Stein’s method for functions of multivariate normal random variables
Stein’s method for functions of multivariate normal random variables
复制标题
多元正态随机变量函数的 Stein 方法
DOI：
10.1214/19-aihp1011
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Robert E. Gaunt
中科院分区：
文献类型：
--
作者：
Robert E. Gaunt
根据连续映射定理，如果一个$d$维随机向量序列$(\mathbf{W}_n)_{n\geq1}$在分布上收敛于一个多元正态随机变量$\Sigma^{1/2}\mathbf{Z}$，那么如果$g:\mathbb{R}^d\rightarrow\mathbb{R}$是连续的，则该随机变量序列$(g(\mathbf{W}_n))_{n\geq1}$在分布上收敛于$g(\Sigma^{1/2}\mathbf{Z})$。在本文中，我们发展了Stein的方法，用于推导关于光滑概率度量的$g(\mathbf{W}_n)$和$g(\Sigma^{1/2}\mathbf{Z})$之间的距离的显式界限问题。我们得到了$\mathbf{W}_n$的$j$ -分量由$W_{n,j}=\frac{1}{\sqrt{n}}\sum_{i=1}^nX_{ij}$给出的情况下的几个边界，其中$X_{ij}$是独立的。特别地，当$g$满足一定的可微性和增长率条件时，对于光滑检验函数，如果$X_{ij}$的第一个$p$矩与正态分布的第一个矩一致，我们得到了一个阶$n^{-(p-1)/2}$界。如果$p$是偶数且$g$是偶函数，则该收敛速度可进一步提高到$n^{-p/2}$阶。这些收敛速率被证明是最优阶的。我们将一般界应用于一些例子，包括渐近卡方分布统计量的分布近似；二项式和泊松随机变量光滑函数的期望近似；delta法的收敛速度；在二元序列的情况下，对无比对序列比较的$D_2^*$统计量进行定量方差-伽马近似。
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.