Edgeworth Expansions When the Parameter Dimension Increases with Sample Size

1. Introduction and Summary

Suppose that we have a estimate $\widehat{w}$ of an unknown parameter $w\in R^q$ of a statistical model, and that

$$\begin{array}{cc}& \mathrm{as} n\to \infty ,X_n=n^{1/2}(\widehat{w}-w)\stackrel{\mathcal{L}}{\to }X\sim {\mathcal{N}}_q(0,V),\hfill \end{array}$$

(1)

the multivariate normal on $R^q$, with density and distribution

$$\begin{array}{cc}& {\varphi }_V\left(x\right)={\left(2\pi \right)}^{-q/2}{\left(detV\right)}^{-1/2}exp(-x^{\prime }{V}^{-1}x/2),{\Phi }_V\left(x\right)={\int }_{-\infty }^x{\varphi }_V\left(x\right)dx.\hfill \end{array}$$

(2)

Here we assume that $\widehat{w}$ is a standard estimate. That is, $E\widehat{w}\to w$ as $n\to \infty$, and for $r\ge 1$, its rth order cumulants have magnitude $n^{1-r}$ and can be expanded in powers of $n^{-1}$. The class of standard estimates includes smooth functions of sample means or empirical distributions, based on one or more random samples, or on samples from a stationary time series. So it has a huge range of potential applications.

For $\widehat{w}$ non-lattice, the density and distribution of $X_n$ of (1) can be expanded in powers of $n^{-1/2}$ about those of X of (1). These are the Edgeworth expansions. To be self-contained, Section 2 summarises these expansions.

How fast can $q=q_n$ increase with n, for these expansions to hold? Section 3 shows that they hold if $q_n=o\left(n^{1/6}\right)$.

Section 4 gives a theorem that reduces the number of terms needed for 2nd and 3rd order Edgeworth expansions. For example, if $q=3$, it reduces the number of terms needed for 2nd and 3rd order Edgeworth expansions by 57% and 94%. This % reduction increases with q. If $q=4$ or 5, it reduces the number of terms by 65% or 69% for the 2nd order Edgeworth expansions, and by 98% for the 3rd order Edgeworth expansions.

Section 5 considers the case when $\widehat{w}=t\left(\overline{X}\right)$ for $t:R^q\to R^p$ a smooth function of a sample mean $\overline{X}$ from a distribution on $R^q$ with finite moments. When $p_n{q}_n=pq\to \infty$ as $n\to \infty$, it shows the Edgeworth expansions for $\widehat{w}$ remain valid if $q_n^8{p}_n^6=o\left(n\right)$. For example, this holds for fixed $p=p_n$ if $q_n=o\left(n^{1/8}\right)$. These may be the 1st CLTs or Edgeworth expansions when, not 1, but 2 parameters are allowed to increase to ∞ with n.

Earlier work done when dimension $q=q_n$ increases with sample size n, has been mainly for sample means, including some CLTs and a 2nd order Edgeworth-type expansion. [10,11] showed asymptotic normality for M-estimators with $q_n$ regression parameters when $q_n^2/n$ is large. [3] gave a CLT for M-estimates in a linear regression model of dimension $q_n$ when $q_n/n\to a\in (0,1).$ Remarkably, (8) of [2] gave a CLT for $\widehat{w}$ the sample mean of bounded random vectors that holds for $q_n=O(exp\left(a{n}^c\right))$ if $c<1/7$. (Substitute into their condition to confirm.) (1.4) of [7], appears to allow for $q_n=O\left(n^c\right)$ if $c<1/3$ to suffice for a CLT for the sample mean, when $X_1,\dots ,X_n$ are log-concave, quoting [4]. It will be interesting to see if this bound can be extended to a broader class of estimates than sample means, and if the log-concave condition can be removed.

Section 2.1 of the very recent paper [7] considers the 2nd order Edgeworth expansion for the distribution of a standardized sample mean. His Theorem 2.1 gives conditions for this which hold when $q_n=O\left(n^c\right)$ for any c, or even when $q_n=O(exp\left(b{n}^c\right))$ and $c<1/3.$ The bounds (1.4) and (2.4) of [7] give a 2nd order Edgeworth expansion for a sample mean, that allows for $q_n$ of magnitude $n^c$ if $c<1/3$, a remarkable result.

[8] considered sampling from vectors, and investigated the simultaneous estimation of the marginal distributions for large $q_n$. [5] considered the asymptotic distributions of the canonical correlations between $X_1\in R^p$ and $X_2\in R^q$ with $p\le q$. They derived asymptotic distributions of the canonical correlations when p is fixed, $q=q_n\to \infty$, and $q_n/n\to a\in [0,1)$, as $n\to \infty$. It assumes that $X_1,$ and $X_2$ have a joint normal distribution. [9] gave a CLT for the sample mean for large $q_n$. [6] gave a number of results for large dimensions. [1] and [7] considered the validity and accuracy of Edgeworth expansions of ${max}_{i=1}^q{X}_n$ for large q when $X_n$ is a standardized sample mean.

2. Multivariate Edgeworth Expansions

Suppose that $\widehat{w}$ is a standard estimate of $w\in R^q$ with respect to n. (n is typically the sample size.) That is, $E\widehat{w}\to w$ as $n\to \infty$, where we use $E$ for expected value, and for $r\ge 1$ and $1\le i_1,\dots ,i_r\le q,$ the rth order cumulants of $\widehat{w}={({\widehat{w}}_1,\dots ,{\widehat{w}}_q)}^{\prime }$ can be expanded as

$$\begin{array}{c}\hfill {\overline{k}}^{1-r}=k^{i_1\dots i_r}=\kappa ({\widehat{w}}_{i_1},\dots ,{\widehat{w}}_{i_r})\approx \sum _{d=r-1}^{\infty }{n}^{-d}{\overline{k}}_d^{1-r}, \mathrm{where} {\overline{k}}_d^{1-r}=k_d^{i_1\dots i_r},\end{array}$$

(3)

≈ indicates an asymptotic expansion, and the cumulant coefficients ${\overline{k}}_d^{1-r}$ may depend on n but are bounded as $n\to \infty$. So the bar replaces each $i_k$ by k. For example, ${\overline{k}}_0^1={\overline{w}}_1=w_{i_1}$ and ${\overline{k}}_1^{12}=k_1^{i_1{i}_2}.$ I reserve $i_k$ for this bar notation to avoid double subscripts. So, (1) holds with $V=\left({\overline{k}}_1^{12}\right),q\times q$. V may depend on n, but I assume that $detV$ is bounded away from 0.

$$\begin{array}{cc}& \mathrm{Let} {\overline{P}}_r^{1-k}=P_r^{i_1\dots i_k} \mathrm{be} \mathit{the} \mathit{kth} \mathit{Edgeworth} \mathit{coefficient} \mathrm{of} \mathrm{order} r \mathrm{for} \widehat{w}.\hfill \end{array}$$

These are Bell polynomials in the cumulant coefficients, ${\overline{k}}_d^{1-r}$ of (3), defined and given in [14] for $1\le r\le 3$. Their importance lies in their central role in the Edgeworth expansions of $X_n$ of (1): see (8) and (14) below.

The ${\overline{P}}_r^{1-k}$ needed for $r=1,2,3$ are given in (19)–(21) of [14]:

$$\begin{array}{cc}& {\overline{P}}_1^1={\overline{k}}_1^1,{\overline{P}}_1^{1-3}={\overline{k}}_2^{1-3}/6,{\overline{P}}_2^{12}={\overline{k}}_2^{12}/2+{\overline{k}}_1^1{\overline{k}}_1^2/2,\hfill \\ & {\overline{P}}_2^{1-4}={\overline{k}}_3^{1-4}/4!+\mathcal{S}{\overline{k}}_1^4{\overline{k}}_2^{1-3}/6,{\overline{P}}_2^{1-6}=\mathcal{S}{\overline{k}}_2^{1-3}{\overline{k}}_2^{4-6}/72,\hfill \\ & {\overline{P}}_3^1={\overline{k}}_2^1,{\overline{P}}_3^{1-3}={\overline{k}}_3^{1-3}/6+\mathcal{S}{\overline{k}}_2^{12}{\overline{k}}_1^3/2+{\overline{k}}_1^1{\overline{k}}_1^2{\overline{k}}_1^3/6,\hfill \\ & {\overline{P}}_3^{1-5}={\overline{k}}_4^{1-5}/5!+S_1/24+S_2/12+S_3/12\hfill \\ & \mathrm{where} S_1=\mathcal{S}{\overline{k}}_3^{1-4}{\overline{k}}_1^5,S_2=\mathcal{S}{\overline{k}}_2^{12}{\overline{k}}_2^{345},S_3=\mathcal{S}{\overline{k}}_1^1{\overline{k}}_1^2{\overline{k}}_2^{345},\hfill \\ & {\overline{P}}_3^{1-7}=\mathcal{S}{\overline{k}}_2^{123}{\overline{k}}_3^{4-7}/144+\mathcal{S}{\overline{k}}_2^{123}{\overline{k}}_2^{4-6}{\overline{k}}_1^7/72,\hfill \end{array}$$

(4)

$$\begin{array}{c}\hfill {\overline{P}}_3^{1-9}=\mathcal{S}{\overline{k}}_2^{1-3}{\overline{k}}_2^{4-6}{\overline{k}}_2^{7-9}/6^4,\end{array}$$

(5)

where $\mathcal{S}$ is the operator $\mathcal{S}$ that symmetrizes ${\overline{f}}^{1-k}$ over $i_1,\dots ,i_k$.

Set $P\left(A\right)=$ Probability that A is true. By [15], or [14], for $\widehat{w}$ non-lattice, the density and distribution of $X_n$ can be expanded as

$$\begin{array}{cc}& p_{X_n}\left(x\right)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{p}_r\left(x\right),P(X_n\le x)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{P}_r\left(x\right),x\in R^q,\hfill \\ & \mathrm{where} p_0\left(x\right)={\varphi }_V\left(x\right),P_0\left(x\right)={\Phi }_V\left(x\right),\mathrm{of} \left(2\right), \mathrm{and} \mathrm{for} r\ge 1,\hfill \end{array}$$

(6)

$$\begin{array}{c}\hfill p_r\left(x\right)/{\varphi }_V\left(x\right)=\sum _{k=1}^{3r}[{\tilde{p}}_{rk}:k-r \mathrm{even}]={\tilde{p}}_r\left(x\right)say,{\tilde{p}}_{rk}={\overline{P}}_r^{1-k}{\overline{H}}^{1-k},\end{array}$$

(7)

$$\begin{array}{c}\hfill P_r\left(x\right)=\sum _{k=1}^{3r}[P_{rk}\left(x\right):k-r \mathrm{even}],P_{rk}\left(x\right)={\overline{P}}_r^{1-k}{\overline{H}}_{*}^{1-k},\end{array}$$

(8)

$$\begin{array}{cc}& \mathrm{and} {\overline{H}}^{1-k}={\overline{H}}^{1-k}(x,V)=H^{i_1\dots i_k}={\varphi }_V{\left(x\right)}^{-1}{\overline{O}}^{1-k}{\varphi }_V\left(x\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill {\overline{H}}_{*}^{1-k}={\overline{H}}_{*}^{1-k}(x,V)={\overline{O}}^{1-k}{\Phi }_V\left(x\right)={\int }_{-\infty }^x{\overline{H}}^{1-k}{\varphi }_V\left(x\right)dx,\end{array}$$

(9)

$$\begin{array}{c}\hfill {\overline{O}}^{1-k}=(-{\overline{\partial }}_1)\dots (-{\overline{\partial }}_k),{\overline{\partial }}_k={\partial }_{i_k},{\partial }_i=\partial /\partial x_i.\end{array}$$

${\overline{H}}^{1-k}(x,V)={\overline{H}}^{1-k}$ and ${\overline{H}}_{*}^{1-k}(x,V)={\overline{H}}_{*}^{1-k}$ are the multivariate Hermite polynomial, and the integrated multivariate Hermite polynomial. By [12],

$$\begin{array}{cc}& {\overline{H}}^{1-k}=E({\overline{y}}_1+\sqrt{-1}{\overline{Y}}_1)\dots ({\overline{y}}_k+\sqrt{-1}{\overline{Y}}_k)\hfill \\ & \mathrm{where} y=V^{-1}x,Y=V^{-1}X\sim {\mathcal{N}}_q(0,V^{-1}),{\overline{y}}_k=y_{i_k}, \mathrm{and} {\overline{Y}}_k=Y_{i_k}.\hfill \end{array}$$

I use the tensor summation convention: repetition of $i_1,\dots ,i_k$ in ${\tilde{p}}_{rk}$ of (7) and $P_{rk}\left(x\right)$ of (8) implies their implicit summation over their range, $1,\dots ,q$. [14] gave ${\overline{H}}^{1-k}$ explicitly for $k\le 6$, and for $k\le 9$ when $q=2$.

$$\begin{array}{cc}& \mathrm{Set} {\overline{\mu }}^{1-2k}=E{\overline{Y}}_1\dots {\overline{Y}}_{2k}=\sum ^{1.3\dots (2k-1)}{\overline{V}}^{12}\dots {\overline{V}}^{2k-1,2k},\hfill \end{array}$$

where ${\sum }^N{\overline{f}}^{1-k}$ sums ${\overline{f}}^{1-k}$ over all N permutations of $i_1,\dots ,i_k$ giving distinct values. For example,

$$\begin{array}{cc}& {\overline{\mu }}^{1-4}=V^{12}{V}^{34}+V^{13}{V}^{24}+V^{14}{V}^{23},{\overline{H}}^1={\overline{y}}_1,{\overline{H}}^{12}={\overline{y}}_1{\overline{y}}_2-{\overline{V}}^{12},\hfill \\ & {\overline{H}}^{1-3}={\overline{y}}_1{\overline{y}}_2{\overline{y}}_3-\sum ^3{\overline{y}}_1{\overline{V}}^{23}={\overline{y}}_1{\overline{y}}_2{\overline{y}}_3-{\overline{y}}_1{\overline{V}}^{23}-{\overline{y}}_2{\overline{V}}^{13}-{\overline{y}}_3{\overline{V}}^{12},\hfill \\ & {\overline{H}}_{*}^1={\overline{J}}^1,{\overline{H}}_{*}^{12}={\overline{J}}^{12}-{\overline{V}}^{12}{\Phi }_V\left(x\right),{\overline{H}}_{*}^{1-3}={\overline{J}}^{123}-\sum ^3{\overline{J}}^1{\overline{V}}^{23}, \mathrm{where}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\overline{J}}^{1-k}={\overline{J}}^{1-k}(x,V)=E{\overline{Y}}_1\dots {\overline{Y}}_{k}I(X\le x)={\overline{V}}^{1,k+1}\dots {\overline{V}}^{k,2k}{\overline{M}}^{k+1-2k},\end{array}$$

(10)

$$\begin{array}{c}\hfill \mathrm{and} {\overline{M}}^{a-b}={\overline{M}}^{a-b}(x,V)=E{\overline{X}}_1\dots {\overline{X}}_{k}I(X\le x)={\int }_{-\infty }^x{\overline{x}}_a\dots {\overline{x}}_b{\varphi }_V\left(x\right)dx,\end{array}$$

(11)

for ${\overline{x}}_a=x_{i_a}.$ So the repeated $i_{k+1},\dots ,i_{2k}$ in (10) implies their repeated summation over $1,\dots ,q$. As x lies in $R^q$, ${\int }_{-\infty }^{x}dx$ in (9) and (11), stands for ${\int }_{-\infty }^{x_1}d{x}_1\dots {\int }_{-\infty }^{x_q}d{x}_q.$ (6) with the ${\overline{P}}_r^{1-k}$ of [14], give the Edgeworth expansions for the density and distribution of $X_n$ of (1) to $O\left(n^{-2}\right)$. ${\tilde{p}}_{rk}$ and $P_{rk}$ each have $q^k$ terms, but many are duplicates as ${\overline{P}}_r^{1-k}$ is symmetric in $i_1,\dots ,i_k$. This is exploited in Section 4 to greatly reduce the number of terms in (8) and (14) below.

By (7), the density of $X_n$ relative to its asymptotic value is

$$\begin{array}{cc}& p_{X_n}\left(x\right)/{\varphi }_V\left(x\right)\approx 1+\sum _{r=1}^{\infty }{n}^{-r/2}{\tilde{p}}_r\left(x\right)=1+n^{-1/2}{\tilde{p}}_1\left(x\right)+O\left(n^{-1}\right),\hfill \end{array}$$

for $x\in R^q.$ For measurable $C\subset R^q$,

$$\begin{array}{cc}& P(X_n\in C)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{P}_r\left(C\right), \mathrm{where} P_0\left(C\right)={\Phi }_V\left(C\right), \mathrm{and} \mathrm{for} r\ge 1,\hfill \end{array}$$

(12)

$$\begin{array}{c}\hfill P_r\left(C\right)=E{p}_r\left(X\right)I(X\in C)={\int }_C{p}_r\left(x\right){\varphi }_V\left(x\right)dx=\sum _{k=1}^{3r}[P_{rk}\left(C\right):k-r \mathrm{even}],\end{array}$$

(13)

$$\begin{array}{c}\hfill P_{rk}\left(C\right)=E{\tilde{p}}_{rk}\left(X\right)I(X\in C)={\int }_C{\tilde{p}}_{rk}\left(x\right){\varphi }_V\left(x\right)dx={\overline{P}}_r^{1-k}{\overline{H}}_{*}^{1-k}\left(C\right),\end{array}$$

(14)

$$\begin{array}{c}\hfill \mathrm{and} {\overline{H}}_{*}^{1-k}\left(C\right)=E{\overline{H}}^{1-k}(X,V)I(X\in C)={\int }_C{\overline{H}}^{1-k}{\varphi }_V\left(x\right)dx.\end{array}$$

This paper focuses on the three Edgeworth expansions, (6) and (12), when $q\to \infty$ as $n\to \infty$.

If $-C=C$, then for r odd, $P_{rk}\left(C\right)=P_r\left(C\right)=0,$ so that

$$\begin{array}{cc}& P(X_n\in C)\approx \sum _{r=0}^{\infty }{n}^{-r}{P}_{2r}\left(C\right)={\Phi }_V\left(C\right)+n^{-1}{P}_2\left(C\right)+O\left(n^{-2}\right).\hfill \end{array}$$

Examples 3 and 4 of [14] gave $P_2\left(C\right)$ for

$C=\{x:x^{\prime }{V}^{-1}x\le u\}$, and $C=\{x:|{\left(V^{-1/2}x\right)}_j|\le u_j,j=1,\dots ,q\}$.

The main take-away here is that for $x\in R^q$, $C\subset R^q,$ ${\tilde{p}}_r\left(x\right),P_r\left(x\right),P_r\left(C\right)$ of (7), (8) and (12), and $s=1,2,\dots$,

$$\begin{array}{cc}& p_{X_n}\left(x\right)/{\varphi }_V\left(x\right)=\sum _{r=0}^{s-1}{n}^{-r/2}{\tilde{p}}_r\left(x\right)+O\left(n^{-s/2}\right),\hfill \\ & P(X_n\le x)=\sum _{r=0}^{s-1}{n}^{-r/2}{P}_r\left(x\right)+O\left(n^{-s/2}\right),\hfill \\ & P(X_n\in C)=\sum _{r=0}^{s-1}{n}^{-r/2}{P}_r\left(C\right)+O\left(n^{-s/2}\right),\hfill \end{array}$$

where, for example, by (4),

$$\begin{array}{cc}& {\tilde{p}}_1\left(x\right)=p_1\left(x\right)/{\varphi }_V\left(x\right)=\sum _{k=1,3}{\tilde{p}}_{1k},{\tilde{p}}_{11}={\overline{k}}_1^1{\overline{H}}^1,{\tilde{p}}_{13}={\overline{k}}_2^{1-3}{\overline{H}}^{1-3}/6,\hfill \\ & P_1\left(x\right)=\sum _{k=1,3}{P}_{1k}\left(x\right),P_{11}\left(x\right)={\overline{k}}_1^1{\overline{H}}_{*}^1,P_{13}\left(x\right)={\overline{k}}_2^{1-3}{\overline{H}}_{*}^{1-3}/6,\hfill \\ & P_1\left(C\right)=\sum _{k=1,3}{P}_{1k}\left(C\right),P_{11}\left(C\right)={\overline{k}}_1^1{\overline{H}}_{*}^1\left(C\right),P_{13}\left(C\right)={\overline{k}}_2^{1-3}{\overline{H}}_{*}^{1-3}\left(C\right)/6.\hfill \end{array}$$

These asymptotic expansions generally diverge, as normal moments and Hermite polynomials increase very rapidly with their degree.

3. The Case $\mathit{q}={\mathit{q}}_{\mathit{n}}\to \infty$ as $\mathbf{n}\to \infty$

Theorem 1.

Let $\widehat{w}$ be a non-lattice estimate of $w\in R^q$, satisfying $E\widehat{w}\to w$, and (3). Set $X_n=n^{1/2}(\widehat{w}-w)$. Take $s\ge 1$. Suppose that as $n\to \infty ,$

$$\begin{array}{cc}& q=q_n\to \infty , \mathrm{and} {\nu }_n=n^{-1/2}{q}_n^3\to 0, \mathrm{that} \mathrm{is},q_n=o\left(n^{1/6}\right).\hfill \end{array}$$

Then, for ${\tilde{p}}_r\left(x\right)$ of (7), $P_r\left(x\right)$ of (8), and $P_r\left(C\right)$ of (13),

$$\begin{array}{cc}& p_{X_n}\left(x\right)/{\varphi }_V\left(x\right)=\sum _{r=0}^{s-1}{n}^{-r/2}{\tilde{p}}_r\left(x\right)+O\left({\nu }_n^s\right),\hfill \end{array}$$

(15)

$$\begin{array}{c}\hfill P(X_n\le x)=\sum _{r=0}^{s-1}{n}^{-r/2}{P}_r\left(x\right)+O\left({\nu }_n^s\right),\end{array}$$

(16)

$$\begin{array}{c}\hfill P(X_n\in C)=\sum _{r=0}^{s-1}{n}^{-r/2}{P}_r\left(C\right)+O\left({\nu }_n^s\right).\end{array}$$

(17)

PROOF Set $Q=q^2$. ${\tilde{p}}_{rk}\left(x\right)$ and $P_{rk}\left(x\right)$ of (8), and ${\tilde{p}}_{rk}\left(C\right)$ of (14), each have $q^k$ terms for $q=q_n$. So $p_r\left(x\right)$ of (7), $P_r\left(x\right)$ of (8), and $P_r\left(C\right)$ of (12), each have $N_r$ terms, where

$$\begin{array}{cc}& \mathrm{for} r \mathrm{odd},N_r=q+q^3+q^5+\dots +q^{3r}=q(Q^{(3r+1)/2}-1)/(Q-1),\hfill \end{array}$$

(18)

$$\begin{array}{c}\hfill \mathrm{for} r \mathrm{even},N_r=q^2+q^4+\dots +q^{3r}=Q(Q^{3r/2}-1)/(Q-1).\end{array}$$

(19)

$$\begin{array}{c}\hfill \mathrm{So},N_r=O\left(q^{3r}\right)\mathrm{as} n\to \infty .\end{array}$$

So $p_r\left(x\right),P_r\left(x\right)$ and $P_r\left(C\right)$ each have magnitude $q_n^{3r}$ as $n\to \infty .$

So $n^{-s/2}(p_s\left(x\right),P_s\left(x\right),P_s\left(C\right))$ has magnitude $n^{-s/2}{q}_n^{3s}={\nu }_n^s$. The theorem follows. □

Example 1.

Let $\widehat{w}$ be the sample mean from a distribution on $R^q$ with finite cross cumulants ${\overline{\kappa }}^{1-r},r\ge 1$. Then $E\widehat{w}=w$, and only the leading coefficient in (3) is non-zero. As in Example 2 of [14], by (4)–(5), the non-zero Edgeworth coefficients $P_r^{1-k}$ needed for the three 4th order Edgeworth expansions of $\widehat{w}$ with $V=\left({\overline{\kappa }}^{12}\right)$, are

$$\begin{array}{cc}& {\overline{P}}_1^{1-3}={\overline{\kappa }}^{1-3}/3!,{\overline{P}}_2^{1-4}={\overline{\kappa }}^{1-4}/4!,{\overline{P}}_2^{1-6}=\mathcal{S}{\overline{\kappa }}^{1-3}{\overline{\kappa }}^{4-6}/72,,\hfill \\ & {\overline{P}}_3^{1-5}={\overline{\kappa }}^{1-5}/5!,{\overline{P}}_3^{1-7}=\mathcal{S}{\overline{\kappa }}^{123}{\overline{\kappa }}^{4-7}/144,{\overline{P}}_3^{1-9}=\mathcal{S}{\overline{\kappa }}^{1-3}{\overline{\kappa }}^{4-6}{\overline{\kappa }}^{7-9}/6^4.\hfill \end{array}$$

Substitution gives $(p_s\left(x\right),P_s\left(x\right),P_s\left(C\right))$ for $s=1,2,3.$ For example, as $q_n=q\to \infty ,$ the coefficients of $n^{-1/2}$ in the 2nd terms in the Edgeworth expansions for $p_{X_n}\left(x\right)/{\varphi }_V\left(x\right),P(X_n\le x)$, and $P(X_n\in C)$, are

$$\begin{array}{cc}& {\tilde{p}}_1\left(x\right)={\tilde{p}}_{13}={\overline{\kappa }}^{1-3}{\overline{H}}^{1-3}/6=O\left(q_n^3\right),\hfill \\ & P_1\left(x\right)=P_{13}\left(x\right)={\overline{\kappa }}^{1-3}{\overline{H}}_{*}^{1-3}/6=O\left(q_n^3\right),\hfill \\ & P_1\left(C\right)=P_{13}\left(C\right)={\overline{\kappa }}^{1-3}{\overline{H}}_{*}^{1-3}\left(C\right)/6=O\left(q_n^3\right).\hfill \end{array}$$

Example 2.

Suppose that $X_1,X_2,\dots ,X_n$ are independent random vectors in $R^q$, with mean $\widehat{w}=\overline{X}$, and that $X_j$ has finite cross-cumulants

$${\overline{\kappa }}_j^{1-r}=\kappa (X_{j{i}_1},\dots ,X_{j{i}_r})forr\ge 1and{i}_1,\dots ,i_{r}in1,\dots ,q.$$

Then for $r\ge 1,$

$$\kappa ({\widehat{w}}_{i_1},\dots ,{\widehat{w}}_{i_r})=n^{1-r}{\overline{\kappa }}^{1-r}where{\overline{\kappa }}^{1-r}=n^{-1}\sum _{j=1}^n{\overline{\kappa }}_j^{1-r}.$$

Suppose also, that $\left\{{\overline{\kappa }}^{1-r}\right\}$ are bounded in n, and that for $V=\left({\overline{\kappa }}^{12}\right)$, $detV$ is bounded away from 0, as n increases. Then the Edgeworth coefficients needed for the three 4th order Edgeworth expansions for $\widehat{w}$, are given by Example 3.1.

4. Further Reduction of Terms

Our next theorem gives a way to reduce the number of terms in ${\tilde{p}}_r\left(x\right)$ of (7), $P_r\left(x\right)$ of (8), and $P_r\left(C\right)$ of (14), from $q^k$, to $L_k=q_k[1+O\left(q^{-1}\right)]$ as $q\to \infty ,$ where

$$q_k=\left({\displaystyle \genfrac{}{}{0pt}{}{q}{k}}\right)=q(q-1)\dots (q-k+1)/k!.$$

As we do not use $q_n$ for q in this section, there is no ambiguity with this different use of $q_k$. k

${\tilde{p}}_{rk}\left(x\right),$$P_{rk}\left(x\right)$ and $P_{rk}\left(C\right)$ have k summations over $1,\dots ,q$, and so have $q^k$ terms. But many are duplicates as ${\overline{P}}_r^{1-k}$ is symmetric in $i_1,\dots ,i_k$. Set $\left({\displaystyle \genfrac{}{}{0pt}{}{k}{a\dots b}}\right)=k!/a!\dots b!$, the multinomial coefficient. For example $\left({\displaystyle \genfrac{}{}{0pt}{}{3}{111}}\right)=6$.

Theorem 2.

Let $T^{i_1\dots i_k}\in R$ be symmetric in $i_1,\dots ,i_k.$

$$\begin{array}{cc}& Set{T}^{i_1^2{i}_2}=T^{i_1{i}_1{i}_2}and so on, and u_k=\sum _{i_1,\dots ,i_k=1}^q{T}^{i_1\dots i_k}.\hfill \\ & Then,u_1=\sum _{i_1=1}^q{T}^{i_1},u_2=\sum _{i_1=1}^q{T}^{i_1^2}+2!\sum _{i_1>i_2}^{q_2}{T}^{i_1{i}_2},\hfill \\ & u_3=\sum _{i_1=1}^q{T}^{i_1^3}+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1}}\right)\sum ^{q(q-1)}{T}^{i_1^2{i}_2}+3!\sum _{i_1>i_2>i_3}^{q_3}{T}^{i_1{i}_2{i}_3},\hfill \\ & u_4=\sum _{i_1=1}^q{T}^{i_1^4}+\left({\displaystyle \genfrac{}{}{0pt}{}{4}{1}}\right)\sum ^{q(q-1)}{T}^{i_1^3{i}_2}+\left({\displaystyle \genfrac{}{}{0pt}{}{4}{2}}\right)\sum _{i_1>i_2}^{q_2}{T}^{i_1^2{i}_2^2}+\left({\displaystyle \genfrac{}{}{0pt}{}{4}{211}}\right)\sum _{i_1>i_2}^{3q_3}{T}^{i_1^2{i}_2{i}_3}\hfill \\ & +4!\sum _{i_1>i_2>i_3>i_4}^{q_4}{T}^{i_1{i}_2{i}_3{i}_4},\hfill \\ & u_5=\sum _{i_1=1}^q{T}^{i_1^5}+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{1}}\right)\sum ^{q(q-1)}{T}^{i_1^4{i}_2}+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2}}\right)\sum ^{q(q-1)}{T}^{i_1^3{i}_2^2}+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{221}}\right)\sum _{i_1>i_2}^{5q_3}{T}^{i_1^2{i}_2^2{i}_3}\hfill \\ & +\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2111}}\right)\sum _{i_2>i_3>i_4}^{4q_4}{T}^{i_1^2{i}_2{i}_3{i}_4}+5!\sum _{i_1>i_2>i_3>i_4>i_5}^{q_5}{T}^{i_1{i}_2{i}_3{i}_4{i}_5},\hfill \\ & u_6=\sum _{i_1=1}^q{T}^{i_1^6}+\left({\displaystyle \genfrac{}{}{0pt}{}{6}{1}}\right)\sum ^{q(q-1)}{T}^{i_1^5{i}_2}+\left({\displaystyle \genfrac{}{}{0pt}{}{6}{2}}\right)\sum ^{q(q-1)}{T}^{i_1^4{i}_2^2}+\left({\displaystyle \genfrac{}{}{0pt}{}{6}{3}}\right)\sum _{i_1>i_2}^{q_2}{T}^{i_1^3{i}_2^3}\hfill \\ & +\left({\displaystyle \genfrac{}{}{0pt}{}{6}{411}}\right)\sum _{i_2>i_3}^{3q_3}{T}^{i_1^4{i}_2{i}_3}+\left({\displaystyle \genfrac{}{}{0pt}{}{6}{321}}\right)\sum ^{6q_3}{T}^{i_1^3{i}_2^2{i}_3}+\left({\displaystyle \genfrac{}{}{0pt}{}{6}{222}}\right)\sum _{i_1>i_2>i_3}^{q_3}{T}^{i_1^2{i}_2^2{i}_3^2}\hfill \\ & +\left({\displaystyle \genfrac{}{}{0pt}{}{6}{3111}}\right)\sum _{i_2>i_3>i_4}^{4q_4}{T}^{i_1^3{i}_2{i}_3{i}_4}+\left({\displaystyle \genfrac{}{}{0pt}{}{6}{2211}}\right)\sum _{i_1>i_2,i_3>i_4}^{6q_4}{T}^{i_1^2{i}_2^2{i}_3{i}_4}\hfill \\ & +\left({\displaystyle \genfrac{}{}{0pt}{}{6}{21111}}\right)\sum _{i_2>i_3>i_4>i_5}^{5q_5}{T}^{i_1^2{i}_2{i}_3{i}_4{i}_5}+6!\sum _{i_1>\dots >i_6}^{q_6}{T}^{i_1\dots i_6},\hfill \end{array}$$

where the sums are for distinct $i_j$. This reduces the number of terms in $u_k$ from $q^k$ to $L_{qk}$ where

$$\begin{array}{cc}& L_{q1}=q,L_{q2}=q+q_2,L_{q3}=q+2q_2+q_3,\hfill \\ & L_{q4}=q+3q_2+3q_3+q_4,L_{q5}=q+4q_2+5q_3+4q_4+q_5,\hfill \\ & L_{q6}=q+5q_2+10q_3+14q_4+5q_5+q_6.\hfill \end{array}$$

For example, $(q,k)=(5,3)\Rightarrow L_{qk}/q^k=0.28;$ that is, Theorem 4.1 reduces the number of terms to 28%.

As a check, $q^k={\sum }_{j=1}^{k}S(k,j){\left(q\right)}_j$ where ${\left(q\right)}_j=j!q_j=q(q-1)\dots (q-j+1)$, and $S(k,j)$ is the Stirling number of the 2nd kind tabled on p310 of Comtet (1974). For example, counting the number of terms to calculate in $u_4$ above gives $q^4=q+(4+3){\left(q\right)}_2+6{\left(q\right)}_3+{\left(q\right)}_4$.

Taking

$$\begin{array}{cc}& T^{i_1\dots i_k}={\overline{P}}_r^{1-k}{\overline{f}}^{1-k},where{\overline{f}}^{1-k}={\overline{H}}^{1-k},or{\overline{H}}_{*}^{1-k},or{\overline{H}}_{*}^{1-k}\left(C\right),\hfill \end{array}$$

(20)

and tensor summation is not used, gives $u_k={\tilde{p}}_{rk}$ of (7), or $u_k=P_{rk}\left(x\right)$ of (8), or $u_k=P_{rk}\left(C\right)$ of (14). For example,

$$\begin{array}{cc}& {\tilde{p}}_{r1}=\sum _{i_1=1}^q{P}_r^{i_1}{H}^{i_1},P_{r1}\left(x\right)=\sum _{i_1=1}^q{P}_r^{i_1}{H}_{*}^{i_1},P_{r1}\left(C\right)=\sum _{i_1=1}^q{P}_r^{i_1}{H}_{*}^{i_1}\left(C\right),\hfill \\ & {\tilde{p}}_{r2}=\sum _{i_1=1}^q{P}_r^{i_1{i}_1}{H}^{i_1{i}_1}+2\sum _{i_1>i_2}^{q_2}{P}_r^{i_1{i}_2}{H}^{i_1{i}_2},P_{r2}\left(x\right)=\sum _{i_1=1}^q{P}_r^{i_1{i}_1}{H}_{*}^{i_1{i}_1}+2\sum _{i_1>i_2}^{q_2}{P}_r^{i_1{i}_2}{H}_{*}^{i_1{i}_2},\hfill \\ & P_{r2}\left(C\right)=\sum _{i_1=1}^q{P}_r^{i_1{i}_1}{H}_{*}^{i_1{i}_1}\left(C\right)+2\sum _{i_1>i_2}^{q_2}{P}_r^{i_1{i}_2}{H}_{*}^{i_1{i}_2}\left(C\right),\hfill \\ & {\tilde{p}}_{r3}=\sum _{i_1=1}^q{P}_r^{i_1{i}_1{i}_1}{H}^{i_1{i}_1{i}_1}+3\sum _{i_1\ne i_2}^{q(q-1)}{P}_r^{i_1{i}_1{i}_2}{H}^{i_1{i}_1{i}_2}+6\sum _{i_1>i_2>i_3}^{q_3}{P}_r^{i_1{i}_2{i}_3}{H}^{i_1{i}_2{i}_3},\hfill \\ & P_{r3}\left(x\right)=\sum _{i_1=1}^q{P}_r^{i_1{i}_1{i}_1}{H}_{*}^{i_1{i}_1{i}_1}+3\sum _{i_1\ne i_2}^{q(q-1)}{P}_r^{i_1{i}_1{i}_2}{H}_{*}^{i_1{i}_1{i}_2}+6\sum _{i_1>i_2>i_3}^{q_3}{P}_r^{i_1{i}_2{i}_3}{H}_{*}^{i_1{i}_2{i}_3},\hfill \\ & P_{r3}\left(C\right)=\sum _{i_1=1}^q{P}_r^{i_1{i}_1{i}_1}{H}_{*}^{i_1{i}_1{i}_1}\left(C\right)+3\sum _{i_1\ne i_2}^{q(q-1)}{P}_r^{i_1{i}_1{i}_2}{H}_{*}^{i_1{i}_1{i}_2}\left(C\right)+6\sum _{i_1>i_2>i_3}^{q_3}{P}_r^{i_1{i}_2{i}_3}{H}_{*}^{i_1{i}_2{i}_3}\left(C\right).\hfill \end{array}$$

This shows that we can replace $q^k$ in the derivation of Theorem 3.1 by $L_{qk}$. So for $r=1,2$, the number of terms, $N_r$ of (18) and (19), in $p_r\left(x\right)$ of (7), $P_r\left(x\right)$ of (8), and $P_r\left(C\right)$ of (12), can be reduced to $N_r^{\prime }$ where

$$\begin{array}{cc}& N_1^{\prime }=L_{q1}+L_{q3}=2q+2q_2+q_3,\hfill \end{array}$$

(21)

$$\begin{array}{c}\hfill N_2^{\prime }=L_{q2}+L_{q4}+L_{q6}=3q+10q_2+10q_3+15q_4+5q_5+q_6.\end{array}$$

(22)

So, $q=1\Rightarrow N_1=2,N_1^{\prime }=2,N_2=3,N_2^{\prime }=3,$

$q=2\Rightarrow N_1=10,N_1^{\prime }=6=.6N_1,N_2=84,N_2^{\prime }=16=.19N_2,$

$q=3\Rightarrow N_1=30,N_1^{\prime }=13=.43N_1,N_2=819,N_2^{\prime }=49=.06N_2,$

$q=4\Rightarrow N_1=68,N_1^{\prime }=24=.35N_1,N_2=4368,N_2^{\prime }=127=.02N_2,$

$q=5\Rightarrow N_1=130,N_1^{\prime }=40=.31N_1,N_2=16275,N_2^{\prime }=295=.02N_2,$

where the values of $N_1^{\prime }/N_1$ and $N_2^{\prime }/N_2$ are approximate.

For example, if $q=3$, Theorem 4.1 reduces the number of terms needed for 2nd and 3rd order Edgeworth expansions by 57% and 94%. If $q=4$ or 5, Theorem 4.1 reduces the number of terms to calculate by 65% or 69% for the 2nd order Edgeworth expansions, and by 98% for the 3rd order Edgeworth expansions.

When $q=2$, this reduction of terms was used in Section 4 of [14].

Example 3.

In Example 3.1, using the expression for $u_3$ in Theorem 4.1,

$$\begin{array}{cc}& 6{\tilde{p}}_1\left(x\right)=\sum _{i_1=1}^q{\kappa }^{i_1^3}{H}^{i_1^3}+3\sum ^{q(q-1)}{\kappa }^{i_1^2{i}_2}{H}^{i_1^2{i}_2}+6\sum _{i_1>i_2>i_3}^{q_3}{\kappa }^{i_1{i}_2{i}_3}{H}^{i_1{i}_2{i}_3},\hfill \\ & 6P_1\left(x\right)=\sum _{i_1=1}^q{\kappa }^{i_1^3}{H}_{*}^{i_1^3}+3\sum ^{q(q-1)}{\kappa }^{i_1^2{i}_2}{H}_{*}^{i_1^2{i}_2}+6\sum _{i_1>i_2>i_3}^{q_3}{\kappa }^{i_1{i}_2{i}_3}{H}_{*}^{i_1{i}_2{i}_3},\hfill \\ & 6P_1\left(C\right)=\sum _{i_1=1}^q{\kappa }^{i_1^3}{H}_{*}^{i_1^3}\left(C\right)+3\sum ^{q(q-1)}{\kappa }^{i_1^2{i}_2}{H}_{*}^{i_1^2{i}_2}\left(C\right)+6\sum _{i_1>i_2>i_3}^{q_3}{\kappa }^{i_1{i}_2{i}_3}{H}_{*}^{i_1{i}_2{i}_3}\left(C\right).\hfill \end{array}$$

As ${\overline{P}}_1^1={\overline{P}}_2^{12}=0$, $L_{q1}$ in (21), and $L_{q2}$ in (22) need to be deleted.

5. Functions of a Vector Sample Mean

Let $\overline{X}$ be the sample mean from a non-lattice distribution on $R^q$ with mean $\mu$, and finite cross cumulants ${\overline{\kappa }}^{1-r}={\kappa }^{j_1\dots j_r},r\ge 1$. Let $t(.):R^q\to R^p$ be a function with $i_k$th component ${\overline{t}}^k(.)=t^{i_k}(.)$ having finite derivatives at $\mu$,

$$\begin{array}{cc}& {\overline{t}}_{r-s}^k={\partial }_{j_r}\dots {\partial }_{j_s}{t}^k\left(\mu \right), \mathrm{where} {\partial }_j=\partial /{\partial }_{{\mu }_j},s\le r.\hfill \end{array}$$

So we expand the bar convention used earlier using $j_1,j_2,\dots$ in $1,\dots ,p$ as well as $i_1,i_2,\dots$ in $1,\dots ,q$ as before. Here we have a notation dilemma. We chose $t(.):R^q\to R^p$, as this allows us to keep the notation ${\overline{k}}_d^{1-r}=k_d^{i_1\dots i_r}$ and ${\overline{P}}_r^{1-k}=P_r^{i_1\dots i_k}$, as used earlier. However, now $\widehat{w}\in R^p$ not $R^q$, so that implicit summation in, say ${\overline{P}}_r^{1-k}=P_r^{i_1\dots i_k}$, as used earlier. So now, implicit summation in $u_k={\overline{P}}_r^{1-k}{\overline{H}}^{1-k}$ is over $i_1,\dots ,i_k$ in $1,\dots ,p$, not in $1,\dots ,q$, and this $u_k$ has $p^k$ terms, reducible to $L_{pk}$ using Theorem 4.1.

If I had chosen $t(.):R^p\to R^q$, then ${\overline{k}}_d^{1-r}$ and ${\overline{P}}_r^{1-k}$ would have had to be reinterpreted as $k_d^{j_1\dots j_r}$ and $P_r^{j_1\dots j_k}$, which would likely be confusing.

Let us use $\sum f^{i_1\dots i_k}\sim p^k,$ to mean $\sum f^{i_1\dots i_k}$ has magnitude $p^k$ from summing $i_1,\dots ,i_k$ over $1,\dots ,p$.

$$Set\sum ^2{\overline{t}}_{a-b}^1{\overline{t}}_{c-d}^2={\overline{t}}_{a-b}^1{\overline{t}}_{c-d}^2+{\overline{t}}_{a-b}^2{\overline{t}}_{c-d}^1.$$

More generally, for ${\pi }_1,\dots ,{\pi }_r$ any partition of $1,\dots ,k$, let ${\sum }^N{\overline{t}}_{{\pi }_1}^1\dots {\overline{t}}_{{\pi }_r}^r$ denote the sum of $t_{{\pi }_1}^{b_1}\dots t_{{\pi }_r}^{b_r}$ over all N permutations $b_1,\dots ,b_r$ of $j_1,\dots ,j_r$, giving distinct values. For example,

$$\sum ^3{\overline{t}}_{13}^1{\overline{t}}_2^2{\overline{t}}_4^3={\overline{t}}_{13}^1{\overline{t}}_2^2{\overline{t}}_4^3+{\overline{t}}_{13}^2{\overline{t}}_2^1{\overline{t}}_4^3+{\overline{t}}_{13}^3{\overline{t}}_2^2{\overline{t}}_4^1.$$

I now give the cumulant coefficients, ${\overline{k}}_d^{1-r}$ of (3), for $\widehat{w}=t\left(\overline{X}\right)$, and track their magnitude in p from summing $i_1,\dots ,i_k$ over $1,\dots ,p$.

Theorem 3.

For $\widehat{w}=t\left(\overline{X}\right):R^q\to R^p,$ (3) holds, where the cumulant coefficients,k the ${\overline{k}}_d^{1-r}=k_d^{i_1\dots i_r}$, needed for the 3rd order Edgeworth expansions of order $s=1,2,3,4$, are given by

$$\begin{array}{cc}& For s=1:{\overline{k}}_1^{12}=V_{i_1{i}_2}={\overline{t}}_1^1{\overline{t}}_2^2{\overline{\kappa }}^{12}=\sum _{j_1,j_2=1}^q{t}_{j_1}^{i_1}{\overline{t}}_{j_2}^{i_2}{\kappa }^{j_1{j}_2}\sim q^2,\hfill \\ & For s=2:{\overline{k}}_1^1={\overline{t}}_{12}^1{\overline{\kappa }}^{12}/2\sim q^2,\hfill \\ & {\overline{k}}_2^{1-3}={\overline{t}}_1^1{\overline{t}}_2^2{\overline{t}}_3^3{\overline{\kappa }}^{1-3}+T_{1-4}^{1-3}{\overline{\kappa }}^{12}{\overline{\kappa }}^{34}\sim q^2+q^4\sim q^4,where T_{1-4}^{1-3}=\sum ^3{\overline{t}}_{13}^1{\overline{t}}_2^2{\overline{t}}_4^3.\hfill \\ & For s=3:{\overline{k}}_2^{12}=T_{1-3}^{12}{\overline{\kappa }}^{1-3}/2+T_{1-4}^{12}{\overline{\kappa }}^{12}{\overline{\kappa }}^{34}/2\sim q^3+q^4\sim q^4,where\hfill \\ & T_{1-3}^{12}=\sum ^2{\overline{t}}_{12}^1{\overline{t}}_3^2,T_{1-4}^{12}=\sum ^2{\overline{t}}_{1-3}^1{\overline{t}}_4^2+{\overline{t}}_{13}^1{\overline{t}}_{24}^2,\hfill \end{array}$$

$$\begin{array}{cc}& {\overline{k}}_3^{1-4}={\overline{t}}_1^1{\overline{t}}_2^2{\overline{t}}_3^3{\overline{t}}_4^4{\overline{\kappa }}^{1-4}+T_{1-5}^{1-4}{\overline{\kappa }}^{1-3}{\overline{\kappa }}^{45}+T_{1-6}^{1-4}{\overline{\kappa }}^{12}{\overline{\kappa }}^{34}{\overline{\kappa }}^{56}\sim q^4+q^5+q^6\sim q^6,\hfill \\ & where T_{1-5}^{1-4}=\sum ^{12}{\overline{t}}_{14}^1{\overline{t}}_2^2{\overline{t}}_3^3{\overline{t}}_5^4,T_{1-6}^{1-4}=\sum ^4{\overline{t}}_{135}^1{\overline{t}}_2^2{\overline{t}}_4^3{\overline{t}}_6^4+\sum ^{12}{\overline{t}}_{13}^1{\overline{t}}_{25}^2{\overline{t}}_4^3{\overline{t}}_6^4.\hfill \\ & For s=4:{\overline{k}}_2^1={\overline{t}}_{1-3}^1{\overline{\kappa }}^{1-3}/6+{\overline{t}}_{1-4}^1{\overline{\kappa }}^{12}{\overline{\kappa }}^{34}/8\sim q^3+q^4\sim q^4,\hfill \\ & {\overline{k}}_3^{1-3}=T_{1-4}^{1-3}{\overline{\kappa }}^{1-4}/2+T_{1-5}^{1-3}{\overline{\kappa }}^{1-3}{\overline{\kappa }}^{45}+T_{1-6}^{1-3}{\overline{\kappa }}^{12}{\overline{\kappa }}^{34}{\overline{\kappa }}^{56}\sim q^4+q^5+q^6\sim q^6,\hfill \\ & where T_{1-4}^{1-3}=\sum ^3{\overline{t}}_{13}^1{\overline{t}}_2^2{\overline{t}}_4^3,\hfill \\ & T_{1-5}^{1-3}=\sum ^6{\overline{t}}_{124}^1{\overline{t}}_3^2{\overline{t}}_5^3/2+\sum ^3{\overline{t}}_{145}^1{\overline{t}}_2^2{\overline{t}}_3^3/2+\sum ^6{\overline{t}}_{12}^1{\overline{t}}_{34}^2{\overline{t}}_5^3/2+\sum ^3{\overline{t}}_{14}^1{\overline{t}}_{25}^2{\overline{t}}_3^3,\hfill \\ & T_{1-6}^{1-3}=\sum ^3{\overline{t}}_{1235}^1{\overline{t}}_4^2{\overline{t}}_6^3/2+\sum ^6{\overline{t}}_{1-3}^1{\overline{t}}_{45}^2{\overline{t}}_6^3+\sum ^6{\overline{t}}_{135}^1{\overline{t}}_{24}^2{\overline{t}}_6^3/2+{\overline{t}}_{13}^1{\overline{t}}_{25}^2{\overline{t}}_{46}^3.\hfill \\ & {\overline{k}}_4^{1-5}={\overline{t}}_1^1\dots {\overline{t}}_5^5{\overline{\kappa }}^{1-5}+T_{1-6}^{1-5}{\overline{\kappa }}^{1-4}{\overline{\kappa }}^{56}+U_{1-6}^{1-5}{\overline{\kappa }}^{1-3}{\overline{\kappa }}^{4-6}\hfill \\ & +T_{1-7}^{1-5}{\overline{\kappa }}^{1-3}{\overline{\kappa }}^{45}{\overline{\kappa }}^{67}+T_{1-8}^{1-5}{\overline{\kappa }}^{12}{\overline{\kappa }}^{34}{\overline{\kappa }}^{56}{\overline{\kappa }}^{78}\sim p^5+\dots +p^8\sim p^8,\hfill \\ & where T_{1-6}^{1-5}=\sum ^{20}{\overline{t}}_{15}^1{\overline{t}}_2^2{\overline{t}}_3^3{\overline{t}}_4^4{\overline{t}}_6^5,U_{1-6}^{1-5}=\sum ^{15}{\overline{t}}_{14}^1{\overline{t}}_2^2{\overline{t}}_3^3{\overline{t}}_5^4{\overline{t}}_6^5,\hfill \\ & T_{1-7}^{1-5}=\sum ^{30}{\overline{t}}_{146}^1{\overline{t}}_2^2{\overline{t}}_3^3{\overline{t}}_5^4{\overline{t}}_7^5+\sum ^{60}{\overline{t}}_{14}^1{\overline{t}}_{26}^2{\overline{t}}_3^3{\overline{t}}_5^4{\overline{t}}_7^5+\sum ^{60}{\overline{t}}_{14}^1{\overline{t}}_{56}^2{\overline{t}}_2^3{\overline{t}}_3^4{\overline{t}}_7^5,\hfill \\ & T_{1-8}^{1-5}=\sum ^5{\overline{t}}_{1357}^1{\overline{t}}_2^2{\overline{t}}_4^3{\overline{t}}_6^4{\overline{t}}_8^5/5+\sum ^{60}{\overline{t}}_{135}^1{\overline{t}}_{27}^2{\overline{t}}_4^3{\overline{t}}_6^4{\overline{t}}_8^5+\sum ^{60}{\overline{t}}_{13}^1{\overline{t}}_{25}^2{\overline{t}}_{47}^3{\overline{t}}_6^4{\overline{t}}_8^5,\hfill \end{array}$$

PROOF This is a special case of Theorem 2 of [13] with ${\overline{k}}_d^{1-r}$ replaced by ${\overline{\kappa }}^{1-r}I(d=r-1).$ □

Lemma 1.

For ${\overline{f}}^{1-k}=f^{i_1\dots i_k}\in R$, and $r=1,2,3,$

$$\begin{array}{cc}& {\overline{P}}_r^{1-k}\sim q^{r+k}.So,{\overline{P}}_r^{1-k}{\overline{f}}^{1-k}\sim q^{r+k}{p}^k.\hfill \end{array}$$

$$\begin{array}{c}\hfill Set P_r=\sum _{k=1}^{3r}[{\overline{P}}_r^{1-k}{\overline{f}}^{1-k}:k-r even]. Then P_r\sim q^{4r}{p}^{3r}.\end{array}$$

(23)

PROOF Use Theorem 5.1 to check that for each of the Edgeworth coefficients given by (4)–(5), ${\overline{P}}_r^{1-k}\sim q^{r+k}.$ The dominant term in $P_r$ is for $k=3r$. So, (23) holds. □

Theorem 4.

Set $\widehat{w}=t\left(\overline{X}\right):R^q\to R^p.$ Suppose that $\widehat{w}$ is non-lattice, that $E\widehat{w}\to w$, and that (3) holds.. Suppose that as $n\to \infty ,$ $p_n{q}_n=pq\to \infty ,$ and that

$$\begin{array}{cc}& {\nu }_n=n^{-1/2}{q}_n^4{p}_n^3\to 0. That is, q_n^8{p}_n^6=o\left(n\right).\hfill \end{array}$$

(24)

Then (15)–(17) hold for $s=1,2,3,4$ with ${\nu }_n$ of (24).

PROOF For $P_r$ of (23), $n^{-r/2}{P}_r\sim {\nu }_n^r$. Now take ${\overline{f}}^{1-k}$ of (20). □ For example, (24) holds for fixed $p=p_n$ if $q_n=o\left(n^{1/8}\right)$, and for fixed $q=q_n$ if $p_n=o\left(n^{1/6}\right)$.

We now apply Theorem 4.1 to the components of Theorem 5.1.

Theorem 5.

$$\begin{array}{cc}& For s=1:{\overline{k}}_1^{12}=V_{i_1{i}_2}\sim L_{q2}.\hfill \\ & For s=2:{\overline{k}}_1^1\sim L_{q2}, reduced from q^2,\hfill \\ & {\overline{k}}_2^{1-3}\sim L_{q3}+3L_{q4}, reduced from q^3+3q^4.\hfill \\ & For s=3:{\overline{k}}_2^{12}\sim 2L_{q3}+3L_{q4},reduced from 2q^3+3q^4,\hfill \\ & {\overline{k}}_3^{1-4}\sim L_{q4}+12L_{q5}+16L_{q6}, reduced from q^4+12q^5+16q^6.\hfill \\ & For s=4:{\overline{k}}_2^1\sim L_{q3}+L_{q4}, reduced from q^3+q^4,\hfill \\ & {\overline{k}}_3^{1-3}\sim 3L_{q4}+18L_{q5}+16L_{q6}, reduced from 3q^4+18q^5+16q^6,\hfill \\ & {\overline{k}}_4^{1-5}\sim L_{q5}+(20+15)L_{q6}+150L_{q7}+125L_{q8},\hfill \\ & reduced from q^5+(20+15)q^6+150q^7+125q^8.\hfill \end{array}$$

Let us write these as

$${\overline{k}}_d^{1-r}\sim N_{rd}.$$

For example, $L_{q2}/q^2=(q+1)/2q\to 1/2$ as $q\to \infty$, and for $q=4$, the number of calculations needed for ${\overline{k}}_2^{1-3}$ is reduced by the factor $N_{32}/(q^3+3q^4)=(L_{q3}+3L_{q4})/(q^3+3q^4)=3/26.$

One can now work out similar results for $T^{i_1\dots i_k}$ of (20, and so for the terms of the Edgeworth expansions, ${\tilde{p}}_r\left(x\right),P_r\left(x\right)$ and $P_r\left(C\right)$. For example,

$$\begin{array}{cc}& {\tilde{p}}_{11}={\overline{P}}_1^1{\overline{H}}^1={\overline{k}}_1^1{\overline{H}}^1\sim p{N}_{11}=p{L}_{q2},\hfill \\ & {\tilde{p}}_{13}={\overline{P}}_1^{1-3}{\overline{H}}^{1-3}\sim k_2^{1-3}{\overline{H}}^{1-3}\sim L_{p3}{N}_{32},\hfill \\ & \Rightarrow {\tilde{p}}_1\left(x\right)\sim p{N}_{11}+L_{p3}{N}_{32}=p{L}_{q2}+L_{p3}(L_{q3}+3L_{q4}),\hfill \end{array}$$

and $T^{i_1{i}_2}=P_2^{i_1{i}_2}{H}^{i_1{i}_2}$ without implicit summation,

$${\tilde{p}}_{22}={\overline{P}}_2^{12}{\overline{H}}^{12}=\sum _{i_1=1}^p{T}^{i_1^2}+2\sum _{i_1>i_2}^{p_2}{T}^{i_1{i}_2}\sim p{L}_{q2}+p_2{L}_{q0}.$$

6. Conclusions

Let $\widehat{w}$ be a standard estimate of an unknown $w\in R^q$. That is, $\widehat{w}$ is a consistent estimate, and for $r\ge 1$, its rth order cumulants have magnitude $n^{1-r}$, and can be expanded in powers of $n^{-1}$. This is a very large class of estimates. It includes functions of sample moments and empirical distributions, samples of independent but not identically distributed random vectors, and samples from stationary series. Then by §2, for fixed q, and non-lattice estimates, Edgeworth-type expansions hold for the density and distribution of $X_n=n^{1/2}(\widehat{w}-w)$, in terms of the Edgeworth coefficients given in [14] for $s\le 4$. For $s\ge 1,$ their first s terms of the three Edgeworth expansions, give the density and distribution of $X_n$, and $P(X_n\in C)$, to $O\left(n^{-s/2}\right)$ as $n\to \infty .$ Theorem 3.1 shows that this remains true when $q_n=q\to \infty$, if the remainder, $O\left(n^{-s/2}\right)$, is replaced by $O\left({\nu }_n^s\right)$, where ${\nu }_n=n^{-1/2}{q}_n^3$. So the three Edgeworth expansions hold when ${\nu }_n\to \infty$, that is, when $q_n=o\left(n^{1/6}\right)$.

Theorem 4.1 gives formulas that dramatically reduce the number of terms needed by 2nd and 3rd order Edgeworth-type expansions, that is, for 1st and 2nd order corrections to the CLT.

When $\widehat{w}=t\left(\overline{X}\right):R^q\to R^p$, and $p_n{q}_n=pq\to \infty$, Theorem 5.2 shows that the three 4th order Edgeworth expansions hold if $q_n^8{p}_n^6=o\left(n\right)$. For example, this holds for fixed $p=p_n$ if $q_n=o\left(n^{1/8}\right)$.

7. Discussion

Theorem 3.1, showed that the three Edgeworth expansions considered here, hold for standard estimates when $q_n=o\left(n^{1/6}\right)$. Is this result optimal? It is certainly not optimal for a sample mean. For, as noted in Section 1, Theorem 2.1 of the recent paper [7] gives conditions for a 2nd order Edgeworth expansion for the distribution of a sample mean, when $q_n=O\left(n^c\right)$ for any c, or even when $q_n=O(exp\left(b{n}^c\right))$ and $c<1/3.$

It will be interesting to see if such weak conditions on $q_n$ can be extended to a wide class of estimates, such as standard estimates.

Theorem 5.2 showed that the three 4th order Edgeworth expansions hold for $\widehat{w}=t\left(\overline{X}\right):R^q\to R^p,$ when $q_n^8{p}_n^6=o\left(n\right)$. It will be interesting to see how much this condition can be weakened. It should not be hard to extend Theorem 5.2 to $\widehat{w}=T\left(F_n\right)\in R^q,$ where $F_n\left(x\right)$ is the empirical distribution of arandom sample of size n from a distribution $F\left(x\right)$ on $R^p$.

One can also extend this to $\widehat{w}$ a function of K independent sample means, $\widehat{w}=t({\overline{X}}_1,\dots ,{\overline{X}}_K):R^{q_1}\times \dots \times R^{q_K}\to R^p.$

The method employed here should also be able to extend many of the results in the references from a sample mean to a standard estimate.