Bounds for distributional approximation in the multivariate delta method by Stein's method
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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.
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