Stein's method for functions of multivariate normal random variables

TL;DR

This paper develops Stein's method to provide explicit bounds on the distance between functions of multivariate normal variables and their limits, with applications to various statistical approximations and convergence rates.

Contribution

It introduces new bounds for functions of multivariate normal vectors, extending Stein's method to derive explicit convergence rates under differentiability and growth conditions.

Findings

Established order $n^{-(p-1)/2}$ convergence rates for smooth functions when moments match normal distribution.

Improved convergence rate to order $n^{-p/2}$ for even functions with even integer $p$.

Applied bounds to chi-square approximations, binomial and Poisson expectations, delta method, and sequence comparison statistics.

Abstract

By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate normal random variable $\Sigma^{1/2}\mathbf{Z}$, then the sequence of random variables $(g(\mathbf{W}_n))_{n\geq1}$ converges in distribution to $g(\Sigma^{1/2}\mathbf{Z})$ if $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. In this paper, we develop Stein's method for the problem of deriving explicit bounds on the distance between $g(\mathbf{W}_n)$ and $g(\Sigma^{1/2}\mathbf{Z})$ with respect to smooth probability metrics. We obtain several bounds for the case that the $j$-component of $\mathbf{W}_n$ is given by $W_{n,j}=\frac{1}{\sqrt{n}}\sum_{i=1}^nX_{ij}$, 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 $n^{-p/2}$. These convergence rates are shown to be of optimal order. We apply our general bounds to some examples, which include the distributional approximation of asymptotically chi-square distributed statistics; the approximation of expectations of smooth functions of binomial and Poisson random variables; rates of convergence in the delta method; and a quantitative variance-gamma approximation of the $D_2^*$ statistic for alignment-free sequence comparison in the case of binary sequences.