The Distribution and Quantiles of the Sample Autocovariance and Other Functions of Sample Moments from a Stationary Process

1. Introduction and Summary

In Withers and Nadarajah (2012), we gave the extended Edgeworth-Cornish-Fisher expansions, the eECF expansions, for smooth functions of the sample cross-moments of a linear process. Here we extend these results to a stationary process.

Suppose that $\widehat{\theta }$ is a standard estimate of an unknown parameter $\theta \in R^q$ of a statistical model, based on a sample of size n. That is, its cross cumulants can be expanded in the form (3.3), or if $q=1$, in the form (2.1). Then eECF expansions are available for its distribution that improve upon the Central Limit Theorem. In order to be self-contained, Section 2 summarises these expansions to $O\left(n^{-3/2}\right)$, and Section 3 gives the leading cumulant coefficients for functions of an unbiased standard estimate. In Section 4 we show that the sample moments of a stationary process are standard estimates. (In fact their cumulant expansions have exactly 2 terms!) So by Withers (1982, 2024), smooth functions of them are also standard estimates, and so have the eECF expansions of Section 2 and 3.

Theorem 4.1 gives 2 choices for the cumulant coefficients needed for these expansions. The examples of Section 5 include the distributions of the sample mean, the sample autocovariance and the sample autocorrelation. Section 6 shows how to extend these results to a multivariate stationary process.

This nonparametric approach removes the need for modelling time series parametrically as a moving average or autoregressive or ARIMA process. One weakness of such approaches is that a wrong model will have the wrong asymptotic variance, so that inference will not be even asymptotically correct. Another is the difficulty of estimating their parameters. A vast collection of software has been developed for economists and others for this purpose, all now unnecessary if one moves to our non-parametric method.

Note 1.1. 

Withers and Nadarajah (2010c, 2012) considered the general stationary linear process

$$\begin{array}{cc}& X_i=\sum _{j=0}^{\infty }{\rho }_j{e}_{i-j}\hfill \end{array}$$

where $\left\{{\rho }_j\right\}$ are constants, and $\left\{e_i\right\}$ are independent and identically distributed random variables from a distribution on R with finite cumulants $\left\{{\tau }_j\right\}$. We showed that its cross-cumulants are

$$\begin{array}{cc}{\kappa }_{i_1\dots i_r}& =\alpha (i_1{i}_2\dots i_r){\tau }_r,where\\ \alpha (i_1{i}_2\dots i_r)& =\alpha (I_1{I}_2\dots I_r)=\sum _{j=0}^{\infty }{\rho }_{j+I_1}{\rho }_{j+I_2}\dots {\rho }_{j+I_r},\end{array}$$

(1.1)

for $I_k$ of (4.3), and that $\alpha (i_1{i}_2\dots i_r)$ is finite for processes like ARMA processes where ${\rho }_j\to 0$ exponentially as $n\to \infty .$ See there for a simulation study. For $\left\{X_j\right\}$ an autoregressive process, see p158 of Withers and Nadarajah (2012).

2. eECF Expansions for Standard Estimates

This section summarises results to be found in Withers (1984, 2025a).

Univariate estimates. Suppose that $\widehat{\theta }$ is a standard estimate of an unknown $\theta \in R$ with respect to n, typically the sample size. That is, $E\widehat{\theta }\to \theta$ as $n\to \infty$, and its cumulants can be expanded as

$$\begin{array}{cc}& {\kappa }_r\left(\widehat{\theta }\right)\approx \sum _{j=r-1}^{\infty }{n}^{-j}{a}_{rj}forr\ge 1,\end{array}$$

(2.1)

where the cumulant coefficients $a_{rj}$ 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. So (2.1) holds in the sense that

$${\kappa }_r\left(\widehat{\theta }\right)=\sum _{j=r-1}^{I-1}{a}_{rj}{n}^{-j}+O\left(n^{-I}\right)forI\ge r\ge 1,$$

where $y_n=O\left(x_n\right)$ means that $y_n/x_n$ is bounded in n. For non-lattice estimates, the distribution and quantiles of

$$\begin{array}{cc}& Y_n={(n/a_{21})}^{1/2}(\widehat{\theta }-\theta )\end{array}$$

(2.2)

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

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

(2.3)

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

(2.4)

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

(2.5)

where $\Phi \left(x\right)=Prob(N\le x),N\sim \mathcal{N}(0,1)$ is a unit normal random variable with density $\varphi \left(x\right)={\left(2\pi \right)}^{-1/2}{e}^{-x^2/2}$, 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 $\left\{A_{ri}\right\}$ where $A_{ri}=a_{ri}/a_{21}^{r/2}$. The expansions (2.3)–(2.5) are given in Withers (1984) to $O\left(n^{-5/2}\right)$ and Withers and Nadarajah (2010a) to $O\left(n^{-3}\right)$, starting

$$\begin{array}{cc}& h_1\left(x\right)=f_1\left(x\right)=g_1\left(x\right)=A_{11}+A_{32}{H}_2/6,{\overline{h}}_1\left(x\right)=A_{11}{H}_1+A_{32}{H}_3/6,\\ & h_2\left(x\right)=(A_{11}^2+A_{22})H_1/2+(A_{11}{A}_{32}+A_{43}/4)H_3/6+A_{32}^2{H}_5/72,\\ & f_2\left(x\right)=(A_{22}/2-A_{11}{A}_{32}/3)H_1+A_{43}{H}_3/24-A_{32}^2(4x^3-7x)/36,\\ & g_2\left(x\right)=A_{22}{H}_1/2+A_{43}{H}_3/24-A_{32}^2(2x^3-5x)/36,\end{array}$$

(2.6)

$$\begin{array}{cc}& {\overline{h}}_2\left(x\right)=(A_{11}^2+A_{22})H_2/2+(A_{11}{A}_{32}+A_{43}/4)H_4/6+A_{32}^2{H}_6/72,\end{array}$$

(2.7)

where for $k\ge 0,H_k$ is the kth Hermite polynomial,

$$\begin{array}{cc}& H_k=H_k\left(x\right)=\varphi {\left(x\right)}^{-1}{(-d/dx)}^k\varphi \left(x\right)\\ & =E{(x+iN)}^{k}fori=\sqrt{-1},N\sim \mathcal{N}(0,1).\\ & So,H_{k+1}=(x-d/dx)H_k,(d/dx)H_{k+1}=(k+1)H_k,\\ & H_0=1,H_1=x,H_2=x^2-1,H_3=x^3-3x,H_4=x^4-6x^2+3,\dots \end{array}$$

(2.8)

So (2.3)–(2.8) give the eECF expansions to $O\left(n^{-3/2}\right)$ for the distribution and quantiles of $Y_n$ of (2.2) in terms of $a_{21},a_{11},a_{32},a_{22},a_{43}$, while . $a_{21},a_{11},a_{32}$ alone only give the eECF expansions to $O\left(n^{-1}\right)$, and $a_{21}$ alone gives the eECF expansions to $O\left(n^{-1/2}\right)$.

Note 2.1. 

Cornish and Fisher (1937) and Cornish and Fisher (1960) gave (2.5) for $r\le 6$ for the special case when $a_{ri}=0$ for $i\ne r-1$. This was extended to standard estimates in Withers (1984). For (2.8), see Withers (2000).

We now reserve $i_1,i_2,\dots$ for any sequence from $1,2,\dots ,p$. To avoid double subscripts, we introduce the bar notation.

The multivariate Hermite polynomial, ${\overline{H}}^{1-k}={\overline{H}}^{1-k}(x,V)$, is

$$\begin{array}{cc}& {\overline{H}}^{1-k}(x,V)={\varphi }_V{\left(x\right)}^{-1}(-{\overline{\partial }}_1)\dots (-{\overline{\partial }}_k){\varphi }_V\left(x\right),where{\partial }_i=\partial /\partial x_{i}and{\overline{\partial }}_k={\partial }_{i_k}.\hfill \end{array}$$

By Withers (2020), for $i=\sqrt{-1}$,

$$\begin{array}{cc}& {\overline{H}}^{1-k}(x,V)=E{\Pi }_{j=1}^k({\overline{y}}_j+i{\overline{Y}}_j)where{\overline{y}}_j=y_{i_j},y=V^{-1}x,\hfill \\ & {\overline{Y}}_j=Y_{i_j},Y\sim {\mathcal{N}}_p(0,V^{-1}).\hfill \\ & So,H^1=y_1,{\overline{H}}^1={\overline{y}}_1,H^{12}=y_1{y}_2-V^{12},{\overline{H}}^{12}={\overline{y}}_1{\overline{y}}_2-{\overline{V}}^{12},\hfill \\ & H^{1-3}=y_1{y}_2{y}_3-\sum ^3{V}^{12}{y}_3,where\sum ^3{V}^{12}{y}_3=V^{12}{y}_3+V^{13}{y}_2+V^{23}{y}_1,\hfill \end{array}$$

$V^{i_1{i}_2}$ is the $(i_1,i_2)$ element of $V^{-1},$ and ${\overline{V}}^{j_1{j}_2}$ is the $(i_{j_1},i_{j_2})$ element of $V^{-1}.$ For their dual form see Withers and Nadarajah (2014). Their integrated form, ${\overline{H}}_{*}^{1-k}={\overline{H}}_{*}^{1-k}(x,V)$, is

$$\begin{array}{cc}& {\overline{H}}_{*}^{1-k}=(-{\overline{\partial }}_1)\dots (-{\overline{\partial }}_k){\Phi }_V\left(x\right)={\int }_{-\infty }^x{\overline{H}}^{1-k}{\varphi }_V\left(x\right)dx.\end{array}$$

(2.9)

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

$$\begin{array}{c}\kappa ({\widehat{w}}_{i_1},\dots ,{\widehat{w}}_{i_r})\approx \sum _{j=r-1}^{\infty }{\overline{k}}_j^{1-r}{n}^{-j}where{\overline{k}}_j^{1-r}=k_j^{i_1\dots i_r},\end{array}$$

(2.10)

and the cumulant coefficients ${\overline{k}}_j^{1-r}=k_j^{i_1\dots i_r}$ may depend on n but are bounded as $n\to \infty$. So ${\overline{k}}_0^1={\overline{w}}_1=w_{i_1}.$ Then

$$\begin{array}{cc}& X_n=n^{1/2}(\widehat{w}-w)\stackrel{\mathcal{L}}{\to }{\mathcal{N}}_p(0,V)whereV=\left(k_1^{i_1{i}_2}\right),p\times p,\end{array}$$

(2.11)

with density and distribution

$${\varphi }_V\left(x\right)={\left(2\pi \right)}^{-q/2}{\left(det\left(V\right)\right)}^{-1/2}exp(-x^{\prime }{V}^{-1}x/2),{\Phi }_V\left(x\right)={\int }_{-\infty }^x{\varphi }_V\left(x\right)dx.$$

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), the distribution and density of $X_n=n^{1/2}(\widehat{w}-w)$ can be expanded as

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

(2.12)

$$\begin{array}{cc}& where{P}_0\left(x\right)={\Phi }_V\left(x\right),p_0\left(x\right)={\varphi }_V\left(x\right),andforr\ge 1,\end{array}$$

(2.13)

$$\begin{array}{cc}& P_r\left(x\right)=\sum _{k=1}^{3r}[{\overline{P}}_r^{1-k}{\overline{H}}_{*}^{1-k}(x,V):k-reven],\end{array}$$

(2.14)

$$\begin{array}{cc}& p_r\left(x\right)/{\varphi }_V\left(x\right)=\sum _{k=1}^{3r}[{\overline{P}}_r^{1-k}{\overline{H}}^{1-k}(x,V):k-reven]={\tilde{p}}_r\left(x\right)say.\end{array}$$

(2.15)

(2.14) and (2.15) use the tensor summation convention of implicitly summing $i_1,\dots ,i_r$ over their range $1,\dots ,p$ and ${\overline{P}}_r^{1-k}$ are the Edgeworth coefficients. These are polynomials in the cumulant coefficients ${\overline{k}}_j^{1-r}$ of (2.10). See Withers (2025a) for their definition. The Edgeworth coefficients needed for Edgeworth expansions to $O\left(n^{-3/2}\right)$ are

$$\begin{array}{cc}& {\overline{P}}_1^1={\overline{k}}_1^1,{\overline{P}}_1^{1-3}={\overline{k}}_2^{1-3}/6,{\overline{P}}_2^{12}={\overline{k}}_1^1{\overline{k}}_1^2+{\overline{k}}_2^{12}/2,\hfill \\ & {\overline{P}}_2^{1-4}={\overline{k}}_3^{1-4}/24+\mathcal{S}{\overline{k}}_1^1{\overline{k}}_2^{2-4}/6+\mathcal{S}{\overline{k}}_1^4{\overline{k}}_2^{1-3}/6,{\overline{P}}_2^{1-6}=\mathcal{S}{\overline{k}}_2^{1-3}{\overline{k}}_2^{4-6}/36,\hfill \end{array}$$

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

$$\begin{array}{cc}& P_1\left(x\right)={\overline{k}}_1^1{\overline{H}}_{*}^1(x,V)+{\overline{k}}_2^{1-3}{\overline{H}}_{*}^{1-3}(x,V)/6,\hfill \\ & {\tilde{p}}_1\left(x\right)={\overline{k}}_1^1{\overline{H}}^1(x,V)+{\overline{k}}_2^{1-3}{\overline{H}}^{1-3}(x,V)/6,\hfill \\ & P_2\left(x\right)=\sum _{k=2,4,6}{\overline{P}}_2^{1-k}(-{\overline{\partial }}_1)\dots (-{\overline{\partial }}_k){\Phi }_V\left(x\right),{\tilde{p}}_2\left(x\right)=\sum _{k=2,4,6}{\overline{P}}_2^{1-k}{\overline{H}}^{1-k}(x,V).\hfill \end{array}$$

For ${\overline{P}}_3^{1-k}$ see Withers (2025a). These give the Edgeworth expansions for the distribution of $X_n=n^{1/2}(\widehat{w}-w)$ to $O\left(n^{-2}\right)$. See Withers (2025a) for more details.

eECF expansions for parametric and nonparametric standard estimates were first given in Withers (1984), and extended to functions of them in Withers (1982, 1983).

In Section 4 we show that for $\widehat{\theta }$ and $\widehat{w}$ sample cross-moments, only the first 2 terms in (2.1) and (2.10) are non-zero for $r\ge 2.$

3. Cumulant Coefficients for Functions of Unbiased Standard Estimates

This section summarises the results needed in Section 5 from Withers (1982, 2024).

Suppose that $\widehat{w}\in R^p$ is a standard estimate satisfying (2.10), and that

$$\begin{array}{c}\theta =t\left(w\right):R^p\to R^q\end{array}$$

(3.1)

is a smooth function with finite derivatives

$$\begin{array}{c}{t}_{\cdot a_1{a}_2\dots }={\partial }_{a_1}{\partial }_{a_2}\dots t\left(w\right)where{\partial }_a=\partial /\partial w_a.\end{array}$$

(3.2)

For $b=1,\dots ,q$, let ${\theta }^b$ be the bth component of $\theta$. Then by Withers (1982, 2024), $\widehat{\theta }=t\left(\widehat{w}\right)\in R$ is a standard estimate of $\theta =t\left(w\right)$. So its rth order cross-cumulants are of magnitude $n^{1-r}$ with an expansion of the form

$$\begin{array}{c}\kappa ({\widehat{\theta }}^{b_1},\dots ,{\widehat{\theta }}^{b_r})\approx \sum _{j=r-1}^{\infty }{n}^{-j}{K}_{nj}^{b_1\dots b_r}\end{array}$$

(3.3)

where $b_1,\dots b_r$ lie in $1,\dots ,q,K_{n0}^b={\theta }^b,$ and the cumulant coefficients for $\widehat{\theta },K_{nj}^{b_1\dots b_r}$, are given in terms of the cumulant coefficients for $\widehat{w},\{{\overline{k}}_j^{1-r}=k_j^{a_1\dots a_r}\}$ of (2.10) and the derivatives $t_{.a_1{a}_2\dots }$, by the appendix to Withers (1982), and in more detail in Theorem 2 of Withers (2024), where we now assume that $E\widehat{w}=w$. So,

$$\begin{array}{c}{Z}_n=n^{1/2}(\widehat{\theta }-\theta )\stackrel{\mathcal{L}}{\to }{\mathcal{N}}_q(0,\tilde{V})asn\to \infty where\tilde{V}=\left(K_{n1}^{b_1{b}_2}\right),\end{array}$$

(3.4)

$$\begin{array}{cc}& K_{n1}^{b_1{b}_2}=t_{\cdot a_1}^{b_1}{k}_1^{a_1{a}_2}{t}_{\cdot a_2}^{b_2},K_{n1}^{b_1}=t_{\cdot a_1{a}_2}^{b_1}{k}_1^{a_1{a}_2}/2,\end{array}$$

(3.5)

where we use the tensor summation convention of implicit summation of the repeated pairs, in this case $a_1,a_2$, over their range $1,\dots ,p$. That is,

$$K_{n1}^{b_1{b}_2}=\sum _{a_1,a_2=1}^p{t}_{\cdot a_1}^{b_1}{k}_1^{a_1{a}_2}{t}_{\cdot a_2}^{b_2},K_{n1}^{b_1}=\sum _{a_1,a_2=1}^p{t}_{\cdot a_1{a}_2}^{b_1}{k}_1^{a_1{a}_2}/2.$$

Similarly,

$$\begin{array}{cc}& K_{n2}^{b_1{b}_2{b}_3}=t_{\cdot a_1}^{b_1}{t}_{\cdot a_2}^{b_2}{t}_{\cdot a_3}^{b_3}{k}_2^{a_1{a}_2{a}_3}+\sum _{b_1{b}_2{b}_3}^3{s}_{a_1}^{b_1}{t}_{\cdot a_1{a}_3}^{b_2}{s}_{a_3}^{b_3}where{s}_{a_1}^{b_1}=k_1^{a_1{a}_2}{t}_{\cdot a_2}^{b_1},\\ & K_{n2}^{b_1{b}_2}=t_{\cdot a_1}^{b_1}{t}_{\cdot a_2}^{b_2}{k}_2^{a_1{a}_2}+\sum _{b_1{b}_2}^2[t_{\cdot a_1{a}_2}^{b_1}{t}_{\cdot a_3}^{b_2}{k}_2^{a_1{a}_2{a}_3}+t_{\cdot a_1{a}_2{a}_3}^{b_1}{s}_{a_3}^{b_2}{k}_1^{a_1{a}_2}]/2\end{array}$$

(3.6)

$$\begin{array}{cc}& +v_{a_3}^{b_1{a}_2}{v}_{a_2}^{b_2{a}_3}/2where{v}_{a_3}^{b_1{a}_2}=t_{\cdot a_1{a}_3}^{b_1}{k}_1^{a_1{a}_2},\\ & K_{n3}^{b_1\dots b_4}=t_{\cdot a_1}^{b_1}\dots t_{\cdot a_4}^{b_4}{k}_3^{a_1\dots a_4}+k_2^{a_1{a}_2{a}_3}\sum ^{12}{t}_{\cdot a_2}^{b_2}{t}_{\cdot a_3}^{b_3}{u}_{a_1}^{b_1{b}_4}+\sum ^4{t}_{\cdot a_1{a}_3{a}_5}^{b_1}{s}_{a_1}^{b_2}{s}_{a_3}^{b_3}{s}_{a_5}^{b_4}\end{array}$$

(3.7)

$$\begin{array}{cc}& +\sum ^{12}{u}_{a_1}^{b_1{b}_3}{k}_1^{a_1{a}_2}{u}_{a_2}^{b_2{b}_4}where{u}_{a_1}^{b_1{b}_4}=t_{\cdot a_1{a}_4}^{b_1}{s}_{a_4}^{b_4},\end{array}$$

(3.8)

These are the coefficients needed for $P_r\left(x\right),p_r\left(x\right)$ of (2.12) for $r=1,2$ when $X_n$ of (2.11) is replaced by $Z_n$ of (3.4).

The case $q=1$. By (2.1), we can replace $K_{nj}^{b_1\dots b_r}$ by $a_{rj}$ where by (3.3)–(3.6),

$$\begin{array}{cc}& a_{10}=\theta ,a_{21}=t_{\cdot a_1}{k}_1^{a_1{a}_2}{t}_{\cdot a_2}=t_{\cdot a_1}{s}_{a_1}for{s}_{a_1}=k_1^{a_1{a}_2}{t}_{\cdot a_2},,\end{array}$$

(3.9)

$$\begin{array}{cc}& a_{11}=t_{\cdot a_1{a}_2}{k}_1^{a_1{a}_2}/2,a_{32}=t_{\cdot a_1}{t}_{\cdot a_2}{t}_{\cdot a_3}{k}_2^{a_1{a}_2{a}_3}+3s_{a_1}{t}_{\cdot a_1{a}_3}{s}_{a_3},\end{array}$$

(3.10)

$$\begin{array}{cc}& a_{22}=t_{\cdot a_1}{k}_2^{a_1{a}_2}{t}_{\cdot a_3}+t_{\cdot a_1{a}_2}{k}_2^{a_1{a}_2{a}_3}{t}_{\cdot a_3}+s_{a_3}{t}_{\cdot a_1{a}_2{a}_3}{k}_1^{a_1{a}_2}+v_{a_3}^{a_2}{v}_{a_2}^{a_3},\\ & a_{43}=t_{\cdot a_1}\dots t_{\cdot a_4}{k}_3^{a_1\dots a_4}+12u_{a_1}{k}_2^{a_1{a}_2{a}_3}{t}_{\cdot a_2}{t}_{\cdot a_3}+4t_{\cdot a_1{a}_3{a}_5}{s}_{a_1}{s}_{a_3}{s}_{a_5}\end{array}$$

(3.11)

$$\begin{array}{cc}& +12u_{a_1}{k}_1^{a_1{a}_2}{u}_{a_2}for{v}_{a_3}^{a_2}=t_{\cdot a_1{a}_3}{k}_1^{a_1{a}_2},u_{a_1}=t_{\cdot a_1{a}_2}{s}_{a_2}.\end{array}$$

(3.12)

This gives the $a_{ri}$ needed for $\{h_r,{\overline{h}}_r,f_r,g_r,r=1,2\}$ of Section 2. So if $a_{21}$ is bounded away from 0 as $n\to \infty$, then $Y_n={(n/a_{21})}^{1/2}(\widehat{\theta }-\theta )$ has the eECF expansions (2.3)–(2.5).

$$\begin{array}{cc}& Forany{f}_{12},set\sum _{12}^2{f}_{12}=f_{12}+f_{21},and\sum _{123}^3{f}_{12}=f_{12}+f_{23}+f_{31}.\end{array}$$

(3.13)

If $p=1$, then $s_1=t_{\cdot 1}{k}_1^{11},u_1=t_{\cdot 11}{s}_1=t_{\cdot 1}{t}_{\cdot 11}{k}_1^{11},v_{11}=t_{\cdot 11}{k}_1^{11}$,

$$\begin{array}{cc}& a_{21}=t_{\cdot 1}^2{k}_1^{11}=t_{\cdot 1}{s}_1,a_{11}=t_{\cdot 11}{k}_1^{11}/2,a_{32}=t_{\cdot 1}^3{k}_2^{111}+3s_1^2{t}_{\cdot 11},\\ & a_{22}=t_{\cdot 1}^2{k}_2^{11}+t_{\cdot 1}{t}_{\cdot 11}{k}_2^{111}+s_1{t}_{\cdot 111}{k}_1^{11}+{\left(v_1^1\right)}^2/2,\\ & a_{43}=t_{\cdot 1}^4{k}_3^{1111}+12k_2^{111}{t}_{\cdot 1}^2{t}_{\cdot 11}{s}_1+4t_{\cdot 111}{s}_1^3+12k_1^{11}{u}_1^2.\end{array}$$

(3.14)

If $p=2$, as in Examples 5.3 and 5.4 below, then $s_j={\sum }_{i=1}^2{k}_1^{ji}{t}_{\cdot i},$

$u_k={\sum }_{j=1}^2{t}_{\cdot kj}{s}_j$, and (3.9)–(3.12) can be written using (3.13) as

$$\begin{array}{cc}& a_{21}=\sum _{i=1}^2{t}_{\cdot i}^2{k}_1^{ii}+2t_{\cdot 1}{t}_{\cdot 2}{k}_1^{12},a_{11}=\sum _{i=1}^2{t}_{\cdot ii}{k}_1^{ii}/2+t_{\cdot 12}{k}_1^{12},\end{array}$$

(3.15)

$$\begin{array}{cc}& a_{32}=\sum _{i=1}^2{t}_{\cdot i}^3{k}_2^{iii}+3\sum _{12}^2{t}_{\cdot 1}^2{t}_{\cdot 2}{k}_2^{112}+3\sum _{j=1}^2{s}_j^2{t}_{\cdot jj}+6s_1{s}_2{t}_{\cdot 12},\\ & a_{22}=t_{\cdot i}{k}_2^{ij}{t}_{\cdot j}+\sum _{i=1}^2{t}_{\cdot i}(t_{\cdot ii}{k}_2^{iii}+2t_{\cdot 12}{k}_2^{12i})+\sum _{12}^2{t}_{\cdot 11}{k}_2^{112}{t}_{\cdot 2}+\sum _{i=1}^2{s}_i{t}_{\cdot iii}{k}_1^{ii}\end{array}$$

(3.16)

$$\begin{array}{cc}& +\sum _{12}^2{s}_1(2t_{\cdot 112}{k}_1^{12}+t_{\cdot 122}{k}_1^{22})+v_j^i{v}_i^j/2=\sum _{k=1}^6{a}_{22k}say,\end{array}$$

(3.17)

$$\begin{array}{cc}& a_{43}=\sum _{k=1}^4{a}_{43k}where{a}_{431}=\sum _{i=1}^2{t}_{\cdot i}^4{k}_3^{1111}+4\sum _{12}^2{t}_{\cdot 1}^3{t}_{\cdot 2}{k}_3^{1112}+6t_{\cdot 1}^2{t}_{\cdot 2}^2{k}_3^{1122},\end{array}$$

(3.18)

$$\begin{array}{cc}& a_{432}=12u_{a_1}{t}_{\cdot a_2}{t}_{\cdot a_3}{k}_2^{a_1{a}_2{a}_3},a_{433}/4=\sum _{i=1}^2{s}_i^3{t}_{\cdot iii}+3s_1^2{s}_2{t}_{\cdot 112}+3s_1{s}_2^2{t}_{\cdot 122},\\ & a_{434}/12=\sum _{i=1}^2{u}_i^2{k}_1^{ii}+2u_1{u}_2{k}_1^{12}.\end{array}$$

If $p=3$, as in Examples 5.5 and 5.6 below, then $s_j={\sum }_{i=1}^3{k}_1^{ji}{t}_{\cdot i}$, and (3.9) and (3.10) can be written using (3.13), as

$$\begin{array}{cc}{a}_{21}& =\sum _{i=1}^3{t}_{\cdot i}^2{k}_1^{ii}+2\sum _{123}^3{t}_{\cdot 1}{t}_{\cdot 2}{k}_1^{12},a_{11}=\sum _{i=1}^3{t}_{\cdot ii}{k}_1^{ii}/2+\sum _{123}^3{t}_{\cdot 12}{k}_1^{12},\\ a_{32}& =\sum _{i=1}^3{t}_{\cdot i}^3{k}_2^{iii}+3\sum _{123}^3{t}_{\cdot 1}^2{t}_{\cdot 2}{k}_2^{112}+6t_{\cdot 1}{t}_{\cdot 2}{t}_{\cdot 3}{k}_2^{123}+3\sum _{j=1}^3{s}_j^2{t}_{\cdot jj}+2\sum _{123}^3{s}_1{s}_2{t}_{\cdot 12}.\end{array}$$

(3.19)

4. The Cumulants of the Sample Cross-Moments

This section uses the notation of Section 2 of Withers (2025b). Let $\dots ,X_{-1},X_0,X_1,\dots$ be any real stationary process with finite mean, cross-moments, and cross-cumulants,

$$\begin{array}{cc}& \mu =E{X}_0,M_{i_1\dots i_r}=E{X}_{i_1}\dots X_{i_r},{\mu }_{i_1\dots i_r}=E(X_{i_1}-\mu )\dots (X_{i_r}-\mu ),\end{array}$$

(4.1)

$$\begin{array}{cc}& k(i_1,\dots ,i_r)=\kappa (X_{i_1},\dots ,X_{i_r}).So,{\mu }_{0^r}={\mu }_r\left(X_0\right),k\left(0^r\right)={\kappa }_r\left(X_0\right),\end{array}$$

(4.2)

where $0^r$ denotes a string of r zeros. For relationships between them see Section 3.30 of Stuart and Ord (1987). Multivariate relations can be written down from their univariate versions. For example for $M_r,{\mu }_r$ and ${\kappa }_r$ the rth central moment and cumulant of $X\in R$,

$$\begin{array}{cc}\hfill M_2& ={\kappa }_2+{\kappa }_1^2\Rightarrow M_{11}={\kappa }_{11}+{\kappa }_{10}{\kappa }_{01},\hfill \\ \hfill {\kappa }_4& ={\mu }_4-3{\mu }_2^2\Rightarrow k\left(1234\right)={\mu }_{1234}-{\mu }_{12}{\mu }_{34}-{\mu }_{13}{\mu }_{24}-{\mu }_{14}{\mu }_{23}.\hfill \end{array}$$

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

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

(4.3)

$$\begin{array}{cc}& So,M_{i_1\dots i_r}=M_{I_1\dots I_r},{\mu }_{i_1\dots i_r}={\mu }_{I_1\dots I_r},{\kappa }_{i_1\dots i_r}={\kappa }_{I_1\dots I_r}.\end{array}$$

(4.4)

These are not changed by permuting subscripts. Also at least one $I_k$ is zero. For $a\ge 0$ and $k(.)$ of (4.2), the ath autocovariance and autocorrelation are

$$\begin{array}{cc}& covar(X_0,X_a)=k(0,a),andcorr(X_0,X_a)=k(0,a)/k\left(0^2\right).\\ & Also,k(0,T)=k(0,|T\left|\right),andif{T}_1<T_2<0or{T}_1<0<T_2,then\\ & k(0,T_1,T_2)=k(0,T_2-T_1,-T_1)=k(0,|T_2-T_1|,|T_1\left|\right).\end{array}$$

(4.5)

Now suppose that we observe only $X_1,\dots ,X_n$. For $I_1,\dots ,I_r$ and $n>I_0=I_0(i_1\dots i_r)$ of (4.3), define the sample non-central cross-moment

$$\begin{array}{c}{\widehat{M}}_{i_1\dots i_r}={\widehat{M}}_{I_1\dots I_r}=N^{-1}\sum _{t=1}^N{X}_{t+I_1}\dots X_{t+I_r}whereN=n-I_0.\end{array}$$

(4.6)

This is an unbiased estimate of $M_{i_1\dots i_r}$ of (4.1). For example $\mu =E{X}_0$ and $M_{0a}=E{X}_0{X}_a$ have unbiased estimates

$$\begin{array}{cc}& \widehat{\mu }=\overline{X}={\widehat{M}}_0=n^{-1}\sum _{j=1}^n{X}_j,\end{array}$$

(4.7)

$$\begin{array}{cc}& {\widehat{M}}_{0a}=N^{-1}\sum _{j=1}^N{X}_j{X}_{j+a}forN=n-a>0,\end{array}$$

(4.8)

and $a\ge 0$ an integer. We can write (4.6) as

$$\begin{array}{c}{\widehat{M}}_{\pi }=N^{-1}\sum _{t=1}^N{X}_t\left(\pi \right)forN=n-I_0\left(\pi \right)>0,that is,forn>I_0\left(\pi \right).\end{array}$$

Given $p\ge 1$ and sequences of integers ${\pi }_1,\dots ,{\pi }_p$, for $j=1,\dots ,p$, set

$$\begin{array}{cc}{w}_j& =M_{{\pi }_j},{\widehat{w}}_j={\widehat{M}}_{{\pi }_j},w=(w_1,\dots ,w_p),\widehat{w}=({\widehat{w}}_1,\dots ,{\widehat{w}}_p),N_j=n-I_0\left({\pi }_j\right).\end{array}$$

(4.9)

We shall see that (2.10) holds with ${\overline{k}}_j^{1-r}=0$ for $j>r\ge 2.$ So $\widehat{w}$ is a standard estimate.

$$\begin{array}{cc}& \kappa ({\widehat{w}}_1,\dots ,{\widehat{w}}_r)={(N_1\dots N_r)}^{-1}\sum _{t_1=1}^{N_1}\dots \sum _{t_r=1}^{N_r}K\left({\displaystyle \genfrac{}{}{0pt}{}{{\pi }_1\dots {\pi }_r}{t_1\dots t_r}}\right)\end{array}$$

(4.10)

$$\begin{array}{cc}& whereK\left({\displaystyle \genfrac{}{}{0pt}{}{{\pi }_1\dots {\pi }_r}{t_1\dots t_r}}\right)=\kappa (X_{t_1}\left({\pi }_1\right),\dots ,X_{t_r}\left({\pi }_r\right)),\\ & and{X}_t(i_1\dots i_r)=X_{t+I_1}\dots X_{t+I_r}.\end{array}$$

(4.11)

Example 4.1. 

If ${\pi }_1=\left(0\right)$ and ${\pi }_2=(0,a),$ then $I_0\left({\pi }_1\right)=0,I_0\left({\pi }_2\right)=a,N_1=n,N_2=n-a$,

$$\begin{array}{cc}& X_t\left({\pi }_1\right)=X_t,X_t\left({\pi }_2\right)=X_t{X}_{t+a},K\left({\displaystyle \genfrac{}{}{0pt}{}{{\pi }_1{\pi }_2}{t_1{t}_2}}\right)=\kappa (X_{t_1},X_{t_2}{X}_{t_2+a}),\hfill \\ & {\kappa }_2\left({\widehat{w}}_1\right)=n^{-2}\sum _{t_1,t_2=1}^n\kappa (X_{t_1},X_{t_2}),{\kappa }_2\left({\widehat{w}}_2\right)=N_2^{-2}\sum _{t_1,t_2=1}^{N_2}\kappa (X_{t_1}{X}_{t_1+a},X_{t_2}{X}_{t_2+a}),\hfill \\ & \kappa ({\widehat{w}}_1,{\widehat{w}}_2)={\left(n{N}_2\right)}^{-1}\sum _{t_1=1}^n\sum _{t_2=1}^{N_2}\kappa (X_{t_1},X_{t_2}{X}_{t_2+a}).\hfill \end{array}$$

$K\left({\displaystyle \genfrac{}{}{0pt}{}{{\pi }_1\dots {\pi }_r}{t_1\dots t_r}}\right)$ can be written in terms of the cross-cumulants of $\left\{X_j\right\}$ using p254–265 of McCullagh (1987) if $L=L_1+\dots +L_r\le 8$ where $L_i$ the length of the sequence ${\pi }_i$. See his p58 and the appendix below for some examples. So this covers ${\kappa }_4\left({\widehat{M}}_{i_1{i}_2}\right)$, which has $L=8$, but not ${\kappa }_3\left({\widehat{M}}_{i_1{i}_2{i}_3}\right)$, which has $L=9$. So at present we can obtain the eECF expansions for ${\widehat{M}}_{i_1{i}_2}$ to $O\left(n^{-3/2}\right)$, but the eECF expansions for ${\widehat{M}}_{i_1{i}_2{i}_3}$ only to $O\left(n^{-1}\right)$.

Under mild conditions, $n^{r-1}\kappa ({\widehat{w}}_{i_1},\dots ,{\widehat{w}}_{i_r})$ is bounded as n increases, as the examples show.

Theorem 4.1. 

Take $r\ge 2.$ Let $K(t_1,\dots ,t_r)$ be any symmetric function of integers $t_1,\dots ,t_r$ such that

$$for any integer,K(t_1,\dots ,t_r)\equiv K(t_1-T,\dots ,t_r-T).$$

Take $n\ge 1$ and integers $a_1<n,\dots ,a_r<n$. Then for $N_j=n-a_j$,

$$\begin{array}{cc}& \sum _{t_1=1}^{N_1}\dots \sum _{t_r=1}^{N_r}K(t_1,\dots ,t_r)=\sum _{-N_1<T_j<N_j,j=2,\dots ,r}{D}_{r\mathbf{NT}}K(0,T_2,\dots ,T_r),\end{array}$$

(4.12)

$$\begin{array}{cc}& where{D}_{r\mathbf{NT}}=min(N_1,N_2-T_2,\dots ,N_r-T_r)+m_{r\mathbf{T}}\end{array}$$

(4.13)

$$\begin{array}{cc}& =n-{\delta }_{r\mathbf{T}}=N_1-{\delta }_{r\mathbf{T}}^{\prime },m_{r\mathbf{T}}=min(0,T_2,\dots ,T_r),\end{array}$$

(4.14)

$$\begin{array}{cc}& {\delta }_{r\mathbf{T}}=max(a_1,T_2+a_2,\dots ,T_r+a_r)-m_{r\mathbf{T}},\end{array}$$

(4.15)

$$\begin{array}{cc}& {\delta }_{r\mathbf{T}}^{\prime }={\delta }_{r\mathbf{T}}-a_1=max(0,T_2+a_2-a_1,\dots ,T_r+a_r-a_1)-m_{r\mathbf{T}}.\end{array}$$

(4.16)

$$\begin{array}{cc}& So LHS\left(4.12\right)=n{a}_{r,r-1}^K+a_{rr}^K=N_1{a}_{r,r-1}^K+a_{rr}^{{}^{\prime }K}where\end{array}$$

(4.17)

$$\begin{array}{cc}& (a_{r,r-1}^K,a_{rr}^K,a_{rr}^{{}^{\prime }K})=\sum _{-N_1<T_j<N_j,j=2,\dots ,r}(1,-{\delta }_{r\mathbf{T}},-{\delta }_{r\mathbf{T}}^{\prime })K(0,T_2,\dots ,T_r)\end{array}$$

(4.18)

$$\begin{array}{cc}& \to (a_{r,r-1},a_{rr},a_{rr}^{\prime })=\sum _{|T_j|<\infty ,j=2,\dots ,r}(1,-{\delta }_{r\mathbf{T}},-{\delta }_{r\mathbf{T}}^{\prime })K(0,T_2,\dots ,T_r)\end{array}$$

(4.19)

as $n\to \infty$, when finite, and

$$\begin{array}{cc}& a_{rr}^K=a_{rr}^{{}^{\prime }K}-a_1{a}_{r,r-1}^K,a_{rr}=a_{rr}^{\prime }-a_1{a}_{r,r-1}^{\prime },\end{array}$$

(4.20)

$$\begin{array}{cc}& a_{21}^K=\sum _{-N_1<T<N_2}K(0,T)\to a_{21}=\sum _{T=-\infty }^{\infty }K(0,T),\end{array}$$

(4.21)

$$\begin{array}{cc}& a_{32}^K=\sum _{-N_1<T_j<N_j,j=2,3}K(0,T_2,T_3)\to a_{32}=\sum _{T_2,T_3=-\infty }^{\infty }K(0,T_2,T_3),\end{array}$$

(4.22)

$$\begin{array}{cc}& a_{22}^{{}^{\prime }K}=-\sum _{-N_1<T<N_2}{\delta }_{2T}^{\prime }K(0,T)\to a_{22}^{\prime }=-\sum _{T=-\infty }^{\infty }{\delta }_{2T}^{\prime }K(0,T),\\ & where{\delta }_{2T}^{\prime }=max(0,T+a_2-a_1)-min(0,T),\end{array}$$

(4.23)

$$\begin{array}{cc}& a_{43}^K=\sum _{-N_1<T_j<N_j,j=2,3,4}K(0,T_2,T_3,T_4)\to a_{43}=\sum _{T_2,T_3,T_4=-\infty }^{\infty }K(0,T_2,T_3,T_4).\end{array}$$

(4.24)

Now suppose that $a_i\equiv a.$ Then $N_i\equiv N=n-a,$

$$\begin{array}{cc}& {\delta }_{r\mathbf{T}}^{\prime }=max(0,T_2,\dots ,T_r)-min(0,T_2,\dots ,T_r),{\delta }_{2\mathbf{T}}^{\prime }=\left|T_2\right|,\\ & {\delta }_{3\mathbf{T}}^{\prime }=T_{3}I(0\le T_2<T_3)+(T_3-T_2)I(T_2\le 0<T_3)-T_{2}I(T_2<T_3\le 0),\end{array}$$

(4.25)

$$\begin{array}{cc}& (a_{r,r-1}^K,a_{rr}^K,a_{rr}^{{}^{\prime }K})=\sum _{|T_j|<N,j=2,\dots ,r}(1,-{\delta }_{r\mathbf{T}},-{\delta }_{r\mathbf{T}}^{\prime })K(0,T_2,\dots ,T_r).\end{array}$$

(4.26)

PROOF Transform from $t_i$ to $T_i=t_i-t_1$ for $i=2,\dots ,r$. So (4.12) holds with

$$\begin{array}{cc}& D_{r\mathbf{NT}}=\sum _{t_1=1}^{N_1}I(-T_j<t_1\le N_j-T_j,j=2,\dots ,r)\hfill \\ & =min(N_1,N_2-T_2,\dots ,N_r-T_r)-max(0,-T_2,\dots ,-T_r).\square \hfill \end{array}$$

For $i=r-1,r$ this will give us two choices for $(a_{r,r-1},a_{rr})$ needed for (2.1)–(2.5), namely $(a_{r,r-1}^K,a_{rr}^K)$ of (4.18) and its limit, $(a_{r,r-1},a_{rr})$ of (4.19). When $a_i\equiv a$, we have another 2 choices for $a_{rr}$, namely $a_{rr}^{{}^{\prime }K}$ of (4.18) and its limit, $a_{rr}^{\prime }$ of (4.19). These limits can be used when the differences, for example $a_{rr}^{{}^{\prime }K}-a_{rr}^{\prime }$, are exponentially small in n, that is, they have magnitude $O\left(e^{-n\lambda }\right)$ for some $\lambda >0$. When $a_i\equiv a$, ${\delta }_{r\mathbf{T}}^{\prime }$ of (4.25) is simpler than ${\delta }_{r\mathbf{T}}=a+{\delta }_{r\mathbf{T}}^{\prime }$, so $a_{rr}^{{}^{\prime }K}$ is simpler than $a_{rr}^K$, and the expansions of Section 2 may be simpler if n is replaced by $N_1$ as in Example 5.2.

Corollary 4.1. 

Take $a_j=I_0\left({\pi }_j\right),N_j=n-a_j,w_j,{\widehat{w}}_j$ of (4.9), and $D_{r\mathbf{NT}},{\delta }_{r\mathbf{T}},{\delta }_{r\mathbf{T}}^{\prime }$ of (4.10)–(4.16). Then for Kof (4.11),

$$\begin{array}{cc}& \kappa ({\widehat{w}}_1,\dots ,{\widehat{w}}_r)=\sum _{j=r-1}^r{n}^{-j}{k}_j^{1\dots r}where{k}_j^{1\dots r}=n^r{(N_1\dots N_r)}^{-1}{a}_{rj}^K,\end{array}$$

(4.27)

$$\begin{array}{cc}& (a_{r,r-1}^K,a_{rr}^K)=\sum _{-N_1<T_j<N_j,j=2,\dots ,r}(1,-{\delta }_{r\mathbf{T}})K\left({\displaystyle \genfrac{}{}{0pt}{}{{\pi }_1{\pi }_2\dots {\pi }_r}{0T_2\dots T_r}}\right).\end{array}$$

(4.28)

So $D_{r\mathbf{NT}}=n-{\delta }_{r\mathbf{T}}=N_{i_1}-{\delta }_{r\mathbf{T}}^{{}^{\prime }*},$ and (4.17)–(4.24) hold with $K(0,T_2,\dots ,T_r)$ replaced by $K\left({\displaystyle \genfrac{}{}{0pt}{}{{\pi }_{i_1}{\pi }_{i_2}\dots {\pi }_{i_r}}{0T_2\dots T_r}}\right)$.

PROOF By (4.10), (4.12), and (4.18), for $k_j^{1\dots r}$ of (4.27), $LHS\left(4.10\right)=$

$${(N_1\dots N_r)}^{-1}RHS\left(4.12\right)={(N_1\dots N_r)}^{-1}(n{a}_{r,r-1}^K+a_{rr}^K)=\sum _{j=r-1}^r{n}^{-j}{k}_j^{1\dots r}.\square$$

For its univariate version, we have the option of expanding in $N^{-1/2}$ rather than in $n^{-1/2}$ as in Section 2.

Corollary 4.2. 

Given a sequence of integers π, set

$$\begin{array}{cc}& N=n-I_0\left(\pi \right),{\pi }^r=(\pi ,\dots ,\pi ).\end{array}$$

$$\begin{array}{cc}& Then{\kappa }_r\left({\widehat{M}}_{\pi }\right)=N^{1-r}{a}_{r,r-1}^K+N^{-r}{a}_{rr}^{{}^{\prime }K}\end{array}$$

(4.29)

$$\begin{array}{cc}& where(a_{r,r-1}^K,a_{rr}^{{}^{\prime }K})=\sum _{|T_j|<N,j=1,\dots ,r}(1,-{\delta }_{r\mathbf{T}}^{\prime })K\left({\displaystyle \genfrac{}{}{0pt}{}{\pi \pi \dots \pi }{0T_2\dots T_r}}\right).\end{array}$$

(4.30)

for $a_{rj}^K$ of (4.28). So the eECF expansions for $Y_n$ of (2.2), that is, (2.3)–(2.5), hold with $(n,Y_n,a_{r,r-1},a_{rr})$ replaced by $(N,Y_N,a_{r,r-1}^K,a_{rr}^{{}^{\prime }K})$,

$$where{Y}_N={(N/a_{21}^K)}^{1/2}({\widehat{M}}_{\pi }-M_{\pi }).$$

5. Examples

Example 5.1. 

The eECF expansions to $O\left(n^{-3/2}\right)$ for the sample mean, $\widehat{\mu }$ of (4.7).

Take $p=1,\pi =\left(0\right)$ so that ${\widehat{M}}_0=\widehat{\mu }=\overline{X}$. For $r\ge 2$ and $K(.)=k(.)$ of (4.2),

$$\begin{array}{cc}& {\kappa }_r\left(\overline{X}\right)=n^{1-r}{a}_{r,r-1}^k+n^{-r}{a}_{rr}^{k}of\left(4.18\right)with{N}_1=n\\ & =n^{1-r}{a}_{r,r-1}+n^{-r}{\tilde{a}}_{rr}+O\left(e^{-n\lambda }\right)of\left(4.19\right)where\lambda >0,\end{array}$$

(5.1)

under mild conditions. (2.1)–(2.5) hold for

$$\begin{array}{cc}& \widehat{w}=\overline{X}with{a}_{10}=\mu ,a_{1j}=0forj\ge 1.\end{array}$$

So (2.3)–(2.7) hold for $Y_n={(n/a_{21})}^{1/2}(\overline{X}-\mu )$ where for $r\ge 2,$

$$(a_{r,r-1},a_{rr})=(a_{r,r-1}^k,a_{rr}^k)of\left(4.18\right)or(a_{r,r-1},a_{rr})of\left(4.19\right),$$

and $a_{ri}=0$ for $i>r.$ This example was the subject of Withers (2025b). See there for extensions to missing values and weighted means.

Example 5.2. 

The eECF expansions to $O\left(n^{-3/2}\right)$ for the sample autocovarianceassuming that $\mu =0$.(This assumption is common in the literature on the grounds that the series can be adjusted by subtracting the estimated mean. But by Example 5.3, it gives the wrong variance if $\mu \ne 0$!) In this case, $p=1$ and for $a\ge 0,w=M_{0a}=E{X}_0{X}_a$ has unbiased estimate $\widehat{w}={\widehat{M}}_{0a}$ of (4.8). So $\pi =(0,a),N=n-a$, and by Corollary 4.2,

$$\begin{array}{cc}& {\kappa }_r\left({\widehat{M}}_{0a}\right)=N^{1-r}{a}_{r,r-1}^K+N^{-r}{a}_{rr}^{{}^{\prime }K}\end{array}$$

$$\begin{array}{cc}& \approx N^{1-r}{a}_{r,r-1}+N^{-r}{a}_{rr}^{\prime }of\left(4.18\right)and\left(4.19\right),\end{array}$$

(5.2)

$$\begin{array}{cc}& whereK(t_1,\dots ,t_r)=\kappa (X_{t_1}\left(\pi \right),\dots ,X_{t_r}\left(\pi \right))for{X}_t\left(\pi \right)=X_t{X}_{t+a}.\end{array}$$

(5.3)

So (2.3)–(2.5) hold with $(n,Y_n)$ replaced by $(N,Y_N)$ where

$$Y_N={(N/a_{21})}^{1/2}({\widehat{M}}_{0a}-M_{0a}),$$

for $a_{ri}=a_{ri}^K$ of (4.26), or its limit when the difference is exponentially small. Alternatively, by (4.27),

$$\begin{array}{cc}& {\kappa }_r\left({\widehat{M}}_{0a}\right)=n^{1-r}{a}_{r,r-1}^K+n^{-r}{a}_{rr}^K\approx n^{1-r}{a}_{r,r-1}+n^{-r}{a}_{rr}\end{array}$$

(5.4)

so that (2.3)–(2.5) hold for $Y_n={(n/a_{21})}^{1/2}({\widehat{M}}_{0a}-M_{0a})$. We now give $K(.)$ needed for (4.21)–(4.24), that is, $K(0,T_2,\dots ,T_r)$ of (5.3) for $2\le r\le 4,$ in terms of the cross-cumulants of the process, $k(.)$ of (4.2). By (A2), since $\mu =0,A_2=A_4=0$, $K(0,T)$ needed by (4.21) and (4.23) for $a_{21}$ and $a_{22}$, is

$$\begin{array}{cc}& K(0,T)=\kappa (X_0{X}_a,X_T{X}_{T+a})=A_1+A_3\\ & where{A}_1=k(0,a,T,T+a),A_3=k{(0,T)}^2+k(0,T+a)k(a,T).\end{array}$$

(5.5)

$a_{32}$ of (4.22) needs

$$\begin{array}{cc}& K(0,T_2,T_3)=\kappa (X_0{X}_a,X_{T_2}{X}_{T_2+a},X_{T_3}{X}_{T_3+a})=\sum _{j=1}^{11}{C}_j\end{array}$$

(5.6)

is given by identifying $(0,a,T_2,T_2+a,T_3,T_3+a)$ with $(1,\dots ,6)$ in (A13). So $C_j=0$ for $j=2,4,7,8,9,11,$ $C_1=k(0,a,T_2,T_2+a,T_3,T_3+a)$,

$$\begin{array}{cc}& C_3=\sum _{23}^2[k(0,T_2)k(0,T_2,T_3,T_3-a)+k(0,T_2+a)k(0,T_2-a,T_3,T_3-a)\hfill \\ & +k(0,T_2-a)k(0,T_2+a,T_3,T_3+a)+k(0,T_2)k(0,T_2,T_3,T_3+a)\hfill \\ & +k(0,T_3-T_2+a)k(0,a,T_2+a,T_3)]+k(0,T_3-T_2)[k(0,a,T_2,T_3)\hfill \\ & +k(0,-a,T_2,T_3)]by\left(A14\right).\hfill \end{array}$$

$$\begin{array}{cc}& C_5=\sum _{23}^2[k(0,a,T_2)k(0,a,T_2-T_3+a)+k(0,a,T_2+a)k(0,a,T_2-T_3)\hfill \\ & +k(0,T_2,T_2+a)k(0,T_3,T_3-a)]by\left(A15\right).\hfill \\ & C_6=k{(0,T_2,T_3)}^2+k(0,T_2+a,T_3+a)k(0,T_2-a,T_3-a)\hfill \\ & +\sum _{23}^{2}k(0,T_2,T_3+a)k(0,T_2,T_3-a)by\left(A16\right).\hfill \\ & By\left(A17\right),C_{10}=k(T_2,T_3)\sum _{23}^{2}k(0,T_2)[k(a,T_3)+k(0,T_3+a)]\hfill \\ & +k(0,T_2)k(0,T_3)\sum _{23}^{2}k(T_2,T_3+a)+\sum _{23}^{2}k(0,T_2+a)k(0,T_3-a)k(T_2,T_3+a).\hfill \end{array}$$

This completes $K(0,T_2,T_3)$ needed for $a_{32}=a_{32}^K$ of (4.22) or its limit. Lastly $a_{43}$ of (4.24) needs

$$\begin{array}{cc}& K(0,T_2,T_3,T_4)=\kappa (X_0{X}_a,X_{T_2}{X}_{T_2+a},X_{T_3}{X}_{T_3+a},X_{T_4}{X}_{T_4+a}).\end{array}$$

(5.7)

This is given by identifying $(0,a,T_2,T_2+a,T_3,T_3+a,T_4,T_4+a)$ with $(1,\dots ,8)$ in (A23), starting with $D_1=\kappa (X_0,X_a,X_{T_2},X_{T_2+a},X_{T_3},X_{T_3+a},X_{T_4},X_{T_4+a}).$

Example 5.3. 

The eECF expansions to $O\left(n^{-3/2}\right)$ for the sample autocovariancewithoutassuming that $\mu =0$.

In this case we take $p=2$ in (3.1), $a\ge 0$,

$$w_1=\mu ,w_2=M_{0a},\theta =t\left(w\right)=\kappa (X_0,X_a)=w_2-w_1^2.$$

The non-zero derivatives are $t_{\cdot 1}=-2\mu ,t_{\cdot 2}=1,t_{\cdot 11}=-2.$ For $i=r-1$ and r, $k_i^{1^r}=a_{ri}^k$ of Example 5.1, and $k_i^{2^r}=a_{ri}^K$ of Example 5.2. So,

$$\begin{array}{cc}& (k_1^{11},k_2^{11},k_2^{111},k_3^{1111})=(a_{21}^k,a_{22}^k,a_{32}^k,a_{43}^k)of\left(4.18\right),\hfill \\ & and(k_1^{22},k_2^{22},k_2^{222},k_3^{2222})=(a_{21}^K,a_{22}^K,a_{32}^K,a_{43}^K)of\left(5.4\right)-\left(5.7\right).\hfill \end{array}$$

By (3.15) and (3.16),

$$\begin{array}{cc}& a_{21}=4{\mu }^2{k}_1^{11}-4\mu k_1^{12}+k_1^{22},a_{11}=-k_1^{11},\end{array}$$

(5.8)

$$\begin{array}{cc}& a_{32}=-8{\mu }^3{k}_2^{111}+12{\mu }^2{k}_2^{112}-6\mu k_2^{122}+k_2^{222}-2s_1^2,s_j=-2\mu k_1^{1j}+k_1^{2j}.\end{array}$$

(5.9)

By (4.27) with $r=2,N_1=n,N_2=N=n-a,$

$$\begin{array}{cc}& \kappa (\widehat{\mu },{\widehat{M}}_{0a})={\left(nN\right)}^{-1}(n{a}_{21}^K+a_{22}^K)=\sum _{j=1}^2{n}^{-j}{k}_j^{12}where\end{array}$$

$$\begin{array}{cc}& k_j^{12}=(n/N)a_{2j}^K,a_{21}^K=\sum _{T=1-n}^{N-1}K(0,T),a_{22}^K=-\sum _{T=1-n}^{N-1}{\delta }_{2T}K(0,T),\\ & {\delta }_{2T}=max(0,T+a)-min(0,T),and\end{array}$$

(5.10)

$$\begin{array}{cc}& K(0,T)=\kappa (X_0,X_T{X}_{T+a})=k(0,T,T+a)+\mu [k(0,T+a)+k(0,T)],\end{array}$$

(5.11)

by (A1). This gives $a_{21}$ of (5.8). By (3.15), $a_{11}=-k_1^{11}.$

$a_{32}$ of (5.9) needs $k_2^{112}$ and $k_2^{122}$.

By (4.27) with $r=3,N_1=N_2=n,N_3=N=n-a,$

$$\begin{array}{cc}& \kappa (\widehat{\mu },\widehat{\mu },{\widehat{M}}_{0a})={\left(n^{2}N\right)}^{-1}(n{a}_{32}^{K_1}+a_{33}^{K_1})=\sum _{j=2}^3{n}^{-j}{k}_j^{112}where\\ & k_j^{112}=(n/N)a_{3j}^{K_1},{\delta }_{3\mathbf{T}}=max(0,T_2,T_3+a)-m_{3\mathbf{T}},andby\left(A3\right),\\ & K_1(0,T_2,T_3)=\kappa (X_0,X_{T_2},X_{T_3}{X}_{T_3+a})=k(0,T_2,T_3,T_3+a)+\mu k(0,T_2,T_3+a)\\ & +\mu k(0,T_2,T_3)+k(0,T_3)k(T_2,T_3+a)+k(0,T_3+a)k(T_2,T_3).\end{array}$$

(5.12)

By (4.27) with $r=3,N_1=n,N_2=N_3=N=n-a,$

$$\begin{array}{cc}& \kappa (\widehat{\mu },{\widehat{M}}_{0a},{\widehat{M}}_{0a})={\left(n{N}^2\right)}^{-1}(n{a}_{32}^{K_2}+a_{33}^{K_2})=\sum _{j=2}^3{n}^{-j}{k}_j^{122}where\\ & k_j^{122}=(n/N)a_{3j}^{K_2},and{\delta }_{3\mathbf{T}}=max(a,T_2,T_3)-min(0,T_2,T_3),\end{array}$$

(5.13)

$$\begin{array}{cc}& and by\left(A6\right),K_2(0,T_2,T_3)=\kappa (X_0,X_{T_2}{X}_{T_2+a},X_{T_3}{X}_{T_3+a})=\sum _{j=1}^6{B}_{j}where\\ & B_1=k(T_2,T_2+a,T_3,T_3+a,0),B_2/\mu =k(0,T_2,T_2+a,T_3)\\ & +k(0,T_2,T_2+a,T_3+a)+k(0,T_2,T_3,T_3+a)+k(0,T_2+a,T_3,T_3+a),\\ & B_3=k(T_2,T_2+a,T_3)k(0,T_3+a)+k(T_2,T_2+a,T_3+a)k(0,T_3)\\ & +k(T_2,T_3,T_3+a)k(0,T_2+a)+k(T_2+a,T_3,T_3+a)k(0,T_2),\\ & B_4+B_5=k(0,T_2,T_3)(k(T_2,T_3)+{\mu }^2)+k(0,T_2,T_3+a)(k(T_2+a,T_3)+{\mu }^2)\\ & +k(0,T_2+a,T_3)(k(T_2,T_3+a)+{\mu }^2)+k(0,T_2+a,T_3+a)(k(T_2,T_3)+{\mu }^2),\end{array}$$

$$\begin{array}{cc}& B_6/\mu =k(T_2,T_3)[k(0,T_2+a)+k(0,T_3+a)]+k(T_2,T_3+a)[k(0,T_2+a)\\ & +k(0,T_3)]+k(0,T_2)[k(T_2+a,T_3)+k(T_2,T_3)]+k(T_2+a,T_3)k(0,T_3+a)\\ & +k(T_2,T_3)k(0,T_3).\end{array}$$

This completes $a_{32}$ of (5.9), giving $h_1,{\overline{h}}_1,f_1,g_1$ of (2.3)–(2.5).

Next we give $a_{22}$. By (3.17),

$$\begin{array}{cc}& a_{22}=\sum (a_{22j}:j=1,2,3,6)where\hfill \\ & a_{221}=4{\mu }^2{k}_2^{11}-4\mu k_2^{12}+k_2^{22},a_{222}=4\mu k_2^{11},a_{223}=-2k_2^{112},a_{226}=2{\left(k_1^{11}\right)}^2.\hfill \end{array}$$

Finally we give $a_{43}$. By (3.18), $a_{43}$ is the sum of

$$\begin{array}{cc}& a_{431}=16{\mu }^4{k}_3^{1111}-32{\mu }^3{k}_3^{1112}+24{\mu }^2{k}_3^{1122}-8\mu k_3^{1222}+k_3^{2222},\hfill \\ & a_{432}=-24s_2(4{\mu }^2{k}_2^{111}-4\mu k_3^{112}+k_2^{122},s_j=-2\mu k_1^{j1}+k_1^{j2},u_1=-2s_2,u_2=0,\hfill \\ & a_{433}=0,a_{434}=12u_1^2{k}_1^{11}=4s_2^2{k}_1^{11}.\hfill \end{array}$$

So we need $k_3^{ijkl}$. Set

$$\begin{array}{cc}& K_{31}=\kappa (\widehat{\mu },\widehat{\mu },\widehat{\mu },{\widehat{M}}_{0a}),K_{22}=\kappa (\widehat{\mu },\widehat{\mu },{\widehat{M}}_{0a},{\widehat{M}}_{0a}),K_{13}=\kappa (\widehat{\mu },{\widehat{M}}_{0a},{\widehat{M}}_{0a},{\widehat{M}}_{0a}).\hfill \end{array}$$

By (4.27) with $r=3,N_1=N_2=N_3=n,N_4=N=n-a,$

$$\begin{array}{cc}& K_{31}=n^{-3}{N}^{-1}(n{a}_{43}^K+a_{44}^K)=\sum _{j=3}^4{n}^{-j}{k}_j^{1112}where{k}_j^{1112}=(n/N)a_{4j}^K,\hfill \\ & {\delta }_{4\mathbf{T}}=max(0,T_2,T_3,T_4+a)-m_{4\mathbf{T}}for{m}_{4\mathbf{T}}of\left(4.14\right).By\left(A10\right),\hfill \\ & K(0,T_2,T_3,T_4)=\kappa (X_0,X_{T_2},X_{T_3},X_{T_4}{X}_{T_4+a})=\sum _{j=1}^3{E}_{j}where\hfill \end{array}$$

$$\begin{array}{cc}& E_1=k(0,T_2,T_3,T_4,T_4+a),E_2/\mu =k(0,T_2,T_3,T_4+a)+k(0,T_2,T_3,T_4),\hfill \\ & E_3=k(0,T_4)k(T_2,T_3,T_4+a)+k(T_2,T_4)k(0,T_3,T_4+a)+k(0,a)k(0,T_2,T_3)\hfill \\ & +k(0,T_4+a)k(T_2,T_3,T_4)+k(T_2,T_4+a)k(0,T_3,T_4)+k(T_3,T_4+a)k(0,T_2,T_4).\hfill \end{array}$$

By (4.27) with $r=3,N_1=N_2=n,N_3=N_4=N=n-a,$

$$\begin{array}{cc}& K_{22}={\left(nN\right)}^{-2}(n{a}_{43}^K+a_{44}^K)=\sum _{j=3}^4{n}^{-j}{k}_j^{1122}where{k}_j^{1112}={(n/N)}^2{a}_{4j}^K,\hfill \\ & {\delta }_{4\mathbf{T}}=max(0,T_2,T_3+a,T_4+a)-m_{4\mathbf{T}},\hfill \end{array}$$

and by (A20),

$$\begin{array}{cc}& K(0,T_2,T_3,T_4)=\sum _{j=1}^{10}{F}_{j}where for example{F}_1=k(0,T_2,T_3,T_3+a,T_4,T_4+a),\hfill \\ & F_2/\mu =\sum _{34}^2[k(0,T_2,T_3+a,T_4,T_4+a)+k(0,T_2,T_3,T_4,T_4+a)].\hfill \end{array}$$

By (4.27) with $r=3,N_1=n,N_2=N_3=N_4=N=n-a,$

$$\begin{array}{cc}& K_{13}={\left(n{N}^3\right)}^{-1}(n{a}_{43}^K+a_{44}^K)=\sum _{j=3}^4{n}^{-j}{k}_j^{1222}where{k}_j^{1122}={(n/N)}^3{a}_{4j}^K,\hfill \\ & {\delta }_{4\mathbf{T}}=max(0,T_2+a,T_3+a,T_4+a)-m_{4\mathbf{T}},\hfill \end{array}$$

and by (A21),

$$\begin{array}{cc}& K(0,T_2,T_3,T_4)=\sum _{j=1}^9{G}_{j}where for example{G}_1=k(0,T_2,T_3,T_3+a,T_4,T_4+a),\hfill \\ & G_2/\mu =\sum _{234}^3[k(0,T_2)k(T_2+a,T_3,T_3+a,T_4,T_4+a)\hfill \\ & +k(0,T_2+a)k(T_2,T_3,T_3+a,T_4,T_4+a)].\hfill \end{array}$$

This completes $a_{43}$, giving $h_2,{\overline{h}}_2,f_3,g_3$ of (2.3)–(2.5).

Example 5.4. 

The eECF expansions to $O\left(n^{-1}\right)$ for the ath sample autocorrelationassuming that $\mu =0$ .

Take $w_1=M_{00},w_2=M_{0a}$ at $a_1=0,a_2=a$. Then $\theta =t\left(w\right)=w_2/w_1$ is the ath autocorrelation and $\widehat{\theta }={\widehat{M}}_{0a}/{\widehat{M}}_{00}$ is the ath sample autocorrelation. So ${\widehat{M}}_{00}=\overline{X^2},$ $p=2$, $a_{21},a_{11},a_{32}$ are given by (3.15) and (3.16), and

$$\begin{array}{cc}& t_{\cdot 1}=-w_2/w_1^2,t_{\cdot 2}=1/w_1,t_{\cdot 11}=2w_2/w_1^3,t_{\cdot 12}=-1/w_1^2,t_{\cdot 22}=0.\hfill \\ & So,a_{21}=(t_1^2{k}_1^{11}-2t{k}_1^{12}+k_1^{22})w_1^{-2},a_{11}=(\theta k_1^{11}-k_1^{12}),w_1^{-2},\hfill \\ & a_{32}=(-{\theta }^3{k}_2^{111}+k_2^{222}+3t^2{k}_2^{112}-3t{k}_2^{122}+6w_2{s}_1^2-6w_1{s}_1{s}_2)w_1^{-3},\hfill \\ \hfill & s_1=(-\theta k_1^{11}+k_1^{12})w_1^{-1},s_2=(-\theta k_1^{21}+k_1^{22})w_1^{-1}.\hfill \end{array}$$

These need $k_1^{ij}$ and $k_2^{ijk}$. $k_i^{2^r}$ is $a_{ri}^K$ of Example 5.2. So,

$$\begin{array}{cc}& k_1^{22}=\sum _{\left|T\right|<N}K(0,T)of\left(5.5\right),k_2^{222}=\sum _{|T_i|<N,i=2,3}K(0,T_2,T_3)of\left(5.6\right).\hfill \end{array}$$

$k_i^{1^r}$ is $k_i^{2^r}$ at $a=0$. So,

$$\begin{array}{cc}& k_1^{11}=\sum _{\left|T\right|<N}K(0,T)whereK(0,T)=k(0,0,T,T)+2k{(0,T)}^2,\hfill \\ & k_2^{111}=\sum _{|T_i|<N,i=2,3}K(0,T_2,T_3)whereK(0,T_2,T_3)=\sum _{j=1,3,5,6,10}{C}_j,\hfill \\ & C_1=k(0,0,T_2,T_2,T_3,T_3),\hfill \\ & C_3=4\sum _{23}^2[k(0,T_2)k(0,T_2,T_3,T_3)+k(0,T_3-T_2)k(0,0,T_2,T_3)].\hfill \\ & C_5=\sum _{23}^2[2k(0,0,T_2)k(0,0,T_2-T_3)+k(0,T_2,T_2)k(0,T_3,T_3)].\hfill \\ & C_6=k{(0,T_2,T_3)}^2+3k{(0,T_2,T_3)}^2.C_{10}=8k(T_2,T_3)k(0,T_2)k(0,T_3).\hfill \end{array}$$

This completes $K(0,T_2,T_3)$ needed for $a_{32}=a_{32}^K$ or its limit. We now give $k_1^{12},k_2^{112},k_2^{122}.$ By (4.27) with $r=2,N_1=n,N_2=N=n-a,$

$$\begin{array}{cc}& \kappa ({\widehat{M}}_{00},{\widehat{M}}_{0a})={\left(nN\right)}^{-1}(n{a}_{21}^K+a_{22}^K)=\sum _{j=1}^2{n}^{-j}{k}_j^{12}where\hfill \\ & k_j^{12}=(n/N)a_{2j}^K,K(0,T)=\sum _{j=1}^4{A}_{j}by\left(A2\right)where\hfill \\ & A_1=k(0,0,T,T+a),A_2=A_4=0,A_3=2k(0,T)k(0,T+a)\hfill \end{array}$$

By (4.27) with $r=3,N_1=N_2=n,N_3=N=n-a,$

$$\begin{array}{cc}& \kappa ({\widehat{M}}_{00},{\widehat{M}}_{00},{\widehat{M}}_{0a})={\left(n^{2}N\right)}^{-1}(n{a}_{32}^K+a_{33}^K)=\sum _{j=2}^3{n}^{-j}{k}_j^{112}where{k}_j^{112}=(n/N)a_{3j}^K,\hfill \\ & andK(0,T_2,T_3)=\kappa (X_0^2,X_{T_2}^2,X_{T_3}{X}_{T_3+a})=\sum _{j=1}^{11}{C}_j\hfill \end{array}$$

of Example 5.5. By (4.27) with $r=3,N_1=n,N_2=N_3=N=n-a,$

$$\begin{array}{cc}& \kappa ({\widehat{M}}_{00},{\widehat{M}}_{0a},{\widehat{M}}_{0a})={\left(n{N}^2\right)}^{-1}(n{a}_{32}^K+a_{33}^K)=\sum _{j=2}^3{n}^{-j}{k}_j^{122}where{k}_j^{122}=(n/N)a_{3j}^K,\hfill \\ & andK(0,T_2,T_3)=\kappa (X_0^2,X_{T_2}{X}_{T_2+a},X_{T_3}{X}_{T_3+a})=\sum _{j=1}^{11}{C}_{j}of\left(A13\right)\hfill \end{array}$$

with $\left(123456\right)$ identified with $(0,0,T_2,T_2+a,T_3,T_3+a)$. Similarly $a_{22},a_{43}$ needed for the eECF expansions to $O\left(n^{-3/2}\right)$, can be obtained from (3.17) and (3.18).

Example 5.5. 

The eECF expansions to $O\left(n^{-1}\right)$ for the ath sample autocorrelationwithoutassuming that $\mu =0$. Take

$$\begin{array}{cc}& w_1=\mu ,w_2=M_{00},w_3=M_{0a},\\ & {\widehat{w}}_1=\overline{X},{\widehat{w}}_2=\overline{X^2},{\widehat{w}}_3={\widehat{M}}_{0a}of\left(4.8\right),\\ & D=var\left(X_0\right)=w_2-w_1^2,E=covar(X_0,X_a)=w_3-w_1^2,\\ & \theta =t\left(w\right)=covar(X_0,X_a)/var\left(X_0\right)=E/D.\end{array}$$

(5.14)

So $p=3$ and $a_{21},a_{11},a_{32}$ are given by (3.19) with

$$\begin{array}{cc}& t_{\cdot 1}=2\mu (\theta -1)/D,t_{\cdot 2}=D^{-1},t_{\cdot 3}=-\theta D^{-1},t_{\cdot 11}=2(\theta -1)w_3{D}^{-2},\hfill \\ & t_{\cdot 12}=2w_1{D}^{-2},t_{\cdot 13}=2w_1(1-2\theta )D^{-2},t_{\cdot 22}=0,t_{\cdot 23}=-D^{-2},t_{\cdot 33}=2\theta D^{-2}.\hfill \end{array}$$

$k_1^{11}$ and $k_2^{111}$ are $a_{21}^k$ and $a_{32}^k$ of (5.1). $k_1^{33}$ and $k_2^{333}$ are $a_{21}^k$ and $a_{32}^k$ of (5.2). $k_1^{22}$ and $k_1^{22}$ are just $k_1^{33}$ and $k_2^{333}$ with $a=0$. $k_1^{13}$ is $k_1^{12}$ of (5.10), $k_1^{12}$ is $k_1^{13}$ with $a=0$. $k_2^{113}$ is $k_2^{112}$ of (5.12). $k_2^{112}$ is $k_2^{113}$ with $a=0$. $k_2^{133}$ is $k_2^{122}$ of (5.13). $k_2^{122}$ is $k_2^{133}$ with $a=0$. $k_1^{23}$ is $k_1^{0a}$ of Example 4.6 of Withers and Nadarajah (2012). (This corrects the derivatives of t given in Example 4.5 there.) $a_{32}$ also needs $k_2^{123},k_2^{223},k_2^{233}.$ Set $N=n-a$. By (4.27) with $r=3,N_1=N_2=n,N_3=N=n-a,$

$$\begin{array}{cc}& \kappa ({\widehat{w}}_1,{\widehat{w}}_2,{\widehat{w}}_3)={\left(n^{2}N\right)}^{-1}(n{a}_{32}^K+a_{33}^K)=\sum _{j=2}^3{n}^{-j}{k}_j^{112}where\hfill \\ & k_j^{123}=(n/N)a_{3j}^K,andK(0,T_2,T_3)=\kappa (X_0,X_{T_2}^2,X_{T_3}{X}_{T_3+a}).\hfill \end{array}$$

Identifying $(12,34,5)$ of (A6) with $(T_2,T_2,T_3,T_3+a,0)$ gives

$$\begin{array}{cc}& K(0,T_2,T_3)=\sum _{j=1}^6{B}_{j}where{B}_1=k(0,T_2,T_2,T_3,T_3+a),\hfill \\ & B_2=\mu [k(0,T_2,T_2,T_3)+k(0,T_2,T_2,T_3+a)+2k(0,T_2,T_3,T_3+a)],\hfill \\ & B_3=k(T_2,T_2,T_3)k(0,T_3+a)+k(T_2,T_2,T_3+a)k(0,T_3)+2k(T_2,T_3,T_3+a)k(0,T_2),\hfill \\ & B_4=2k(T_2,T_3)k(0,T_2,T_3+a)+2k(T_2,T_3+a)k(0,T_2,T_3),\hfill \\ & B_5=2{\mu }^2[k(0,0,T_3+a)+k(T_2,T_3,T_3+a)],\hfill \\ & B_6=2\mu k(0,T_2)[k(T_2,T_3)+k(T_2,T_3+a)]+\mu k(0,T_3)[k(0,0)+k(T_2,T_3+a)]\hfill \\ & +\mu k(0,T_3+a)[k(0,0)+k(T_2,T_3)].\hfill \end{array}$$

This gives $k_2^{123}$. Next we give $k_2^{223}$. By (4.27) with $r=3,N_1=N_2=n,N_3=N=n-a,$

$$\begin{array}{cc}& \kappa ({\widehat{w}}_2,{\widehat{w}}_2,{\widehat{w}}_3)={\left(n^{2}N\right)}^{-1}(n{a}_{32}^K+a_{33}^K)=\sum _{j=2}^3{n}^{-j}{k}_j^{223}where{k}_j^{223}=(n/N)a_{3j}^K,\hfill \\ & andK(0,T_2,T_3)=\kappa (X_0^2,X_{T_2}^2,X_{T_3}{X}_{T_3+a}).\hfill \end{array}$$

Identifying $(12,34,56)$ of (A13) with $(0,0,T_2,T_2,T_3,T_3+a)$ gives

$$\begin{array}{cc}& K(0,T_2,T_3)=\kappa (X_0^2,X_{T_2}^2,X_{T_3}{X}_{T_3+a})=\sum _{j=1}^{11}{C}_j,C_1=k(0,0,T_2,T_2,T_3,T_3+a),\hfill \\ & C_2=\mu [k(0,0,T_2,T_2,T_3)+k(0,0,T_2,T_2,T_3+a)+2k(0,0,T_2,T_3,T_3+a)\hfill \\ & +2k(0,T_2,T_2,T_3,T_3+a)],\hfill \\ & C_3=2k(0,T_2)k(0,T_2,T_3,T_3+a)+2k(0,T_3)k(0,T_2,T_2,T_3+a)\hfill \\ & +2k(0,T_3+a)k(0,T_2,T_2,T_3)+2k(0,T_2)k(0,T_2,T_3,T_3+a)\hfill \\ & +2k(T_2,T_3)k(0,0,T_2,T_3+a)+2k(T_2,T_3+a)k(0,0,T_2,T_3),\hfill \end{array}$$

$$\begin{array}{cc}& C_4={\mu }^2[2k(0,T_2,T_3,T_3+a)+k(0,T_2,T_2,T_3+a)+k(0,T_2,T_2,T_3)\hfill \\ & +k(0,0,T_2,T_3+a)+k(0,0,T_2,T_3),\hfill \\ & C_5=2\sum _{23}^{2}k(0,0,T_2)k(T_2,T_3,T_3+a)+k(0,0,T_3)k(T_2,T_2,T_3+a)\hfill \\ & +k(0,0,T_3+a)k(T_2,T_2,T_3),\hfill \\ & C_6=4k(0,T_2,T_3)k(0,T_2,T_3+a),\hfill \\ & C_7=\mu [2k(T_2,T_3)k(0,0,T_2)+2k(0,T_3)k(0,T_2,T_2)+2k(T_2,T_3+a)k(0,0,T_2)\hfill \\ & +k(0,T_3+a)(2k(0,T_2,T_2)+k(0,T_2,T_3))+k(T_3,T_3+a)k(0,0,T_2)\hfill \\ & +k(T_2,T_3+a)(k(0,0,T_3)+k(0,T_2,T_3))].\hfill \\ & C_8/4\mu =k(0,T_2,T_3)[k(0,T_2)+k(0,T_3+a)+k(T_2,T_3+a)]\hfill \\ & +k(0,T_2,T_3+a)[k(0,T_2)+k(0,T_3)+k(T_2,T_3)],\hfill \\ & C_9=4{\mu }^3[k(0,T_2,T_3)+k(0,T_2,T_3+a)],\hfill \\ & C_{10}=4k(0,T_2)[k(0,T_3)k(T_2,T_3+a)+k(0,T_3+a)k(T_2,T_3)],\hfill \\ & C_{11}/4{\mu }^2=k(0,T_2)[k(0,T_3)+k(0,T_3+a)+k(T_2,T_3)+k(T_2,T_3+a)]\hfill \\ & +k(0,T_3)k(T_2,T_3+a)+k(0,T_3+a)k(T_2,T_3).\hfill \end{array}$$

Finally $a_{32}$ needs $k_2^{233}$. By (4.27) with $r=3,N_1=n,N_2=N_3=N=n-a,$

$$\begin{array}{cc}& \kappa ({\widehat{w}}_2,{\widehat{w}}_3,{\widehat{w}}_3)={\left(n{N}^2\right)}^{-1}(n{a}_{32}^K+a_{33}^K)=\sum _{j=2}^3{n}^{-j}{k}_j^{233}where{k}_j^{233}={(n/N)}^2{a}_{3j}^K,\hfill \\ & andK(0,T_2,T_3)=\kappa (X_0^2,X_{T_2}{X}_{T_2+a},X_{T_3}{X}_{T_3+a}).\hfill \end{array}$$

Identifying $(12,34,56)$ of (A13) with $(0,0,T_2,T_2+a,T_3,T_3+a)$ gives

$$\begin{array}{cc}& K(0,T_2,T_3)=\kappa (X_0^2,X_{T_2}{X}_{T_2+a},X_{T_3}{X}_{T_3+a})=\sum _{j=1}^{11}{C}_j,C_1=k(0,0,T_2,T_2+a,T_3,T_3+a),\hfill \\ & C_2=\mu [k(0,0,T_2,T_2+a,T_3)+k(0,0,T_2,T_2+a,T_3+a)+k(0,0,T_2,T_3,T_3+a)\hfill \\ & +k(0,0,T_2+a,T_3,T_3+a)+2k(0,T_2,T_2+a,T_3,T_3+a)],\hfill \\ & C_3=2k(0,T_2)k(0,T_2+a,T_3,T_3+a)+2k(0,T_2+a)k(0,T_2,T_3,T_3+a)\hfill \\ & +2k(0,T_3)k(0,T_2,T_2+a,T_3+a)+2k(0,T_3+a)k(0,T_2,T_2+a,T_3)\hfill \\ & +k(T_2,T_3)k(0,0,T_2+a,T_3+a)+k(T_2,T_3+a)k(0,0,T_2+a,T_3)\hfill \\ & +k(T_2+a,T_3)k(0,0,T_2,T_3+a)+k(T_2,T_3)k(0,0,T_2,T_3),\hfill \end{array}$$

$$\begin{array}{cc}& C_4/{\mu }^2=2k(0,T_2+a,T_3,T_3+a)+2k(0,T_2,T_3,T_3+a)\hfill \\ & +2k(0,T_2,T_2+a,T_3+a)+2k(0,T_2,T_2+a,T_3)+k(0,0,T_2+a,T_3)\hfill \\ & +k(0,0,T_2+a,T_3)+k(0,0,T_2,T_3+a)+k(0,0,T_2,T_3),\hfill \end{array}$$

and similarly for $C_5,\dots ,C_{11}$.

Example 5.6. 

Fix $a_1\ge 0$ and $a_2\ge 0$. For $i=1,2,$ set $N_i=n-a_i>0.$ We give $a_{21}$ needed for the eECF expansions for $\widehat{\theta }=t\left(\widehat{w}\right)$ when

$$\begin{array}{cc}& \theta =t\left(w\right)=\kappa (X_0{X}_{a_1},X_0{X}_{a_2})=w_3-w_1{w}_2,\hfill \\ & w_1=M_{0a_1},w_2=M_{0a_2},w_3=M_{00a_1{a}_2}=E{X}_0^2{X}_{a_1}{X}_{a_2}.\hfill \\ & So,{\widehat{w}}_3=N_3^{-1}\sum _{j=1}^{N_3}{X}_j^2{X}_{j+a_1}{X}_{j+a_2},where{N}_3=min(N_1,N_2).\hfill \end{array}$$

So $p=3$, and $a_{21},a_{11},a_{32}$ are given by (3.19) with non-zero derivatives

$$\begin{array}{c}\hfill t_{\cdot 1}=-w_2,t_{\cdot 2}=-w_1,t_{\cdot 3}=1,t_{\cdot 12}=-1.\end{array}$$

The eECF expansions for $\widehat{\theta }=t\left(\widehat{w}\right)$ to $O\left(n^{-1/2}\right)$ is given by $a_{21}$. By (3.19), $a_{21}$ needs $k_1^{ij}$. For $i=1,2,(k_1^{ii},k_2^{iii})=(a_{21}^K,a_{32}^K)$ of (5.4) with $a=a_i$. By (4.27) with $r=2$,

$$\begin{array}{cc}& \kappa ({\widehat{w}}_1,{\widehat{w}}_2)={\left(N_1{N}_2\right)}^{-1}(n{a}_{21}^K+a_{22}^K)=\sum _{j=1}^2{n}^{-j}{k}_j^{12}where{k}_j^{12}=(n^2/N_1{N}_2)a_{2j}^K,\hfill \\ & K(0,T)=\kappa (X_0{X}_{a_1},X_T{X}_{T+a_2})=\sum _{j=1}^4{A}_{j}by\left(A2\right)where\hfill \\ & A_1=k(0,a_1,T,T+a_2),A_2/\mu =k(a_1,T,T+a_2)+k(0,T,T+a_2)\hfill \\ & +k(0,a_1,T+a_2)+k(0,a_1,T),\hfill \\ & A_3=k(0,T)k(a_1,T+a_2)+k(0,T+a_2)k(a_1,T),\hfill \\ & A_4/{\mu }^2=k(a_1,T+a_2)+k(a_1,T)+k(0,T+a_2)+k(0,T).\hfill \end{array}$$

By (4.27), for $i=1,2$,

$$\begin{array}{cc}& \kappa ({\widehat{w}}_i,{\widehat{w}}_3)={\left(N_i{N}_3\right)}^{-1}(n{a}_{21}^K+a_{22}^K)=\sum _{j=1}^2{n}^{-j}{k}_j^{i3}where{k}_j^{i3}=(n^2/N_i{N}_3)a_{2j}^K,\hfill \\ & K(0,T)=\kappa (X_0{X}_{a_i},X_T^2{X}_{T+a_1}{X}_{T+a_2})=\sum _{j=1}^{19}{A}_{j}by\left(A11\right)where for example\hfill \end{array}$$

$$\begin{array}{cc}& A_1=k(0,a_i,T,T,T+a_1,T+a_2),\hfill \\ & A_2/\mu =k(0,T,T,T+a_1,T+a_2)+k(a_i,T,T,T+a_1,T+a_2),\hfill \\ & A_3/\mu =k(0,a_i,T,T,T+a_1)+k(0,a_i,T,T,T+a_2)+2k(0,a_i,T,T+a_1,T+a_2).\hfill \end{array}$$

By (4.27),

$$\begin{array}{cc}& {\kappa }_2\left({\widehat{w}}_3\right)=N_3^{-2}(n{a}_{21}^K+a_{22}^K)=\sum _{j=1}^2{n}^{-j}{k}_j^{33}where{k}_j^{33}={(n/N_3)}^2{a}_{2j}^K,\hfill \\ & and by\left(A22\right),K(0,T)=\kappa (X_0^2{X}_{a_1}{X}_{a_2},X_T^2{X}_{T+a_1}{X}_{T+a_2})=\sum _{j=1}^{16}{H}_j\hfill \\ & whereforexample{H}_1=k(0,0,a_1{a}_2,T,T,T+a_1,T+a_2).\hfill \end{array}$$

This gives $k_1^{33}$. $a_{21}$ is now given by (3.19). However $a_{32}$ needs the joint cumulants of length $L=9$, which are not given in McCullagh (1987).

Example 5.7. 

Fix $a_1,a_2$. Let us find $a_{21}$ for $w=M_{0a_1{a}_2},\widehat{w}={\widehat{M}}_{0a_1{a}_2}$ of (4.6):

$$\begin{array}{cc}& \widehat{w}=N^{-1}\sum _{t=1}^N{X}_{t+I_1}\dots X_{t+I_r}whereN=n-I_0,\end{array}$$

(5.15)

and $I_0=max(0,a_1,a_2)-min(0,a_1,a_2).$ By Corollary 4.2, we can take

$$\begin{array}{cc}& a_{21}^K={(n/N)}^2\sum \{K^{*}(0,T):\left|T\right|<N\}where\hfill \\ & K^{*}(0,T)=\kappa (X_0{X}_{a_1}{X}_{a_2},X_T{X}_{T+a_1}{X}_{T+a_2})=\sum _{i=1}^{15}{A}_i\hfill \end{array}$$

is given by identifying $0,a_1,a_2,T,T+a_1,T+a_2$ with $1,\dots ,6$ in (A12). But it has 1+6+6.4+9.5+18.6=178 terms, so software is needed.

Example 5.8. 

Take $p=5$,

$$\begin{array}{cc}& w_1=\mu ,w_2=M_{0a_1},w_3=M_{0a_2},w_4=M_{a_1{a}_2},w_5=M_{0a_1{a}_2},\hfill \\ & \theta =t\left(w\right)=\kappa (X_0,X_{a_1},X_{a_2})=w_5-w_1(w_2+w_3+w_4)+2w_1^3.\hfill \\ & So,t_{\cdot 1}=6w_1^2-w_2-w_3-w_4,t_{\cdot 2}=t_{\cdot 3}=t_{\cdot 4}=-w_1,t_{\cdot 5}=1.\hfill \end{array}$$

$a_{21}$ is given by (3.9) in terms of $k_1^{a_1{a}_2},a_1,a_2=1,\dots ,5$. By Example 5.1, $k_1^{11}=\sum \{k(0,T):|T|<n\}.$ For $a\ge 0$, set $k_1^{22}\left(a\right)=a_{21}$ of (5.8). Then for $i=2,3,k_1^{ii}=k_1^{22}\left(\right|a_i\left|\right),k_1^{44}=k_1^{22}\left(\right|a_1-a_2\left|\right).k_1^{55}=a_{21}$ of Example 5.8. Set $k_1^{12}\left(a\right)=k_1^{12}=(n/N)\sum \{K(0,T):-n<T<N\},$ for $N=n-\left|a\right|$ and $K(0,T)$ of (5.11). Then $k_1^{12}=k_1^{12}\left(\right|a_1\left|\right),k_1^{13}=k_1^{12}\left(\right|a_2\left|\right),k_1^{14}=k_1^{12}\left(\right|a_1-a_2\left|\right).$ By (4.27), for N of (5.15), $k_1^{15}=(n/N)\sum \{K(0,T):-n<T<N\},$ where now $K(0,T)=\kappa (X_0,X_T{X}_{T+a_1}{X}_{T+a_2})=M_{0,T,T+a_1,T+a_2}-\mu M_{0a_1{a}_2}$.

$k_1^{23}=k_1^{12}$ of Example 5.6. $k_1^{24}$ needs $\kappa ({\widehat{M}}_{0a_1},{\widehat{M}}_{0a_2})$, a special case of

$$\begin{array}{cc}& \kappa ({\widehat{M}}_{0a_1},{\widehat{M}}_{a_2{a}_3})=\sum _{j=1}^2{n}^{-j}{k}_j^{24},where\\ & k_j^{24}=(n^2/N_2{N}_4)a_{2j}^K,N_2=n-|a_1|,N_4=n-|a_2-a_3|,\end{array}$$

(5.16)

$$\begin{array}{cc}& K(0,T)=\kappa (X_0{X}_{a_1},X_{T+a_2}{X}_{T+a_3})=\sum _{j=1}^4{A}_j\end{array}$$

(5.17)

by (A2) is given by identifying $(0,a_1,T+a_2,T+a_3)$ with $(1,2,3,4)$. By (4.27),

$$\begin{array}{cc}& \kappa (\widehat{\mu },{\widehat{M}}_{0a_1{a}_1})=(n/N)\sum _{-n<T<N}{K}^{*}(0,T)where\\ & N=n-max(0,a_1,a_2)+min(0,a_1,a_2),K^{*}(0,T)=\kappa (X_0,X_T{X}_{T+a_1}{X}_{T+a_2}),\end{array}$$

(5.18)

given by identifying $0,T,T+a_1,T+a_2$ with $1,2,3,4$ in (A4).

Next we obtain $k_1^{34},k_1^{35},k_1^{45}.$ $k_1^{34}$ is a special case of (5.16). $k_1^{35}$ is covered by $k_1^{25}$ of (5.18). $k_1^{45}$ is a special case of the lead coefficient in $\kappa ({\widehat{M}}_{a_1{a}_2},{\widehat{M}}_{a_3{a}_4{a}_5}).$ In this case, for $N_1=max(a_1,a_2)-min(a_1,a_2),N_2=max(a_3,a_4,a_5)-min(a_3,a_4,a_5)$,

$$\begin{array}{cc}& k_1^{45}=(n^2/N_1{N}_2)\sum _{-N_1<T<N_2}{K}^{*}(0,T)where{K}^{*}(0,T)=\kappa (X_{a_1}{X}_{a_2},X_{T+a_3}{X}_{T+a_4}{X}_{T+a_5}),\hfill \end{array}$$

is given by identifying $a_1,a_2,T+a_3,T+a_4,T+a_5$ with 12345 in (A5). This completes the terms needed for $a_{21}$.

Note how Example 5.5 built on Example 5.1, Example 5.6 built on Example 5.5, and Example 5.9 built on Example 5.7. So the above results form a catalogue upon which other examples can build.

Example 5.1 gave $a_{21}=k_1^{11},a_{32}=k_2^{111},a_{22}=k_2^{11},a_{43}=k_3^{1111}$ for $\widehat{\mu }$.

Example 5.3 gave $a_{21}=k_1^{11},a_{32}=k_2^{111}$ for ${\widehat{M}}_{0a}$.

Example 5.5 gave $k_1^{ij},k_2^{ijk}$ for ${\widehat{w}}_1=\overline{X},{\widehat{w}}_2=\overline{X^2},{\widehat{w}}_3={\widehat{M}}_{0a}$.

Example 5.6 gave $k_1^{ij}$ for ${\widehat{w}}_1={\widehat{M}}_{0a_1},{\widehat{w}}_2={\widehat{M}}_{0a_2},{\widehat{w}}_3={\widehat{M}}_{00a_1{a}_2}$.

Example 5.7 gave $a_{21}=k_1^{11}$ for ${\widehat{w}}_1={\widehat{M}}_{0a_1{a}_2}.$

Example 5.8 gave $a_{21}=k_1^{11}$ for $\kappa (X_0,X_{a_1},X_{a_2})$.

6. Multivariate Stationary Processes

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

$$\begin{array}{cc}& \mu =E{X}_0,{\mu }^j=E{X}_0^j,M_{i_1\dots i_r}^{j_1\dots j_r}=E{X}_{i_1}^{j_1}\dots X_{i_r}^{j_r},\hfill \\ & {\mu }_{i_1\dots i_r}^{j_1\dots j_r}=E(X_{i_1}^{j_1}-{\mu }^{j_1})\dots (X_{i_r}^{j_r}-{\mu }^{j_r}),k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\dots j_r}{i_1,\dots ,i_r}}\right)=\kappa (X_{i_1}^{j_1},\dots ,X_{i_r}^{j_r}).\hfill \end{array}$$

Given a sequence of integers $i_1,\dots ,i_r$, define $i_0,I_k$ as in (4.3). (4.4) becomes

$$\begin{array}{c}\hfill M_{i_1\dots i_r}^{j_1\dots j_r}=M_{I_1\dots I_r}^{j_1\dots j_r},{\mu }_{i_1\dots i_r}^{j_1\dots j_r}={\mu }_{I_1\dots I_r}^{j_1\dots j_r},k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\dots j_r}{i_1\dots i_r}}\right)=k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\dots j_r}{I_1\dots I_r}}\right).\end{array}$$

However in general $k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1{j}_2}{0,I}}\right)\ne k\left({\displaystyle \genfrac{}{}{0pt}{}{j_1{j}_2}{0,-I}}\right)$. An unbiased estimate of $M\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\dots j_r}{i_1\dots i_r}}\right)$ is

$$\begin{array}{c}\hfill {\widehat{M}}_{i_1\dots i_r}^{j_1\dots j_r}=N^{-1}\sum _{t=1}^N{X}_{t+I_1}^{j_1}\dots X_{t+I_r}^{j_r}forN=n-I_0>0.\end{array}$$

We can abbreviate 2 sequences of integers of the same length r, as $\pi =(i_1\dots i_r),\tau =(j_1\dots j_r).$ Given $P\ge 1$ and finite sequences of integers ${\pi }_1,\dots ,{\pi }_P$ and ${\tau }_1,\dots ,{\tau }_P$ not depending on n with ${\tau }_i$ of the same length as ${\pi }_i$,

$$\begin{array}{cc}& set{w}_a=M_{{\pi }_a}^{{\tau }_a},{\widehat{w}}_a={\widehat{M}}_{{\pi }_a}^{{\tau }_a},w=(w_1,\dots ,w_p),\widehat{w}=({\widehat{w}}_1,\dots ,{\widehat{w}}_p).\\ & Then\kappa ({\widehat{w}}_{a_1},\dots ,{\widehat{w}}_{a_r})=\sum _{j=r-1}^r{n}^{-j}{k}_j^{a_1\dots a_r}for{a}_1,\dots ,a_r\in \{1,\dots ,p\}.\end{array}$$

(6.1)

Let $\theta =t\left(w\right):R^p\to R^q$ be a function with finite derivatives (3.2) at $w=E{X}_0$. Then $\widehat{\theta }=t\left(\widehat{w}\right)$ satisfies (3.3), and $Y_n=n^{1/2}(\widehat{w}-w)$ and $Z_n=n^{1/2}(\widehat{\theta }-\theta )$ have multivariate Edgeworth expansions. The results of Section 5 apply with $K(t_1\dots t_r)$ replaced by its extension $K\left({\displaystyle \genfrac{}{}{0pt}{}{j_1\dots j_r}{t_1\dots t_r}}\right)$.

7. Discussion

For the mean of a random sample, the coefficients of the Edgeworth expansions are functions of the cumulants. However when the observations are dependent, the usual unbiased estimate of $var\left(X_0\right)$ fails, so we use empirical estimates. The role of the empirical distribution is eclipsed by having to deal with the cross-cumulants.

Winterbottom (1979, 1980, 1984) showed that Cornish-Fisher expansions gave substantial improvements to the Central Limit Theorem, even for lattice estimates like the binomial and the Poisson. Phillips (1987) gave an extension to nonstationary processes. Bose (1988) obtained an Edgeworth correction by bootstrap in autoregressions. Loh (1996) gave an Edgeworth expansion for U-statistics with dependent observations. Kitamura (1997) used an Edgeworth expansion to study empirical likelihood methods for dependent processes. Lieberman et al. (2001) gave an Edgeworth expansion for the sample autocorrelation function under long range dependence. For a review of a paper by Taniguchi and Kakizawa on Edgeworth and saddlepoint expansions for time series, see Lieberman (2002). Andrews and Lieberman (2005) gave an Edgeworth expansion for maximum likelihood estimator for stationary long-memory Gaussian time series. Lieberman et al. (2003) gave expansions for the maximum likelihood estimator for a stationary Gaussian process.

However until now, there has been no non-parametric theory for the distribution of the general standard estimate based on a sample from a stationary process. Consequently, econometrics, the foremost user of time series, has been dominated by software contructing estimates for parametric autoregressive (AR), moving average (MA), ARMA, and integrated ARMA processes. But parametric models suffer from a severe drawback: if the model is wrong, the results will generally not even give 1st order (Central Limit Theorem) accuracy. This is where a non-parametric method is far superior. Software is still needed when the required $a_{ri}$ of (2.1) have a large numbers of terms.

Future directions.

1. These results can easily be extended to both weighted estimates, and estimates based on missing data, as done in Withers (2025b).

2. Lahiri (2003) gave $a_{21}$ and a Central Limit Theorem for weighted sums of a spatial process. It should not be too hard to extend this to give the other leading $a_{rj}$.

3. Software to implement the results of Section 3 for arbitrary $t\left(w\right)$ would also be very useful.

4. There is a need to extend the results of McCullagh (1987), and to spell out his succinct notation as done in the appendix below. As noted, his results cover $a_{43}$ for ${\kappa }_4\left({\widehat{M}}_{i_1{i}_2}\right)$ but not $a_{32}$ for ${\kappa }_3\left({\widehat{M}}_{i_1{i}_2{i}_3}\right)$.

5. To extend these results to confidence intervals for a general function of cross-cumulants, say $t\left(w\right)$, would be a huge advance. The 1st step is to extend them to Edgeworth expansions for its Studentized form, $\widehat{\theta }=(t\left(\widehat{w}\right)-t\left(w\right))/{\widehat{a}}_{21}^{1/2}$, where ${\widehat{a}}_{21}=\sum \{\widehat{K}(0,T):\left|T\right|<l_n$ is a consistent estimate of $a_{21}$ of (4.21), and one can take $l_n=K{n}^{1/2}$ for some $K>0$. Lahiri (2010) gave Edgeworth expansions for the case $t\left(w\right)=\mu$, so that $K(0,T)=\kappa (X_0,X_T)$ and $\widehat{K}(0,T)={\widehat{M}}_{0T}$ of Example 5.3. The expansions are a superposition of three distinct series, respectively, given by one in powers of $n^{-1/2}$, one in powers of ${(n/l_n)}^{-1/2}$ (resulting from the standard error of the studentizing factor), and one in powers of the bias of ${\widehat{a}}_{21}$. But this is typicallly exponentially small, taking us to the choice $l_n=K{n}^{1/2}$.

6. It may be useful to replace our empirical estimate of the cross-cumulant, by estimates with lower bias. While a jack-knife or bootstrap could be used, analytic results are more easily obtained by adjusting it with an estimate of its bias.