  We obtain bounds to quantify the distributional approximation in the delta
method for vector statistics (the sample mean of $n$ independent random
vectors) for normal and non-normal limits, measured using smooth test
functions. For normal limits, we obtain bounds of the optimal order $n^{-1/2}$
rate of convergence, but for a wide class of non-normal limits, which includes
quadratic forms amongst others, we achieve bounds with a faster order $n^{-1}$
convergence rate. We apply our general bounds to derive explicit bounds to
quantify distributional approximations of an estimator for Bernoulli variance,
several statistics of sample moments, order $n^{-1}$ bounds for the chi-square
approximation of a family of rank-based statistics, and we also provide an
efficient independent derivation of an order $n^{-1}$ bound for the chi-square
approximation of Pearson's statistic. In establishing our general results, we
generalise recent results on Stein's method for functions of multivariate
normal random vectors to vector-valued functions and sums of independent random
vectors whose components may be dependent. These bounds are widely applicable
and are of independent interest.
