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Stein's method for functions of multivariate normal random variables

Research paper by Robert E. Gaunt

Indexed on: 30 Jul '15Published on: 30 Jul '15Published in: Mathematics - Probability



Abstract

It is a well-known fact that if the random vector $\mathbf{W}$ converges in distribution to a multivariate normal random variable $\Sigma^{1/2}\mathbf{Z}$, then $g(\mathbf{W})$ converges in distribution to $g(\Sigma^{1/2}\mathbf{Z})$ if $g$ is continuous. In this paper, we develop a general method for deriving bounds on the distributional distance between $g(\mathbf{W})$ and $g(\Sigma^{1/2}\mathbf{Z})$. To illustrate this method, we obtain several bounds for the case that the $j$-component of $\mathbf{W}$ is given by $W_j=\frac{1}{\sqrt{n}}\sum_{i=1}^nX_{ij}$, where the $X_{ij}$ 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 $X_{ij}$ 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.