Edgeworth-Cornish-Fisher Expansions for the Mean When Sampling from a Stationary Process

1. Introduction and Summary

The behaviour of a standard estimate, as described by its Edgeworth-Cornish-Fisher expansions, is governed by the coefficients obtained by expanding its cumulants. For the simplest case, the mean of independent identically distributed random variables, the cumulant expansion has only one term. In Section 2 we summarise Edgeworth-Cornish-Fisher expansions for a standard estimate.

In Section 3 and Section 4 we apply this to the mean of a sample from a stationary process for univariate and multivariate series. Remarkably, we show that for the sample mean of a stationary process, its cumulant expansion has exactly 2 terms.

Suppose that $\overline{X}$ is the mean of a sample $X_1,\cdots ,X_n$ from a stationary process $\left\{X_i\right\}$ in $R^p$. So $\overline{X}$ is an unbiased estimate of $\mu =E{X}_0$. In Section 2 we show that when $p=1$, for $r>1,$ its rth cumulant has the form

$$\begin{array}{cc}& {\kappa }_r\left(\overline{X}\right)=a_{nr,r-1}{n}^{1-r}+a_{nrr}{n}^{-r}.\hfill \end{array}$$

(1)

where $a_{nri}$ are bounded as n increases, and $a_{n21}$ is bounded away from 0. This makes it a special case of a standard estimate, so that Section 2 applies with $a_{ri}=a_{nri}$ for $i=r-1,r$ and $a_{ri}=0$ for $i>r$.

If $a_{nri}=a_{ri}+O\left(e^{-n{\lambda }_r}\right)$ where ${\lambda }_{ri}>0,$ then $a_{nri}$ can be replaced by $a_{ri}.$ Here $x_n=O\left(y_n\right)$ means that $x_n/y_n$ is bounded.

We also consider the case where observations are not sequential, as for missing data. And we consider unbiased weighted means.

2. Edgeworth-Cornish-Fisher Theory

Here we summarise the expansions of Withers (1984) for the distribution and quantiles of a standard estimate. In Section 3 we shall show that (1) holds, so that the sample mean is a special case of a standard estimate.

Univariate estimates. An estimate $\widehat{w}$ of an unknown $w\in R$ is said to be a standard estimate with respect to n, if $E\widehat{w}\to w$ as $n\to \infty$, and for $r\ge 1$, its rth cumulant can be expanded as

$$\begin{array}{cc}& {\kappa }_r\left(\widehat{w}\right)\approx \sum _{i=r-1}^{\infty }{a}_{ri}{n}^{-i}.\hfill \end{array}$$

(1)

The cumulant coefficients $a_{ri}$ may depend on n but are bounded as $n\to \infty$, and $a_{21}$ is bounded away from 0. Here and below ≈ indicates an asymptotic expansion that need not converge. That is, (2) holds in the sense that

$$forI\ge r,{\kappa }_r\left(\widehat{w}\right)=\sum _{i=r-1}^{I-1}{a}_{ri}{n}^{-i}+O\left(n^{-I}\right).$$

For non-lattice estimates, the distribution and quantiles of

$$Y_n={(n/a_{21})}^{1/2}(\widehat{w}-w)$$

have asymptotic expansions in powers of $n^{-1/2}$:

$$\begin{array}{cc}& P_n\left(x\right)=P(Y_n\le x)\approx \mathrm{\Phi }\left(x\right)-\varphi \left(x\right)\sum _{r=1}^{\infty }{h}_r\left(x\right)n^{-r/2},\hfill \end{array}$$

(2)

$$\begin{array}{cc}& p_n\left(x\right)=d{P}_n\left(x\right)/dx\approx \varphi \left(x\right)[1+\sum _{r=1}^{\infty }{\overline{h}}_r\left(x\right)n^{-r/2}],\hfill \end{array}$$

(3)

$$\begin{array}{cc}& {\mathrm{\Phi }}^{-1}\left(P_n\left(x\right)\right)\approx x-\sum _{r=1}^{\infty }{f}_r\left(x\right)n^{-r/2},P_n^{-1}\left(\mathrm{\Phi }\left(x\right)\right)\approx x+\sum _{r=1}^{\infty }{g}_r\left(x\right)n^{-r/2},\hfill \end{array}$$

(4)

where $\mathrm{\Phi }$ and $\phi$ are the unit normal distribution and density of $N\sim \mathcal{N}(0,1)$, and $h_r\left(x\right),{\overline{h}}_r\left(x\right),f_r\left(x\right),g_r\left(x\right)$ are polynomials in x and the standardized cumulant coefficients $\left\{A_{ri}\right\}$,

$$\begin{array}{cc}& A_{ri}=a_{ri}/a_{21}^{r/2}.\hfill \end{array}$$

(5)

The expansions (2), (4), are given in Withers (1984):

$$\begin{array}{cc}& h_1\left(x\right)=f_1\left(x\right)=g_1\left(x\right)=A_{11}+A_{32}{H}_2/6,\hfill \\ & {\overline{h}}_1\left(x\right)=A_{11}{H}_1+A_{32}{H}_3/6,\hfill \\ & h_2\left(x\right)=(A_{11}^2+A_{22})H_1+(A_{11}{A}_{32}+A_{43})H_3/6+A_{32}^2{H}_5/72,\hfill \\ & f_2\left(x\right)=(A_{22}/2-A_{11}{A}_{32}/3)H_1+A_{43}{H}_3/24-A_{32}^2(4x^3-7x)/36,\hfill \\ & g_2\left(x\right)=A_{22}{H}_1/2+A_{43}{H}_3/24-A_{32}^2(4x^3-7x)/36,\hfill \\ & {\overline{h}}_2\left(x\right)=(A_{11}^2+A_{22})H_2+(A_{11}{A}_{32}+A_{43})H_4/6+A_{32}^2{H}_6/72,\hfill \end{array}$$

(6)

where $H_k$ is the kth Hermite polynomial,

$$\begin{array}{cc}& H_k=H_k\left(x\right)=\varphi {\left(x\right)}^{-1}{(-d/dx)}^k\varphi \left(x\right)\hfill \\ & =E{(x+iN)}^{k}fork\ge 0,i=\sqrt{-1}:\hfill \\ & H_0=1,H_1=x,H_2=x^2-1,H_3=x^3-3x,H_4=x^4-6x^2+3,\hfill \\ & H_5=x^5-10x^3+15x,H_6=x^6-15x^4+45x^2-15,\cdots \hfill \end{array}$$

(7)

See Withers (2000) for (7). The log density has a simpler form than the density:

$$\begin{array}{cc}& ln[p_n\left(x\right)/\varphi \left(x\right)]=\sum _{r=1}^{\infty }{b}_r\left(x\right)n^{-r/2},b_1\left(x\right)=h_1\left(x\right),\hfill \\ & b_2\left(x\right)=-A_{11}^2/2+(A_{22}-A_{32}{A}_{11})H_2/2-A_{32}^2(3x^4-12x^2+5)/24+A_{43}{H}_4/24,\hfill \end{array}$$

and for $r>1,b_r\left(x\right)$ is a polynomial of order only $r+2$, while ${\overline{h}}_r\left(x\right)$ is of order $3r$. See Withers and Nadarajah (2010a) for ${\overline{h}}_r\left(x\right)$ and $b_r\left(x\right)$.

Notation 1. 

The original Edgeworth expansion was for $\widehat{w}$ the mean of n independent identically distributed random variables from a distribution with rth cumulant ${\kappa }_r$. So (1) holds with $a_{ri}={\kappa }_{r}I(i=r-1)$, and $A_{ii}\equiv 0$. An explicit formula for its general term was given in Withers and Nadarajah (2009).

Ordinary Bell polynomials. For a sequence $e=(e_1,e_2,\cdots ),$ the partial ordinary Bell polynomial ${\tilde{B}}_{rs}={\tilde{B}}_{rs}\left(e\right)$, is defined by the identity

$$\begin{array}{cc}& S^s=\sum _{r=s}^{\infty }{z}^r{\tilde{B}}_{rs}\left(e\right)whereS=\sum _{r=1}^{\infty }{z}^r{e}_r,z\in R.\hfill \\ & So,{\tilde{B}}_{r0}={\delta }_{r0},{\tilde{B}}_{r1}=e_r,{\tilde{B}}_{rr}=e_1^r,{\tilde{B}}_{21}=2e_1{e}_2,\hfill \end{array}$$

(8)

where ${\delta }_{00}=1,{\delta }_{r0}=0$ for $r\ne 0.$ They are tabled on p309 of Comtet (1974). The complete ordinary Bell polynomial, ${\tilde{B}}_r\left(e\right)$ is defined in terms of S by

$$\begin{array}{cc}& e^S=\sum _{r=0}^{\infty }{z}^r{\tilde{B}}_{rs}\left(e\right).So{\tilde{B}}_r\left(e\right)=\sum _{s=0}^r{\tilde{B}}_{rs}\left(e\right)/s!:\hfill \end{array}$$

(9)

$${\tilde{B}}_0\left(e\right)=1,{\tilde{B}}_1\left(e\right)=e_1,{\tilde{B}}_2\left(e\right)=e_2+e_1^2/2,{\tilde{B}}_3\left(e\right)=e_3+e_1{e}_2+e_1^3/6$$

(10)

Multivariate estimates. Suppose that $\widehat{w}$ is a standard estimate of $w\in R^p$ with respect to n. That is, $E\widehat{w}\to w$ as $n\to \infty$, and for $r\ge 1$, $1\le i_1,\dots ,i_r\le p,$ the rth order cumulants of $\widehat{w}$ can be expanded as

$$\begin{array}{c}\hfill {\overline{k}}^{1-r}=\kappa ({\widehat{w}}^{i_1},\dots ,{\widehat{w}}^{i_r})=\sum _{e=r-1}^{\infty }{\overline{k}}_e^{1-r}{n}^{-e},{\overline{k}}_e^{1-r}=k_e^{i_1-i_r},\end{array}$$

(11)

where the cumulant coefficients ${\overline{k}}_e^{1-r}=k_e^{j_1-j_r}$ may depend on n but are bounded as $n\to \infty$. So ${\overline{k}}_0^1=w^{i_1}.$ Set $V=\left(k_2^{i_1{i}_2}\right),p\times p$. $Y_n$ converges in law to the multivariate normal ${\mathcal{N}}_p(0,V)$ with $p\times p$ covariance V and distribution and density ${\mathrm{\Phi }}_V\left(x\right)$ and ${\varphi }_V\left(x\right)$. So V may depend on n, but we assume that $det\left(V\right)$ is bounded away from 0. By Withers and Nadarajah (2010b) or Withers (2024), $Y_n=n^{1/2}(\widehat{w}-w)$ has distribution and density

$$\begin{array}{cc}& Prob.(Y_n\le x)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{P}_r\left(x\right),p_{Y_n}\left(x\right)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{p}_r\left(x\right),x\in R^p,\hfill \end{array}$$

(12)

$$\begin{array}{cc}& \mathrm{where}{\left(b_1\right)}_i=b_1^i,P_0\left(x\right)={\mathrm{\Phi }}_V\left(x\right),p_0\left(x\right)={\varphi }_V\left(x\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}& P_r\left(x\right)={\tilde{B}}_r\left(e(-\partial /\partial x)\right){\mathrm{\Phi }}_V\left(x\right),p_r\left(x\right)={\tilde{B}}_r\left(e(-\partial /\partial x)\right){\varphi }_V\left(x\right),r\ge 1,\hfill \end{array}$$

(14)

$$\begin{array}{cc}& e_j\left(s\right)=\sum _{r=1}^{j+2}{\overline{b}}_{r+j}^{1\cdots r}{s}_{i_1}\cdots s_{i_r}/r!,\hfill \\ & {\overline{b}}_{r+j}^{1\cdots r}=b_{r+j}^{i_1\cdots i_r},b_{2d+1}^{i_1\cdots i_r}=0,b_{2d}^{i_1\cdots i_r}=k_d^{i_1\cdots i_r}:\hfill \\ & e_1={\overline{k}}_1^1{\overline{t}}_1+{\overline{k}}_2^{1-3}{\overline{t}}_1{\overline{t}}_2{\overline{t}}_3/6,e_2={\overline{k}}_2^{12}{\overline{t}}_1{\overline{t}}_2/2+{\overline{k}}_3^{1-4}{\overline{t}}_1\cdots {\overline{t}}_4/24,\hfill \end{array}$$

(15)

This gives the Edgeworth expansion for the distribution of $Y_n$ to $O\left(n^{-3/2}\right)$. See Withers (2024) for more terms. (15) uses the tensor summation convention of implicitly summing $i_1,\cdots ,i_r$ over their range $1,\cdots ,p$. For example,

$$\begin{array}{cc}& for{\partial }_i=\partial /\partial x_{i}and{\overline{\partial }}_k={\partial }_{i_k},\hfill \\ & P_1\left(x\right)=e_1(-\partial /\partial x)){\mathrm{\Phi }}_V\left(x\right)=\sum _{r=1}^3{\overline{b}}_{r+1}^{1\cdots r}(-{\overline{\partial }}_1)\cdots (-{\overline{\partial }}_r){\mathrm{\Phi }}_V\left(x\right)/r!\hfill \\ & ={\overline{k}}_1^1(-{\overline{\partial }}_1){\mathrm{\Phi }}_V\left(x\right)+{\overline{k}}_2^{1-3}(-{\overline{\partial }}_1)(-{\overline{\partial }}_2)(-{\overline{\partial }}_3){\mathrm{\Phi }}_V\left(x\right)/6,\hfill \\ & p_1\left(x\right)={\overline{k}}_1^1(-{\overline{\partial }}_1){\varphi }_V\left(x\right)+{\overline{k}}_2^{1-3}(-{\overline{\partial }}_1)(-{\overline{\partial }}_2)(-{\overline{\partial }}_3){\varphi }_V\left(x\right)/6.\hfill \\ & (-{\overline{\partial }}_1)\cdots (-{\overline{\partial }}_k){\varphi }_V\left(x\right)={\overline{H}}^{1-k}(x,V){\varphi }_V\left(x\right),\hfill \end{array}$$

for ${\overline{H}}^{1-k}={\overline{H}}^{1-k}(x,V)$ the multivariate Hermite polynomial. For their dual form see Withers and Nadarajah (2014). By Withers (2020), for $i=\sqrt{-1}$,

$$\begin{array}{cc}& {\overline{H}}^{1-k}(x,V)=E{\Pi }_{j=1}^k({\overline{y}}_j+i{\overline{Y}}_j)where{\overline{y}}_j=y_{i_j},{\overline{Y}}_j=Y_{i_j},y=V^{-1}x,\hfill \\ & Y\sim {\mathcal{N}}_p(0,V^{-1}).So,H^1=y_1,{\overline{H}}^1={\overline{y}}_1,H^{12}=y_1{y}_2-V^{12},{\overline{H}}^{12}={\overline{y}}_1{\overline{y}}_2-{\overline{V}}^{12},\hfill \\ & H^{1-3}=y_1{y}_2{y}_3-\sum ^3{V}^{12}{y}_3,\hfill \\ & H^{1-4}=y_1\cdots y_4-\sum ^6{V}^{12}{y}_3{y}_4+\sum ^3{V}^{12}{V}^{34},\hfill \\ & H^{1-5}=y_1\cdots y_5-\sum ^{10}{V}^{12}{y}_3\cdots y_5+\sum ^{15}{V}^{12}{V}^{34}{y}_5,\hfill \\ & H^{1-6}=y_1\cdots y_6-\sum ^{15}{V}^{12}{y}_3\cdots y_6+\sum ^{45}{V}^{12}{V}^{34}{y}_5{y}_6-\sum ^{45}{V}^{12}{V}^{34}{V}^{56},\hfill \end{array}$$

where $V^{i_1{i}_2}$ is the $(i_1,i_2)$ element of $V^{-1},$ ${\overline{V}}_2^{j_{1}j}$ is the $(i_{j_1},i_{j_2})$ element of $V^{-1},$ and for example,

$$\sum ^3{V}^{12}{y}_3=V^{12}{y}_3+V^{13}{y}_2+V^{23}{y}_1.$$

For $r\ge 1$, we can write

$$\begin{array}{cc}& {\tilde{B}}_r\left(e\left(s\right)\right)=\sum _{k=1}^{3r}[{\overline{B}}_r^{1-k}{\overline{s}}_1\cdots {\overline{s}}_k:k-reven],\hfill \\ & where{\overline{B}}_1^1={\overline{k}}_1^1,{\overline{B}}_1^{1-3}={\overline{k}}_2^{1-3}/6,{\overline{B}}_2^{12}={\overline{k}}_1^1{\overline{k}}_1^2+{\overline{k}}_2^{12}/2,\hfill \\ & {\overline{B}}_2^{1-4}={\overline{k}}_3^{1-4}/24+{\overline{k}}_1^1{\overline{k}}_2^{2-4}/6+{\overline{k}}_1^4{\overline{k}}_2^{1-3}/6,{\overline{B}}_2^{1-6}={\overline{k}}_2^{1-3}{\overline{k}}_2^{4-6}/36.\hfill \\ & So,P_r\left(x\right)=\sum _{k=1}^{3r}[{\overline{B}}_r^{1-k}(-{\overline{\partial }}_1)\cdots (-{\overline{\partial }}_k){\mathrm{\Phi }}_V\left(x\right):k-reven],\hfill \\ & p_r\left(x\right)/{\varphi }_V\left(x\right)=\sum _{k=1}^{3r}[{\overline{B}}_r^{1-k}{\overline{H}}^{1-k}(x,V):k-reven]={\tilde{p}}_r\left(x\right)say.\hfill \\ & So,{\tilde{p}}_1\left(x\right)={\overline{k}}_1^1{\overline{H}}^1(x,V)+{\overline{k}}_2^{1-3}{\overline{H}}^{1-3}(x,V)/6,\hfill \\ & P_2\left(x\right)=\sum _{k=2,4,6}{\overline{B}}_2^{1-k}(-{\overline{\partial }}_1)\cdots (-{\overline{\partial }}_k){\mathrm{\Phi }}_V\left(x\right),\hfill \\ & {\tilde{p}}_2\left(x\right)=\sum _{k=2,4,6}{\overline{B}}_2^{1-k}{\overline{H}}^{1-k}(x,V).\hfill \end{array}$$

The log density can be expanded as

$$\begin{array}{cc}& ln[p_n\left(x\right)/{\varphi }_V\left(x\right)]\approx \sum _{r=1}^{\infty }{n}^{-r/2}{b}_r\left(x\right).\hfill \end{array}$$

(16)

$$\begin{array}{cc}& So{p}_n\left(x\right)/{\varphi }_V\left(x\right)]\approx \sum _{r=0}^{\infty }{n}^{-r/2}{\tilde{B}}_r\left(b\left(x\right)\right)whereb=(b_2,b_2,\cdots ).\hfill \end{array}$$

(17)

See Withers and Nadarajah (2016). Cornish-Fisher expansions for parametric and nonparametric standard estimates were first given in Withers (1984) and Withers (1983), and extended to functions of them in Withers (1982). In Withers and Nadarajah (2012) we gave Cornish-Fisher expansions for smooth functions of the sample cross-moments of a linear process. In Section 3 we show that this extends easily to a stationary process.

3. The Cumulants of the Sample Mean

Consider the general real stationary process $\cdots ,X_{-1},X_0,X_1,\cdots$ with finite mean and cross-cumulants,

$$\begin{array}{cc}\hfill & \mu =E{X}_0,k(i_1,\cdots ,i_r)=\kappa (X_{i_1},\cdots ,X_{i_r}).\hfill \end{array}$$

(1)

Given a sequence of integers $i_1,\cdots ,i_r,$ set

$$\begin{array}{c}\hfill i_0=\underset{k=1}{\overset{r}{min}}{i}_k,I_k=i_k-i_0\ge 0,n(i_1\cdots i_r)=I_0=\underset{k=1}{\overset{r}{max}}{I}_k=\underset{k=1}{\overset{r}{max}}{i}_k-i_0.\end{array}$$

(2)

Since $\left\{X_i\right\}$ is stationary,

$$\begin{array}{c}\hfill k(i_1\cdots i_r)=k(I_1\cdots I_r).\end{array}$$

(3)

These are not changed by permuting subscripts. Also at least one $I_k$ is zero. $E\overline{X}=\mu .$ For $r\ge 2,$ transforming from $i_k$ to $T_k=i_k-i_1$ for $k=2,\cdots ,r$,

$$\begin{array}{cc}& n^r{\kappa }_r\left(\overline{X}\right)=\sum _{i_1,\cdots ,i_r=1}^{n}k(i_1,\cdots ,i_r)=\sum _{|T_k|<n,k=2,\cdots ,r}(n-{\delta }_r\left(\mathbf{T}\right))k(0,T_2,\cdots ,T_r)\hfill \end{array}$$

(4)

$$\mathrm{where}{\delta }_r\left(\mathbf{T}\right)=max(0,T_2,\cdots ,T_r)-min(0,T_2,\cdots ,T_r).$$

(5)

$$\begin{array}{cc}& \mathrm{For\; example,},{\delta }_2\left(\mathbf{T}\right)=\left|T_2\right|,\hfill \\ & {\delta }_3\left(\mathbf{T}\right)=T_{3}I(0\le T_2<T_3)+(T_3-T_2)I(T_2<0<T_3)-T_{2}I(T_2<T_3\le 0).\hfill \\ & \mathrm{So\; for}r\ge 2,{\kappa }_r\left(\overline{X}\right)=\sum _{i=r-1}^r{a}_{nri}{n}^{-i}\mathrm{where}\hfill \\ & a_{nr,r-1}=\sum _{|T_i|<n,i=2,\cdots ,r}k(0,T_2,\cdots ,T_r),\hfill \\ & a_{nrr}=-\sum _{|T_i|<n,i=2,\cdots ,r}{\delta }_r\left(\mathbf{T}\right)k(0,T_2,\cdots ,T_r).\hfill \end{array}$$

(6)

This proves that (1) holds. So the Edgeworth-Cornish-Fisher expansions of Section 2 apply to $(\widehat{w},w)=(\overline{X},\mu )$ with $a_{ri}$ in (5) replaced by these $a_{nri}$.

If the cross-cumulants $k(0,T_2,\cdots ,T_r)$ decrease exponentially in $T_2$, as is true for a stationary ARMA process by Withers and Nadarajah (2012), then for $r\ge 2$,

$$\begin{array}{cc}& a_{nr,r-1}=a_{r,r-1}+O\left(e^{-n{\lambda }_r}\right),a_{nrr}=a_{rr}+O\left(e^{-n{\lambda }_r}\right)where{\lambda }_r>0,\hfill \\ & a_{r,r-1}=\sum _{|T_i|<\infty ,i=2,\cdots ,r}k(0,T_2,\cdots ,T_r),\hfill \\ & a_{rr}=-\sum _{|T_i|<\infty ,i=2,\cdots ,r}{\delta }_r\left(\mathbf{T}\right)k(0,T_2,\cdots ,T_r),\hfill \end{array}$$

so that the Edgeworth-Cornish-Fisher expansions of Section 2 apply to

$(\widehat{w},w)=(\overline{X},\mu )$ with $a_{ri}$ in (5) replaced by these $a_{ri}$.

For convergence in law of $n^{1/2}(\overline{X}-\mu ))$ to $\mathcal{N}(0,a_{21})$ with

$a_{21}={\sum }_{T=-\infty }^{\infty }k(0,T)<\infty$, under mixing conditions on a stationary process, see Section 18.4, 18.5 of Ibragimov and Linnik (1971). They also show how to express $a_{21n}$ and $a_{21}$ in terms of the spectral distribution and density.

Missing values. Now suppose that we only have observations at times $t_1,\cdots ,t_n$. Our estimate of $\mu =E{X}_0$ is then

$$\begin{array}{cc}& \widehat{\mu }=\widehat{\mu }(t_1,\cdots ,t_n)=n^{-1}\sum _{i=1}^n{X}_{t_i}.SoE\widehat{\mu }=\mu ,andforr\ge 2,S_k=t_{i_k}-t_{i_1},\hfill \\ & n^r{\kappa }_r\left(\widehat{\mu }\right)=\sum _{i_1,\cdots ,i_r=1}^{n}k(t_{i_1},\cdots ,t_{i_r})=\sum _{i_1,\cdots ,i_r=1}^{n}k(0,S_2,\cdots ,S_r)=n{a}_{r,r-1}say.\hfill \end{array}$$

So if $a_{21}$ is bounded away from 0 and $a_{r,r-1}$ is bounded in n, we can apply Section 2 with $a_{ri}=0$ for $i\ge r$.

Weighted means. Let $w_{n1},\cdots ,w_{nn}$ be given real numbers adding to n. For example the standardized form of the Chernoff weight $i/n$ is $w_{ni}=2i/(n+1).$ See Chernoff and Zacks(1964). An unbiased estimate of $\mu$ is the weighted sample mean, ${\widehat{\mu }}_w={\sum }_{i=1}^n{w}_{ni}{X}_i.$

$$\begin{array}{cc}& Forr\ge 2,n^r{\kappa }_r\left({\widehat{\mu }}_w\right)=\sum _{i_1,\cdots ,i_r=1}^n{w}_{n{i}_1}\cdots w_{n{i}_r}k(t_{i_1},\cdots ,t_{i_r})=n{a}_{r,r-1}\hfill \end{array}$$

say. So if $a_{21}$ is bounded away from 0 and $a_{r,r-1}$ is bounded in n, we can apply Section 2 with $a_{ri}=0$ for $i\ge r$.

4. Multivariate Edgeworth Expansions for $\overline{X}$

Suppose that $\cdots ,X_{-1},X_0,X_1,\cdots$ lie in $R^p$ and are stationary with finite moments. For $j=1,\cdots ,p$, denote the jth component of $X_i$ by $X_i^j$ and the and the cross-cumulants, by

$$\begin{array}{cc}& \mu =E{X}_0,{\mu }^j=E{X}_0^j,\mathrm{and\; for}1\le j_1,\cdots ,j_r\le p,\%M_{i_1\cdots i_r}^{j_1\cdots j_r}=E{X}_{i_1}^{j_1}\cdots X_{i_r},\hfill \\ & k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{i_1,\cdots ,i_r}}\right)=\kappa (X_{i_1}^{j_1},\cdots ,X_{i_r}^{j_r}).\hfill \end{array}$$

(1)

Given a sequence of integers $i_1,\cdots ,i_r$, define $i_0,I_k$ as in (2), and again transform from $i_k$ to $T_k=i_k-i_1$ for $k=2,\cdots ,r$. (3) becomes

$$\begin{array}{c}\hfill k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{i_1\cdots i_r}}\right)=k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{I_1\cdots I_r}}\right).\end{array}$$

(2)

In general $k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1{j}_2}{0,I}}\right)\ne k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1{j}_2}{0,-I}}\right)$. By (4),

$$\begin{array}{cc}& n^r\kappa ({\overline{X}}^{j_1},\cdots ,{\overline{X}}^{j_r})=\sum _{i_1,\cdots ,i_r=1}^{n}k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{i_1,\cdots ,i_r}}\right)\hfill \\ & =\sum _{|T_k|<n,k=2,\cdots ,r}(n-{\delta }_r\left(\mathbf{T}\right))k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{0,T_2,\cdots ,T_r}}\right).\hfill \\ & \mathrm{So\; for}r\ge 2,\kappa ({\overline{X}}^{j_1},\cdots ,{\overline{X}}^{j_r})=\sum _{e=r-1}^r{k}_{ne}^{j_1\cdots j_r}{n}^{-e}\mathrm{where}\hfill \\ & k_{n,r-1}^{j_1\cdots j_r}=\sum _{|T_i|<n,i=2,\cdots ,r}k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{0,T_2,\cdots ,T_r}}\right),\hfill \\ & \mathrm{and}{k}_{nr}^{j_1\cdots j_r}=-\sum _{|T_i|<n,i=2,\cdots ,r}{\delta }_r\left(\mathbf{T}\right)k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{0,T_2,\cdots ,T_r}}\right).\hfill \end{array}$$

(3)

This proves that a 2 term version of (11) holds. So the expansions (12)–(14) and (16) hold for the distribution and density of $n^{1/2}(\overline{X}-\mu )$.

If the cross-cumulants $\left({\displaystyle \genfrac{}{}{0pt}{}{k(j_1\cdots j_r}{0,T_2,\cdots ,T_r)}}\right)$ decrease exponentially in $T_2$, then for $r\ge 2$,

$$\begin{array}{cc}& k_{n,r-1}^{j_1\cdots j_r}=k_{r-1}^{j_1\cdots j_r}+O\left(e^{-n{\lambda }_r}\right),\mathrm{and}{k}_{nr}^{j_1\cdots j_r}=k_r^{j_1\cdots j_r}+O\left(e^{-n{\lambda }_r}\right)\mathrm{where}{\lambda }_r>0,\hfill \\ & k_{r-1}^{j_1\cdots j_r}=\sum _{|T_i|<\infty ,i=2,\cdots ,r}k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{0,T_2,\cdots ,T_r}}\right),\hfill \\ & \mathrm{and}{k}_r^{j_1\cdots j_r}=-\sum _{|T_i|<\infty ,i=2,\cdots ,r}{\delta }_r\left(\mathbf{T}\right)k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\cdots j_r}{0,T_2,\cdots ,T_r}}\right).\hfill \end{array}$$