Stein's method for functions of multivariate normal random variables - Gaunt, R (2015)
It is a well-known fact that if the random vector W converges in distribution to a multivariate normal random variable Σ1/2Z, the g(W) converges in distribution to g(Σ1/2Z) if g is continuous. In this paper, we develop a general method for deriving bounds on the distributional distance between g(W) and g(Σ1/2Z). To illustrate this method, we obtain several bounds for the case that the j-component of W is given by Wj = n-1/2 Σni=1 Xij, where the Xij are independent. In particular, provided g satisfies certain differentiability and growth rate conditions, we obtain an order n-(p-1)/2 bound, for smooth test functions, if the first p moments of the Xij agree with those of the normal distribution. If p is an even integer and g is an even function, this convergence rate can be improved further to order n-p/2. We apply these general bounds to some examples concerning asymptotically chi-square, variance-gamma and chi distributed statistics.