured using smooth test functions, between the distributions of
$g(\mathbf{W}_n)$ and $g(\mathbf{Z})$, where $\mathbf{W}_n$ is a standardised
sum of random vectors with independent components and $\mathbf{Z}$ is a
standard $d$-dimensional multivariate normal random vector. We apply these
general bounds to obtain bounds for the chi-square approximation of the family
of power divergence statistics (special cases include the Pearson and
likelihood ratio statistics), for the case of two cell classifications, that
improve on existing results in the literature.
