MS 584 Expansions for the Conditional Distribution and Density of a Standard Estimate

1. Introduction and Summary

Suppose that we have a non-lattice estimate $\widehat{w}$ of an unknown parameter $w\in R^q$ of a statistical model, based on a sample of size n. The distribution of a standard estimate is determined by the coefficients obtained by expanding its cumulants in powers of $n^{-1}$. In §2 we summarise the extended Edgeworth-Cornish-Fisher expansions of Withers (1984) for $\widehat{w}$ when $q=1$. Then we give the multivariate Edgeworth expansions to $O\left(n^{-2}\right)$. We show that the distribution of $X_n=n^{1/2}(\widehat{w}-w)$ has the form $Prob.(X_n\le x)\approx {\sum }_{r=0}^{\infty }{n}^{-r/2}{P}_r\left(x\right)$ where $P_0\left(x\right)={\Phi }_V\left(x\right),$ the normal distribution with $V=\left(k_1^{i_1{i}_2}\right)$, and for $r\ge 1,P_r\left(x\right)={\sum }_{k=1}^{3r}[P_{rk}\left(x\right):k-reven]$ where $P_{rk}\left(x\right)$ has $q^k$ terms, reducible using symmetry. Its density has a similarly form. We argue that these expansions may be valid even if $q=q_n\to \infty$ if $q_n/n^{1/6}$ is bounded.

§3 gives these expansions in complete detail when $q=2$.

In §4 we suppose that $q\ge 2$ and partition $X_n$ as $\left(\genfrac{}{}{0pt}{}{{\mathbf{X}}_{n1}}{{\mathbf{X}}_{n2}}\right)$ of dimensions $q_1\ge 1$ and $q_2\ge 1.$ We derive expansions for the conditional density and distribution of ${\mathbf{X}}_{n1}$ given ${\mathbf{X}}_{n2}$ to $O\left(n^{-2}\right)$ . §5 specialises to bivariate estimates.

§6 gives the extended Cornish-Fisher expansions for the quantiles of the conditional distribution when $q_1=1$. An example is the distribution of a sample mean given the sample variance.

2. Extended Edgeworth-Cornish-Fisher theory

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

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

(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 (1) holds in the sense that

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

where $y_n=O\left(x_n\right)$ means that $y_n/x_n$ is bounded in n. Withers (1984) extended Cornish and Fisher (1937) and Fisher and Cornish (1960) to give the distribution and quantiles of

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

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

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

(2)

(3)

(4)

where $\Phi \left(x\right)=Prob(N\le x),N\sim \mathcal{N}(0,1)$ is a unit normal random variable with density $\varphi \left(x\right)={(2\pi )}^{-1/2}{e}^{-x^2/2}$, 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 the standardized cumulant coefficients

(5)

and so on, where $H_k$ is the kth Hermite polynomial,

(6)

See Withers (1984) for $g_r,r\le 6,$, Withers (2000) for (6) and §6 for relations between $h_r,f_r,g_r$. Also,

$$\begin{array}{cc}& ln[p_n\left(x\right)/\varphi \left(x\right)]\approx \sum _{r=1}^{\infty }{n}^{-r/2}{b}_r\left(x\right)where{b}_1\left(x\right)=h_1\left(x\right),\hfill \\ & b_2\left(x\right)=-A_{11}^2/2+(A_{22}-A_{32}{A}_{11})H_2/2-A_{32}^2(3x^4-12x^2+5)/24+A_{43}{H}_4/24.\hfill \end{array}$$

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 $3r$.

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 other $a_{ri}=0$. An explicit formula for its general term was given in Withers and Nadarajah (2009) using Bell polynomials.

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

$$\begin{array}{cc}& fors\ge 0,S^s=\sum _{r=s}^{\infty }{z}^r{\tilde{B}}_{rs}\left(e\right)whereS=\sum _{r=1}^{\infty }{z}^r{e}_r,z\in R.\hfill \end{array}$$

(7)

(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

$$\begin{array}{cc}& e^S=\sum _{r=0}^{\infty }{z}^r{\tilde{B}}_r\left(e\right).So{\tilde{B}}_0\left(e\right)=1andforr\ge 1,{\tilde{B}}_r\left(e\right)=\sum _{s=1}^r{\tilde{B}}_{rs}\left(e\right)/s!:\hfill \end{array}$$

(9)

(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

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

(11)

where the cumulant coefficients ${\overline{k}}_d^{1-r}=k_d^{i_1-i_r}$ may depend on n but are bounded as $n\to \infty$. So the bar replaces $i_j$ by j: ${\overline{k}}_0^1=w^{i_1},{\overline{k}}_1^{12}=k_1^{i_1{i}_2}.$

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

(12)

with density and distribution

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

V may depend on n, but we assume that $det\left(V\right)$ is bounded away from 0. Set

$$\begin{array}{cc}& {\overline{t}}_r=t_{i_r},{\overline{b}}_{2d}^{1-r}={\overline{k}}_d^{1-r},{\overline{b}}_{2d+1}^{1-r}=0,e_j\left(t\right)=\sum _{r=1}^{j+2}{\overline{b}}_{r+j}^{1-r}{\overline{t}}_1\cdots {\overline{t}}_r/r!.\hfill \\ & So{e}_1={\overline{k}}_1^1{\overline{t}}_1+{\overline{k}}_2^{1-3}{\overline{t}}_1{\overline{t}}_2{\overline{t}}_3/6,e_2={\overline{k}}_2^{12}{\overline{t}}_1{\overline{t}}_2/2+{\overline{k}}_3^{1-4}{\overline{t}}_1\cdots {\overline{t}}_4/24.\hfill \end{array}$$

(13)

(14)

where for $r\le 3,{\overline{P}}_r^{1-k}$ is a function of ${\overline{k}}_e^{1-r}$ given for the 1st time in the appendix. In (13), (14) and below, we use the tensor summation convention of implicitly summing $i_1,\cdots ,i_r$ over their range $1,\cdots ,q$. We make ${\overline{P}}_r^{1-k}$ symmetric in $i_1,\cdots ,i_k$ using the operator $\mathcal{S}$ that symmetrizes over $i_1,\cdots ,i_k$:

$$\begin{array}{cc}& {\overline{P}}_1^1={\overline{k}}_1^1,{\overline{P}}_1^{1-3}={\overline{k}}_2^{1-3}/6,{\overline{P}}_2^{12}={\overline{k}}_2^{12}/2+k_1^1{\overline{k}}_1^2/2,\hfill \end{array}$$

(15)

(16)

$$\begin{array}{cc}& {\overline{P}}_3^{1-5}={\overline{k}}_4^{1-5}/120+\mathcal{S}{\overline{k}}_3^{1-4}{\overline{k}}_1^5/24+\mathcal{S}{\overline{k}}_2^{12}{\overline{k}}_2^{3-5}/12+\mathcal{S}{\overline{k}}_2^{123}{\overline{k}}_1^4{\overline{k}}_1^5/12,\hfill \\ & {\overline{P}}_3^{1-7}=\mathcal{S}{\overline{k}}_2^{123}{\overline{k}}_3^{4-7}/144+\mathcal{S}{\overline{k}}_2^{123}{\overline{k}}_2^{4-6}{\overline{k}}_1^7/72,\hfill \\ & {\overline{P}}_3^{1-9}=\mathcal{S}{\overline{k}}_2^{1-3}{\overline{k}}_2^{4-6}{\overline{k}}_2^{7-9}/6^3.\hfill \end{array}$$

(17)

The terms involving $\mathcal{S}$ are given in the Appendix A. By Withers and Nadarajah (2010b) or Withers (2024), $X_n$ has distribution and density

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

(18)

(19)

(20)

(21)

(22)

(23)

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

$$\begin{array}{cc}& {\overline{H}}^{1-k}=E{\Pi }_{j=1}^k({\overline{y}}_j+i{\overline{Y}}_j)where{\overline{y}}_j=y_{i_j},{\overline{Y}}_j=Y_{i_j},y=V^{-1}x,\hfill \\ & Y\sim {\mathcal{N}}_p(0,V^{-1}).So,H^1=y_1,{\overline{H}}^1={\overline{y}}_1,H^{12}=y_1{y}_2-V^{12},{\overline{H}}^{12}={\overline{y}}_1{\overline{y}}_2-{\overline{V}}^{12},\hfill \\ & H^{1-3}=y_1{y}_2{y}_3-\sum ^3{V}^{12}{y}_3,\sum ^3{V}^{12}{y}_3=V^{12}{y}_3+V^{13}{y}_2+V^{23}{y}_1,\hfill \\ & H^{1-4}=y_1\cdots y_4-\sum ^6{V}^{12}{y}_3{y}_4+\sum ^3{V}^{12}{V}^{34},\hfill \\ & H^{1-5}=y_1\cdots y_5-\sum ^{10}{V}^{12}{y}_3\cdots y_5+\sum ^5{y}_5\sum ^3{V}^{12}{V}^{34},\hfill \\ & H^{1-6}=y_1\cdots y_6-\sum ^{15}{V}^{12}{y}_3\cdots y_6+\sum ^{15}{y}_5{y}_6\sum ^3{V}^{12}{V}^{34}-\sum ^{45}{V}^{12}{V}^{34}{V}^{56},\hfill \end{array}$$

where $V^{i_1{i}_2}$ is the $(i_1,i_2)$ element of $V^{-1},$ and ${\overline{V}}^{j_1{j}_2}$ is the $(i_{j_1},i_{j_2})$ element of $V^{-1}.$ This gives ${\overline{H}}^{1-k}$ in terms of the moments of Y. For example

$$\begin{array}{cc}& P_1\left(x\right)=e_1(-\partial /\partial x)){\Phi }_V\left(x\right)=\sum _{r=1}^3{\overline{b}}_{r+1}^{1-r}{\overline{O}}^{1-r}{\Phi }_V\left(x\right)/r!=\sum _{k=1,3}{P}_{1k}\left(x\right),\hfill \\ & P_{11}\left(x\right)={\overline{k}}_1^1(-{\overline{\partial }}_1){\Phi }_V\left(x\right),P_{13}\left(x\right)={\overline{k}}_2^{1-3}{\overline{O}}^{1-3}{\Phi }_V\left(x\right)/6,\hfill \\ & p_1\left(x\right)={\overline{k}}_1^1(-{\overline{\partial }}_1){\varphi }_V\left(x\right)+{\overline{k}}_2^{1-3}{\overline{O}}^{1-3}{\varphi }_V\left(x\right)/6,\hfill \end{array}$$

(24)

(25)

(26)

This gives the Edgeworth expansion for the distribution of $Y_n$ to $O\left(n^{-2}\right)$. See Withers (2024) for more terms.

For large q, $P_{rk},{\tilde{p}}_{rk}$ of (21) and (23) have $\sim q^{3r}$ terms. So if $q=q_n\to \infty$ and $M_{rn}=ma{x}_k\left|{\overline{P}}_r^{1-k}\right|$, then $n^{-r/2}(P_r\left(x\right),p_r\left(x\right)\sim M_{rn}{\nu }_n^r$ where ${\nu }_n=n^{-1/2}{q}_n^3$. So if for example $M_{rn}$ is bounded, then the Edgeworth series should converge if $q_n/n^{1/6}\to 0.$

The log density can be expanded as

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

(27)

$$\begin{array}{cc}& Soby\left(18\right),{\tilde{p}}_r\left(x\right)={\tilde{B}}_r\left(b\left(x\right)\right)whereb=(b_2,b_2,\cdots ).\hfill \end{array}$$

(28)

See Withers and Nadarajah (2016). Also for $H_k\left(x\right)$ of (6),

$$\begin{array}{cc}& H^{j^k}=E{(y_j+i{Y}_j)}^k={\tau }_j^k{H}_k\left({\tau }_j^{-1}{y}_j\right)where{\tau }_j={\left(V^{jj}\right)}^{1/2}:\hfill \\ & H^j=y_j,H^{j^2}=y_j^2-V^{jj},H^{j^3}=y_j^3-3V^{jj}{y}_j,H^{j^4}=y_j^4-6V^{jj}{y}_j^2+3{\left(V^{jj}\right)}^2.\hfill \end{array}$$

(29)

(30)

Example 1. 

Let $\widehat{w}$ be a sample mean. Then $E\widehat{w}=w$, and only the leading coefficient in (11) are non-zero. So ${\overline{k}}_1^1={\overline{P}}_1^1={\tilde{p}}_{11}=0$. In order needed, the non-zero $P_r^{1-k}$ are

$$\begin{array}{cc}& {\overline{P}}_1^{1-3}={\overline{k}}_2^{1-3}/3!,{\overline{P}}_2^{1-4}={\overline{k}}_3^{1-4}/4!,{\overline{P}}_2^{1-6}of\left(16\right),\hfill \\ & {\overline{P}}_3^{1-5}={\overline{k}}_4^{1-5}/5!,{\overline{P}}_3^{1-7}=\mathcal{S}{\overline{k}}_2^{123}{\overline{k}}_3^{4-7}/144,{\overline{P}}_3^{1-9}of\left(17\right).\hfill \end{array}$$

${\tilde{p}}_{rk}$ and $P_{rk}$ have $q^k$ terms but many are duplicates. We now show how symmetry reduces this to $\sim \left(\genfrac{}{}{0pt}{}{q}{k}\right)$ terms. We use the multinomial coefficient $\left(\genfrac{}{}{0pt}{}{k}{a\cdots b}\right)=k!/a!\cdots b!$. For example $\left(\genfrac{}{}{0pt}{}{3}{111}\right)=6$.

Set $T_r^{i_1\cdots i_k}={\overline{P}}_r^{1-k}{\overline{H}}^{1-k}$ where tensor summation is not used. By (22),

$$\begin{array}{cc}& {\tilde{p}}_{rk}=\sum _{i_1,\cdots ,i_k=1}^q{T}_r^{i_1\cdots i_k}.So,\hfill \\ & {\tilde{p}}_{r1}=\sum _{i_1=1}^q{T}_r^{i_1},{\tilde{p}}_{r2}=\sum _{i_1=1}^q{T}_r^{i_1{i}_1}+2\sum _{i_1>i_2}^{\left(\genfrac{}{}{0pt}{}{q}{2}\right)}{T}_r^{i_1{i}_2},\hfill \\ & {\tilde{p}}_{r3}=\sum _{i_1=1}^q{T}_r^{i_1{i}_1{i}_1}+\left(\genfrac{}{}{0pt}{}{3}{1}\right)\sum _{i_1\ne i_2}^{q(q-1)}{T}_r^{i_1{i}_1{i}_2}+\left(\genfrac{}{}{0pt}{}{3}{111}\right)\sum _{i_1>i_2>i_3}^{\left(\genfrac{}{}{0pt}{}{q}{3}\right)}{T}_r^{i_1{i}_2{i}_3},\hfill \\ & {\tilde{p}}_{r4}=\sum _{i_1=1}^q{T}_r^{i_1{i}_1{i}_1{i}_1}+\left(\genfrac{}{}{0pt}{}{4}{1}\right)\sum _{i_1\ne i_2}^{q(q-1)}{T}_r^{i_1{i}_1{i}_1{i}_2}+\left(\genfrac{}{}{0pt}{}{4}{2}\right)\sum _{i_1>i_2}^{\left(\genfrac{}{}{0pt}{}{q}{2}\right)}{T}_r^{i_1{i}_1{i}_2{i}_2}+\left(\genfrac{}{}{0pt}{}{4}{211}\right)\sum _{i_1>i_2}^{q\left(\genfrac{}{}{0pt}{}{q}{2}\right)}{T}_r^{i_1{i}_1{i}_2{i}_3}\hfill \\ & +4!\sum _{i_1>i_2>i_3>i_4}^{\left(\genfrac{}{}{0pt}{}{q}{4}\right)}{T}_r^{i_1{i}_2{i}_3{i}_4},\hfill \\ & {\tilde{p}}_{r6}=\sum _{i_1=1}^q{T}_r^{i_1^6}+\left(\genfrac{}{}{0pt}{}{6}{1}\right)\sum _{i_1\ne i_2}^{q(q-1)}{T}_r^{i_1^5{i}_2}+\left(\genfrac{}{}{0pt}{}{6}{2}\right)\sum _{i_1\ne i_2}^{q(q-1)}{T}_r^{i_1^4{i}_2^2}+\left(\genfrac{}{}{0pt}{}{6}{3}\right)\sum _{i_1>i_2}^{\left(\genfrac{}{}{0pt}{}{q}{2}\right)}{T}_r^{i_1^3{i}_2^3}\hfill \\ & +\left(\genfrac{}{}{0pt}{}{6}{411}\right)\sum ^{3\left(\genfrac{}{}{0pt}{}{q}{3}\right)}{T}_r^{i_1^4{i}_2{i}_3}+\left(\genfrac{}{}{0pt}{}{6}{321}\right)\sum ^{6\left(\genfrac{}{}{0pt}{}{q}{3}\right)}{T}_r^{i_1^3{i}_2^2{i}_3}+\left(\genfrac{}{}{0pt}{}{6}{222}\right)\sum _{i_1>i_2>i_3}^{\left(\genfrac{}{}{0pt}{}{q}{3}\right)}{T}_r^{i_1^2{i}_2^2{i}_3^2}\hfill \\ & +\left(\genfrac{}{}{0pt}{}{6}{3111}\right)\sum _{i_2>i_3>i_4}^{4\left(\genfrac{}{}{0pt}{}{q}{4}\right)}{T}_r^{i_1^3{i}_2{i}_3{i}_4}+\left(\genfrac{}{}{0pt}{}{6}{2211}\right)\sum _{i_1>i_2,i_3>i_4}^{6\left(\genfrac{}{}{0pt}{}{q}{4}\right)}{T}_r^{i_1^2{i}_2^2{i}_3{i}_4}\hfill \\ & +\left(\genfrac{}{}{0pt}{}{6}{21111}\right)\sum _{i_2>i_3>i_4>i_5}^{4\left(\genfrac{}{}{0pt}{}{q}{4}\right)}{T}_r^{i_1^2{i}_2{i}_3{i}_4{i}_5}+6!\sum _{i_1>\cdots >i_6}^{\left(\genfrac{}{}{0pt}{}{q}{6}\right)}{T}_r^{i_1\cdots i_6},\hfill \end{array}$$

where all $i_j$ are distinct. Similarly we can write out ${\tilde{p}}_{rk}$ for $k=5,7,9.$ This reduces the number of terms in ${\tilde{p}}_{rk}$ from $q^k$ to $q+\left(\genfrac{}{}{0pt}{}{q}{2}\right)$ for $k=2$, to $q+q(q-1)+\left(\genfrac{}{}{0pt}{}{q}{3}\right)$ for $k=3$, to $q+q(q-1)+\left(\genfrac{}{}{0pt}{}{q}{2}\right)+q\left(\genfrac{}{}{0pt}{}{q}{2}\right)+\left(\genfrac{}{}{0pt}{}{q}{4}\right)$ for $k=4,$ and to $q+5\left(\genfrac{}{}{0pt}{}{q}{2}\right)+10\left(\genfrac{}{}{0pt}{}{q}{3}\right)+14\left(\genfrac{}{}{0pt}{}{q}{4}\right)+\left(\genfrac{}{}{0pt}{}{q}{6}\right)$ for $k=6.$

If we reinterpret $T_r^{i_1\cdots i_k}$ as ${\overline{P}}_r^{1-k}(-{\overline{\partial }}_1)\cdots (-{\overline{\partial }}_k){\Phi }_V\left(x\right)$, where again tensor summation is not used, then we can reinterpret the above expression for ${\tilde{p}}_{rk}$, as an expression for $P_{rk}\left(x\right)$. For example,

$$P_{r2}\left(x\right)=\sum _{i_1=1}^q{P}_r^{i_1{i}_1}{(-{\partial }_{i_1})}^2{\Phi }_V\left(x\right)+2\sum _{i_1>i_2}^{\left(\genfrac{}{}{0pt}{}{q}{2}\right)}{P}_r^{i_1{i}_2}(-{\partial }_{i_1})(-{\partial }_{i_2}){\Phi }_V\left(x\right).$$

These results can be extended to Type B estimates, that is to $\widehat{w}$ with cumulant expansions not of type (11), but

$$\begin{array}{c}\hfill \kappa ({\widehat{w}}^{i_1},\dots ,{\widehat{w}}^{i_r})\approx \sum _{d=2r-2}^{\infty }{b}_d^{i_1-i_r}{n}^{-d/2}.\end{array}$$

3. The Distribution of ${\mathbf{n}}^{\mathbf{1}/\mathbf{2}}(\widehat{\mathbf{w}}-\mathbf{w})$ for $\mathbf{q}=\mathbf{2}$

We first give ${\tilde{p}}_{rk}$ of (22) for $r\le 3$, and then $P_{rk}\left(x\right)$ of (21).

$$\begin{array}{cc}& {\tilde{p}}_{rk}=\sum _{b=0}^k{P}_r\left(1^{k-b}{2}^b\right)H_{k-b,b}where{P}_r\left(1^a{2}^b\right)=\left(\genfrac{}{}{0pt}{}{a+b}{a}\right)P_r^{1^a{2}^b},\hfill \end{array}$$

(31)

for ${\overline{P}}_r^{1-k}$ of (16), (17), where we use the dual notation,

$$\begin{array}{cc}& H_{ab}=H^{1^a{2}^b}=E{(y_1+i{Y}_1)}^a{(y_2+i{Y}_2)}^b\hfill \end{array}$$

(32)

So $H_{k0}$ and $H_{0k}$ are given by (29) with $j=1$ and 2,

$$\begin{array}{cc}& H_{11}=y_1{y}_2-V^{12},H_{21}=y_1^2{y}_2-V^{11}{y}_2-2V^{12}{y}_1,H_{12}=y_1{y}_2^2-V^{22}{y}_1-2V^{12}{y}_2.\hfill \end{array}$$

For more examples see Withers (2000). $P_r\left(1^b{2}^a\right)$ is just $P_r\left(1^a{2}^b\right)$ with 1 and 2 reversed. The other $P_r\left(1^a{2}^b\right)$ needed in (31) for ${\tilde{p}}_{rk}$ are as follows.

$$\begin{array}{cc}& For{\tilde{p}}_{11},P_1\left(j\right)=k_1^j.For{\tilde{p}}_{13},P_1\left(j^3\right)=k_2^{jjj}/6,P_1(1^{2}2)=k_2^{112}/2.\hfill \\ & For{\tilde{p}}_{31},P_3\left(j\right)=k_2^j.For{\tilde{p}}_{33},P_3\left(j^3\right)=k_3^{j^3}/6+k_2^{jj}{k}_1^j/2+{\left(k_1^j\right)}^3/6,\hfill \\ & P_3(1^{2}2)=[k_3^{112}+2k_1^1{k}_2^{12}+k_1^2{k}_2^{11}+{\left(k_1^1\right)}^2{k}_1^2]/6.\hfill \\ & For{\tilde{p}}_{22},P_2\left(j^2\right)=k_2^{jj}/2+{\left(k_1^j\right)}^2/2,P_2(12)=k_2^{12}+k_1^1{k}_1^2.\hfill \\ & For{\tilde{p}}_{24},P_2\left(j^4\right)=k_3^{j^4}/24+k_1^j{k}_2^{j^3}/6,P_2(1^{3}2)=k_3^{1112}/6+k_1^1{k}_2^{112}/3+k_1^2{k}_2^{111}/3,\hfill \\ & P_2\left(1^2{2}^2\right)=k_3^{1122}/4+k_1^1{k}_2^{112}/2+k_1^2{k}_2^{122}/2,\hfill \end{array}$$

$$\begin{array}{cc}& For{\tilde{p}}_{26},P_2\left(j^6\right)={\left(k_2^{jjj}\right)}^2/72,P_2(1^{5}2)=k_2^{111}{k}_2^{112}/12,\hfill \\ & P_2\left(1^4{2}^2\right)=k_2^{111}{k}_2^{122}/12+{\left(k_2^{112}\right)}^2/8,P_2\left(1^3{2}^3\right)=k_2^{111}{k}_2^{222}/36+k_2^{112}{k}_2^{122}/4,\hfill \\ & For{\tilde{p}}_{35},P_3\left(j^5\right)=k_4^{j^5}/120+k_3^{j^4}{k}_1^j/24+k_2^{jjj}[k_2^{jj}+{\left(k_1^j\right)}^2]/12,\hfill \\ & P_3(1^{4}2)/5=P_3^{1^{4}2}=k_4^{1^{4}2}/120+S_1/24+S_2/12+S_3/12where\hfill \\ & S_1=(4k_1^1{k}_3^{1112}+k_1^2{k}_3^{1111})/5by\left(\right),S_2=(3k_2^{11}{k}_2^{112}+2k_2^{12}{k}_2^{111})/5by\left(\right),\hfill \\ & S_3=[2k_1^1{k}_1^2{k}_2^{111}+3{\left(k_1^1\right)}^2{k}_2^{112}]/5by\left(\right),\hfill \\ & P_3\left(1^3{2}^2\right)=10P_3^{1^3{2}^2}=k_4^{1^3{2}^2}/12+k_3^{1^{3}2}{k}_1^2/3+k_3^{1122}{k}_1^1/4+k_3^{1^3}{\left(k_1^2\right)}^2/12\hfill \\ & +k_3^{112}{k}_1^2{k}_1^2/2+k_3^{122}{\left(k_1^1\right)}^2/4.\hfill \\ & For{\tilde{p}}_{37},P_3\left(j^7\right)=k_3^{j^4}{k}_2^{j^3}/144+{\left(k_2^{j^3}\right)}^2{k}_1^j/72,\hfill \\ & P_3(1^{6}2)=7(S_4/144+S_5/72)whereby\left(\right),S_4=(3k_2^{112}{k}_3^{1111}+4k_2^{111}{k}_3^{1112})/7,\hfill \\ & S_5=[k_1^2{\left(k_2^{111}\right)}^2+6k_1^1{k}_2^{111}{k}_2^{112}]/7,sothat\hfill \\ & P_3(1^{6}2)=k_2^{112}{k}_3^{1111}/48+k_2^{111}{k}_3^{1112}/36+k_1^2{\left(k_2^{111}\right)}^2/72+k_1^1{k}_2^{111}{k}_2^{112}/12.\hfill \\ & P_3\left(1^5{2}^2\right)=k_3^{1^4}{k}_2^{122}/48+k_3^{1^{3}2}{k}_2^{112}/12+k_3^{1122}{k}_2^{111}/24+5k_2^{111}{k}_2^{112}{k}_1^2/24\hfill \\ & +k_2^{111}{k}_2^{122}{k}_1^1/12+{\left(k_2^{112}\right)}^2{k}_1^1/12,\hfill \\ & P_3\left(1^4{2}^3\right)=35P_3^{1^4{2}^3}=A/144+B/72forA=k_3^{1^4}{k}_2^{222}+13k_3^{1^{3}2}{k}_2^{122}+17k_3^{1122}{k}_2^{112}\hfill \\ & +k_3^{1222}{k}_2^{111},B=(k_2^{111}{k}_2^{222}+15k_2^{112}{k}_2^{122})k_1^1+(9k_2^{111}{k}_2^{122}+10k_2^{112}{k}_2^{112})k_1^2.\hfill \\ & For{\tilde{p}}_{39},P_3\left(j^9\right)={\left(k_2^{j^3}\right)}^3/6^3,P_3(1^{8}2)=9{\left(k_2^{111}\right)}^2{k}_2^{112}/6^3,\hfill \\ & andfor{\overline{a}}^{123}={\overline{k}}_2^{123},6^3{P}_3\left(1^7{2}^2\right)=9{\left(a^{111}\right)}^2{a}^{122}+3a^{111}{\left(a^{112}\right)}^2,\hfill \\ & 6^3{P}_3\left(1^6{2}^3\right)=3[{\left(a^{111}\right)}^2{a}^{222}+18a^{111}{a}^{112}{a}^{122}+9{\left(a^{112}\right)}^3],\hfill \\ & 6^3{P}_3\left(1^5{2}^4\right)=9[a^{111}{a}^{112}{a}^{222}+15a^{111}{\left(a^{122}\right)}^2+27{\left(a^{112}\right)}^2{a}^{122}].\hfill \end{array}$$

(18) and (22) now give the distribution and density of $X_n=n^{1/2}(\widehat{w}-w)$ to $O\left(n^{-2}\right)$. Set

$$\begin{array}{cc}& H_{ab}^{*}={(-{\partial }_1)}^a{(-{\partial }_2)}^b{\Phi }_V\left(x\right).So{H}_{ab}^{*}=H_{a-1,b-1}{\varphi }_V\left(x\right)ifa>0,b>0,\hfill \\ & H_{a0}^{*}={\int }_{-\infty }^{x_2}{H}_{a-1,0}{\varphi }_V\left(x\right)d{x}_{2}ifa>0,H_{0b}^{*}={\int }_{-\infty }^{x_1}{H}_{0,b-1}{\varphi }_V\left(x\right)d{x}_{1}ifb>0.\hfill \end{array}$$

Then $P_{rk}\left(x\right)$ of (21) is given by replacing $H_{ab}$ by $H_{ab}^{*}$ in the expressions above for ${\tilde{p}}_{rk}$. That is,

$$\begin{array}{cc}& P_{rk}\left(x\right)=\sum _{b=0}^k{P}_r\left(1^{k-b}{2}^b\right)H_{k-b,b}^{*}.\hfill \end{array}$$

(33)

(18) and (21) now give $Prob(X_n\le x)$ to $O\left(n^{-2}\right)$ for $X_n=n^{1/2}(\widehat{w}-w)$.

4. The Conditional Density and Distribution

For $q=q_1+q_2,q_1\ge 1,$ and $q_2\ge 1,$ partition $x,y=V^{-1}x,X_n=n^{1/2}(\widehat{w}-w)$ and $X\sim {\mathcal{N}}_q(0,V)$, as $\left(\genfrac{}{}{0pt}{}{{\mathbf{x}}_1}{{\mathbf{x}}_2}\right),\left(\genfrac{}{}{0pt}{}{{\mathbf{y}}_1}{{\mathbf{y}}_2}\right),\left(\genfrac{}{}{0pt}{}{{\mathbf{X}}_{n1}}{{\mathbf{X}}_{n2}}\right),\left(\genfrac{}{}{0pt}{}{{\mathbf{X}}_1}{{\mathbf{X}}_2}\right),$ where ${\mathbf{x}}_i,{\mathbf{y}}_i,{\mathbf{X}}_{ni}$ are vectors of length $q_i$. Partition $V,V^{-1}$ as $\left({\mathbf{V}}_{ij}\right),\left({\mathbf{V}}^{ij}\right)$ where ${\mathbf{V}}_{ij},{\mathbf{V}}^{ij}$ are $q_i\times q_j$.

The conditional density of ${\mathbf{X}}_{n1}$ given $({\mathbf{X}}_{n2}={\mathbf{x}}_2)$, is

$$\begin{array}{cc}& p_{1\cdot 2}\left({\mathbf{x}}_1\right)=p_{X_n}\left(x\right)/p_{X_{n2}}\left({\mathbf{x}}_2\right)={\varphi }_{1\cdot 2}\left({\mathbf{x}}_1\right)(1+S)/(1+S_2)\hfill \end{array}$$

(34)

(35)

(36)

$p_r^{*}\left({\mathbf{x}}_2\right)$ is ${\tilde{p}}_r\left(x\right)$ of (22) for ${\mathbf{X}}_{n2}$, and ${\varphi }_{1\cdot 2}\left({\mathbf{x}}_1\right)$ is the density of ${\mathbf{X}}_1|({\mathbf{X}}_2={\mathbf{x}}_2)$. By (37)–(39), §2.5 of Anderson (1958),

$$\begin{array}{cc}& {\varphi }_{1\cdot 2}\left({\mathbf{x}}_1\right)={\varphi }_V\left(x\right)/{\varphi }_{{\mathbf{V}}_{22}}\left({\mathbf{x}}_2\right)={\varphi }_{V_{1\cdot 2}}({\mathbf{x}}_1-{\mu }_{1\cdot 2}),\hfill \end{array}$$

(37)

(38)

The distribution of ${\mathbf{X}}_1|({\mathbf{X}}_2={\mathbf{x}}_2)$ is

$$\begin{array}{cc}& {\Phi }_{1\cdot 2}\left({\mathbf{x}}_1\right)={\Phi }_{V_{1\cdot 2}}({\mathbf{x}}_1-{\mu }_{1\cdot 2}).\hfill \end{array}$$

(39)

(40)

By (22), for $r\ge 1$ and ${\overline{P}}_r^{1-k}$ of (14)–(16),

$$\begin{array}{cc}& p_r^{*}\left({\mathbf{x}}_2\right)=\sum _{k=1}^{3r}[p_{rk}^{*}:k-reven],where{p}_{rk}^{*}={\overline{P}}_r^{1-k}{\overline{H}}_{q_2}^{1-k}\hfill \end{array}$$

(41)

and ${\overline{H}}_{q_2}^{1-k}$ is given by replacing $y=V^{-1}x$ and $\left(V^{ij}\right)=V^{-1}$ in ${\overline{H}}_q^{1-k}$ by

$$\begin{array}{cc}& z={\mathbf{V}}_{22}^{-1}{\mathbf{x}}_{2}and\left(U^{ij}\right)={\mathbf{V}}_{22}^{-1},\hfill \end{array}$$

(42)

and now implicit summation in (41) is for $i_1,\cdots ,i_k$ over $q_1+1,\cdots ,q$. So,

$$\begin{array}{cc}& p_1^{*}\left({\mathbf{x}}_2\right)=\sum _{k=1,3}{p}_{1k}^{*},p_{11}^{*}=\sum _{i_1=q_1+1}^q{\overline{k}}_1^1{\overline{H}}_{q_2}^1,p_{13}^{*}=\sum _{i_1,i_2,i_3=q_1+1}^q{\overline{k}}_2^{1-3}{\overline{H}}_{q_2}^{1-3}/6,\hfill \\ & {\overline{H}}_{q_2}^1={\overline{z}}_1,{\overline{H}}_{q_2}^{1-3}={\overline{z}}_1{\overline{z}}_2{\overline{z}}_3-\sum ^3{\overline{U}}^{12}{\overline{z}}_3,\hfill \end{array}$$

(43)

(44)

$p_3^{*}\left({\mathbf{x}}_2\right)={\sum }_{k=1,3,5,7,9}{p}_{3k}^{*}$, and so on. For ${\tilde{B}}_{rs}\left(e\right)$ of (7), set

$$\begin{array}{cc}& B_r^{*}=B_r^{*}\left(e\right)=\sum _{s=0}^r{\tilde{B}}_{rs}\left(e\right).\hfill \\ & So,B_0^{*}=1B_1^{*}=e_1,B_2^{*}=e_2+e_1^2,B_3^{*}=e_3+2e_1{e}_2+e_1^3,\hfill \\ & and{(1+S_2)}^{-1}=\sum _{r=0}^{\infty }{n}^{-r/2}{C}_{r}where{C}_r=B_r^{*}(-f)for{f}_{r}of\left(36\right):\hfill \end{array}$$

(45)

(46)

So the conditional density $p_{1\cdot 2}\left({\mathbf{x}}_1\right)$ of (34), relative to ${\varphi }_{V_{1\cdot 2}}\left({\mathbf{x}}_1\right)$ of (37), is

$$\begin{array}{cc}& p_{1\cdot 2}\left({\mathbf{x}}_1\right)/{\varphi }_{1\cdot 2}\left({\mathbf{x}}_1\right)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{D}_{r}where\hfill \end{array}$$

(47)

(48)

(49)

So now we have the conditional density to $O\left(n^{-2}\right)$. The expansion for the conditional distribution about ${\Phi }_{1\cdot 2}\left({\mathbf{x}}_1\right)$ of (39), is

$$\begin{array}{cc}& P_{1\cdot 2}\left({\mathbf{x}}_1\right)=Prob.({\mathbf{X}}_{n1}\le {\mathbf{x}}_1|{\mathbf{X}}_{n2}={\mathbf{x}}_2)\approx {\Phi }_{1\cdot 2}\left({\mathbf{x}}_1\right)+\sum _{r=1}^{\infty }{n}^{-r/2}{G}_r\hfill \\ & where{G}_r={\int }_{-\infty }^{{\mathbf{x}}_1}{D}_{r}d{\Phi }_{1\cdot 2}\left({\mathbf{x}}_1\right)={\int }_{-\infty }^{{\mathbf{x}}_1}{D}_{r}forshort\hfill \end{array}$$

(50)

(51)

(52)

(53)

(54)

This gives $g_r$ in terms of ${\overline{I}}^{1-k}$, given by (54) in terms of $\theta$ and derivatives of ${\Phi }_V\left(x\right)$. (51) now gives $G_r$ in terms of $f_r$ of (36). So $G_1,G_2,G_3$ and (50) give the conditional distribution to $O\left(n^{-2}\right)$. Alternatively, as ${\overline{H}}_q^{1-k}$ is a polynomial in ${\mathbf{x}}_1={(x_{11},\cdots ,x_{1q_1})}^{\prime }$, by (37), ${\overline{I}}^{1-k}={\int }_{-\infty }^{{\mathbf{x}}_1}{\overline{H}}_q^{1-k}d{\Phi }_{1\cdot 2}\left({\mathbf{x}}_1\right)$ is linear in

$${\int }_{-\infty }^{{\mathbf{x}}_1}{x}_{1j_1}\cdots x_{1j_s}d{\Phi }_{1\cdot 2}\left({\mathbf{x}}_1\right)={\int }_{-\infty }^{\mathbf{u}}{({\mu }_{1\cdot 2}+u)}_{1j_1}\cdots {({\mu }_{1\cdot 2}+u)}_{1j_s}d{\Phi }_{V_{1\cdot 2}}\left(\mathbf{u}\right)$$

for $0\le s\le k$ where $\mathbf{u}={\mathbf{x}}_1-{\mu }_{1\cdot 2}.$ We now illustrate this.

The case$q_1=1.$ So

$$\begin{array}{cc}& {\mathbf{x}}_1=x_1,{\mathbf{X}}_1=X_1,{\mathbf{X}}_{n1}=X_{n1},{\mathbf{V}}_{11}=V_{11}.Set{\sigma }_{1\cdot 2}=V_{1\cdot 2}^{1/2},\hfill \end{array}$$

(55)

By (39), $U_n|({\mathbf{X}}_{n2}={\mathbf{x}}_2)\stackrel{\mathcal{L}}{\to }U\sim \mathcal{N}(0,1).$ By (50), for $C_r$ of (46),

$$\begin{array}{cc}& P_n\left(u\right)=Prob(U_n\le u|({\mathbf{X}}_{n2}={\mathbf{x}}_2))\approx \sum _{r=0}^{\infty }{n}^{-r/2}{G}_r\left(u\right)where\hfill \\ & G_r\left(u\right)=C_r\otimes g_r\left(u\right),G_0\left(u\right)=g_0\left(u\right)=\Phi \left(u\right),andforr\ge 1,g_r\left(u\right)=g_r\hfill \end{array}$$

(56)

given by (52) in terms of $G_{rk}\left(u\right)=G_{rk}$. For $\theta$ of (53), by (37),

${\theta }^{-1}{\varphi }_V\left(x\right)d{x}_1=\varphi \left(u\right)du.$ Set

$$\begin{array}{cc}& H^{1-k}\left[y\right]=h^{1-k}\left(u\right)=\sum _{s=0}^k{h}_s^{1-k}{u}^s=H_q^{1-k}aty=V^{-1}x,\hfill \end{array}$$

(57)

(58)

By (53),

$$\begin{array}{cc}& G_{rk}={\overline{P}}_r^{1-k}{\overline{I}}^{1-k}where{\overline{I}}^{1-k}=\sum _{s=0}^k{\overline{h}}_s^{1-k}{\gamma }_{s}and{\gamma }_s={\int }_{-\infty }^u{u}^s\varphi \left(u\right)du:\hfill \end{array}$$

(59)

(60)

(61)

$$\begin{array}{cc}& G_{r3}={\overline{P}}_r^{1-3}{\overline{I}}^{1-3}for{\overline{I}}^{1-3}=\sum _{s=0}^3{\overline{h}}_s^{1-3}{\gamma }_s,\hfill \\ & where{\overline{h}}_0^{1-3}={\Pi }_{j=1}^3{\overline{a}}_j-\sum ^3{\overline{a}}_1{\overline{V}}^{23},{\overline{h}}_1^{1-3}=\sum ^3{\overline{b}}_1({\overline{a}}_2{\overline{a}}_3-{\overline{V}}^{23}),\hfill \end{array}$$

(62)

$$\begin{array}{cc}& {\overline{h}}_2^{1-3}=\sum ^3{\overline{a}}_1{\overline{b}}_2{\overline{b}}_3,{\overline{h}}_3^{1-3}={\Pi }_{j=1}^3{\overline{b}}_j.\hfill \end{array}$$

(63)

5. The Case ${\mathbf{q}}_{\mathbf{1}}={\mathbf{q}}_{\mathbf{2}}=\mathbf{1}$

In this case $q=2$ and for $z,U^{11}$ of (42) and u of (55),

$$\begin{array}{cc}& {\mathbf{x}}_j=x_j,{\mathbf{V}}_{i_1{i}_2}=V_{i_1{i}_2},{\mathbf{V}}^{i_1{i}_2}=V^{i_1{i}_2},V_{1\cdot 2}=V_{11}-V_{12}^2{V}_{22}^{-1},\hfill \\ & z=V_{22}^{-1}{x}_2,U^{11}=V_{22}^{-1},{\Phi }_{1\cdot 2}\left(x_1\right)={\Phi }_{V_{1\cdot 2}}(x_1-{\mu }_{1\cdot 2})=\Phi \left(u\right).\hfill \\ & By\left(41\right),forr\ge 1,p_r^{*}\left(x_2\right)=\sum _{k=1}^{3r}[p_{rk}^{*}:k-reven],where{p}_{rk}^{*}=P_r\left(2^k\right)H_k^{*},\hfill \\ & H_k^{*}=H_k(x_2,V_{22})={\sigma }_2^{-k}{H}_k\left({\sigma }_2{x}_2\right)for{\sigma }_2=V_{22}^{1/2},\hfill \end{array}$$

and $H_k\left(x\right)$ of (6). For example by (30), $H_3^{*}=z^3-3V_{22}^{-1}z\ne H_3\left(x\right).$ ${\tilde{p}}_{rk}$ and $P_{rk}\left(x\right)$ are given in §3 in terms of $P_r\left(1^a{2}^b\right)=\left(\genfrac{}{}{0pt}{}{a+b}{a}\right)P_r^{1^a{2}^b}.$

(47) gives $D_r$ in terms of ${\tilde{p}}_r$ and $p_r^{*}$, which are given for $r\le 3$ by (22) in terms of ${\tilde{p}}_{rk}$ of §3. For $H_{ab}$ of (32), set

$$\begin{array}{cc}& H_{ab}\left(u\right)=\sum _{s=0}^{a+b}{H}_{abs}{u}^s=H_{ab}aty=V^{-1}x,x_1={\mu }_{1\cdot 2}+{\sigma }_{1\cdot 2}u.\hfill \end{array}$$

(64)

For example,

$$\begin{array}{cc}& H_{10}\left(u\right)=y_1,H_{01}\left(u\right)=y_2,y_i=h_0^i+h_1^{i}uof\left(\right).So{H}_{10s}=h_s^1,H_{01s}=h_s^2.\hfill \end{array}$$

(65)

for ${\gamma }_s$ of (59) and (60). For example

$$\begin{array}{cc}& G_{r1}=P_r(1)I_{10}^1+P_r(2)I_{01}^{1}for{I}_{10}^1=I^1,I_{01}^1=I^{2}of\left(62\right).\hfill \\ & G_{r3}=\sum _{a=0}^3{P}_r\left(1^a{2}^{3-a}\right)I_{a,3-a}^3.Soweneed{I}_{ab}^3=\sum _{s=0}^3{H}_{abs}{\gamma }_{s}forab=30,21,12,03.\hfill \end{array}$$

$$\begin{array}{cc}& H_{30}=H_2^{111}=\sum _{s=0}^3{h}_s^{111}{u}^{s}of\left(57\right)whereby\left(63\right),\hfill \\ & h_0^{111}=a_1^3-3a_1{V}^{11},h_1^{111}=3b_1[a_1^2-V^{11}],h_2^{111}=3a_1{b}_1^2,h_3^{111}=b_1^3.\hfill \\ & H_{21}=H_2^{112}\sum _{s=0}^3{h}_s^{112}{u}^{s}where{h}_0^{112}=a_1^2{a}_2-2a_1{V}^{12}-a_2{V}^{11},\hfill \\ & h_1^{112}=2h_0^1[h_0^1{h}_0^2-V^{12}]+h_0^2[{\left(h_0^1\right)}^2-V^{11}],h_2^{112}=2a_1{b}_1{b}_2+a_2{b}_1^2,h_3^{112}=b_1^2{b}_2,\hfill \end{array}$$

and $H_{bas}$ is giving by reversing 1 and 2 in $H_{abs}$. Alternatively, we can use

Theorem 1. 

Set $\theta ={\varphi }_{V_{22}}\left(x_2\right).$ For $k-r$ even, $G_{rk}$ of (53) is given by

$$\begin{array}{cc}& G_{rk}={\theta }^{-1}[P_r\left(2^k\right){\int }_{-\infty }^{x_1}{H}_{0k}{\varphi }_V\left(x\right)d{x}_1-b_{rk}{\varphi }_V\left(x\right)],forr\ge 1,k\ge 1,\hfill \end{array}$$

(66)

(67)

(68)

PROOF For ${\overline{I}}_{1-k}$ of (54),

$$\begin{array}{cc}& G_{rk}={\overline{P}}_r^{1-k}{\overline{I}}_{1-k}=P_r\left(2^k\right)I\left(2^k\right)+\sum _{a=1}^k{P}_r\left(1^a{2}^{k-a}\right)I\left(1^a{2}^{k-a}\right),where\hfill \\ & \theta I\left(2^k\right)={\int }_{-\infty }^{x_1}{H}_{0k}{\varphi }_V\left(x\right)d{x}_1,andfora\ge 1,\theta I\left(1^a{2}^b\right)=-H_{a-1,b}{\varphi }_V\left(x\right).\square \hfill \end{array}$$

This gives $G_{rk}$, and so $g_r$ of (52) and so $P_{1\cdot 2}$ to $O\left(n^{-(r+1)/2}\right)$, in terms of the coefficients $P_r\left(1^a{2}^b\right).$ So (67) gives $g_1,g_2,g_3$ in terms of $P_r\left(1^a{2}^b\right)$ of §3 via

$$\begin{array}{cc}& B_1=\sum _{k=1,3}{b}_{1k}where{b}_{11}=P_1(1)=k_1^1,\hfill \\ & b_{13}=P_1\left(1^3\right)H_{20}+P_1(112)H_{11}+P_1(122)H_{02}\hfill \\ & =(k_2^{111}{H}_{20}+3k_2^{112}{H}_{11}+3k_2^{122}{H}_{02})/6,\hfill \\ & B_2=\sum _{k=2,4,6}{b}_{2k}where{b}_{22}=P_2(11)H_{10}+P_2(12)H_{01},\hfill \\ & b_{24}=P_2\left(1^4\right)H_{30}+P_2(1^{3}2)H_{21}+P_2\left(1^2{2}^2\right)H_{12}+P_2(12^3)H_{03}.\hfill \\ & b_{26}=P_2\left(1^6\right)H_{50}+P_2(1^{5}2)H_{41}+P_2(12^5)H_{05}+\sum _{12}^2{P}_2\left(1^4{2}^2\right)H_{32}+P_2\left(1^3{2}^3\right)H_{23},\hfill \end{array}$$

$$\begin{array}{cc}& B_3=\sum _{k=1,3,5,7,9}(P_3\left(2^k\right)H_{0k},b_{3k})for{b}_{3k}of\left(66\right).\hfill \end{array}$$

The explicit form for (66), despite the work needed to obtain $H_{abs}$ of (64).

Example 2. 

If the distribution of $\widehat{w}$ is symmetric about w, then for r odd, $p_r\left(x\right)=P_r\left(x\right)=0$, and the non-zero ${\overline{P}}_2^{1-k}$ are

${\overline{P}}_2^{12}={\overline{k}}_2^{12},{\overline{P}}_2^{1-4}={\overline{k}}_3^{1-4}/24.$

Example 3. 

Let $\widehat{w}$ be a sample mean. Then $E\widehat{w}=w$, and only the leading coefficients in (11) are non-zero. So ${\overline{k}}_1^1={\overline{P}}_1^1={\tilde{p}}_{11}=p_{11}^{*}=0$ . The non-zero $P_r^{1-k}$ were given in Example 2.1. For $q=2$${\tilde{p}}_{rk},P_{rk}\left(x\right)$ are given by §3 with these non-zero $P_r^{1-k}$, and ${\tilde{p}}_{11}={\tilde{p}}_{22}={\tilde{p}}_{31}=0.$

$$\begin{array}{cc}& For{\tilde{p}}_{13},P_1\left(j^3\right)=k_2^{j^3}/3!,P_1(1^{2}2)=k_2^{112}/2,P_1(12^2)=k_2^{122}/2.\hfill \\ & For{\tilde{p}}_{24},P_2\left(j^4\right)=k_3^{j^4}/4!,P_2(1^{3}2)=k_3^{1112}/6,P_2(12^3)=k_3^{1222}/6,\hfill \\ & P_2\left(1^2{2}^2\right)=k_3^{1122}/4.\hfill \\ & For{\tilde{p}}_{35},P_3\left(j^5\right)=k_4^{j^5}/5!,P_3(1^{4}2)/5=k_4^{1^{4}2}/5!,P_3\left(1^3{2}^2\right)=k_4^{1^3{2}^2}/12.\hfill \\ & For{\tilde{p}}_{37},P_3\left(j^7\right)=k_3^{j^4}{k}_2^{j^3}/144,P_3(1^{6}2)=k_2^{112}{k}_3^{1111}/48+k_2^{111}{k}_3^{1112}/36,\hfill \\ & P_3\left(1^5{2}^2\right)=k_3^{1^4}{k}_2^{122}/48+k_3^{1^{3}2}{k}_2^{112}/12+k_3^{1122}{k}_2^{111}/24,\hfill \\ & P_3\left(1^4{2}^3\right)=A/144forA=k_3^{1^4}{k}_2^{222}+13k_3^{1^{3}2}{k}_2^{122}+17k_3^{1122}{k}_2^{112}+k_3^{1222}{k}_2^{111}.\hfill \end{array}$$

${\overline{P}}_2^{1-6}$ needed for ${\tilde{p}}_{26}$ of §3 does not simplify. Nor does ${\overline{P}}_3^{1-9}$ of §3 needed for ${\tilde{p}}_{35}$.

Example 4. 

Consider the classical problem of the distribution of a sample mean, given the sample variance. So $q=2.$ Let ${\widehat{w}}_2,{\widehat{w}}_2$ be the usual unbiased estimates of the 1st 2 cumulants $w_1,w_2$ from a univariate random sample of size n from a distribution with rth cumulant ${\kappa }_r$. So $w_1={\kappa }_1,w_2={\kappa }_2.$ By the last 2 equations of §12.15 and (12.35)–(12.38) of Stuart and Ord (1991), the cumulant coefficients needed for ${\overline{P}}_r^{1-k}$ of (14) for $r\le 3$, that is, the coefficients needed for the conditional density to $O\left(n^{-2}\right)$ are

$$\begin{array}{cc}& k_1^{11}={\kappa }_2,k_1^{12}={\kappa }_3,k_1^{22}={\kappa }_4+2{\kappa }_2^2,\Rightarrow V=\left(\begin{array}{cc}{\kappa }_2& {\kappa }_3\\ {\kappa }_3& {\kappa }_4+2{\kappa }_2^2\end{array}\right),\hfill \\ & k_1^1=k_1^2=0,k_2^{111}={\kappa }_3,k_2^{112}=k_2^{122}=0,k_2^{222}=(6)+12(24)+4\left(3^2\right)+8\left(2^3\right),\hfill \end{array}$$

$$\begin{array}{cc}& k_2^{11}=k_2^{12}=0,k_2^{22}=2\left(2^2\right),k_3^{1111}=(4),k_3^{1112}=(5),\hfill \\ & k_3^{1122}=k_3^{1222}=0,k_3^{2222}=(8)+24(26)+32(35)+32\left(4^2\right)+144(2^{2}4)+96\left({23}^2\right)\hfill \\ & +48\left(2^4\right),k_2^1=k_2^2=k_3^{111}=k_3^{112}=k_3^{122}=0,k_3^{222}=12(24)+16\left(2^3\right),\hfill \\ & k_4^{1^5}=k_4^{1^3{2}^2}=k_4^{1^2{2}^3}=k_4^{12^4}=0,k_4^{1^{4}2}=(6),k_4^{2^5}=(10)+40(28)+80(37)+200(46)\hfill \\ & +96\left(5^2\right)+480(2^{2}6)+1280(235)+1280\left({24}^2\right)+960(3^{2}4)+1920(2^{3}4)\hfill \\ & +1920\left(2^2{3}^2\right)+384\left(2^5\right),where(i_1^{j_1}{i}_2^{j_2}\cdots )={\kappa }_{i_1}^{j_1}{\kappa }_{i_2}^{j_2}\cdots .\hfill \end{array}$$

(47) gives $D_r$ in terms of ${\tilde{p}}_r$ and $p_r^{*}$, that is, in terms of ${\tilde{p}}_{rk}$ and $p_{rk}^{*}$ of §3 in terms of $P_r\left(1^a{2}^b\right)$ of (31). In this example, many of these are 0. By (15)–(17) and the Appendix A, the non-zero $P_r\left(1^a{2}^b\right)$ are in order needed,

$$\begin{array}{cc}& P_1\left(j^3\right)=k_2^{jjj}/6,P_2\left(2^2\right)={\kappa }_2^2,P_2\left(j^4\right)=k_3^{jjjj}/24,P_2(1^{3}2)=k_3^{1112}/6={\kappa }_5/6,\hfill \\ & P_2\left(j^6\right)={\left(k_2^{jjj}\right)}^2/72,P_2\left(1^3{2}^3\right)=k_2^{111}{k}_2^{222}/36.P_3\left(j^3\right)=k_3^{jjj}/6.\hfill \\ & P_3\left(2^5\right)=k_4^{2^5}/120+k_2^{22}{k}_2^{222}/12.P_3\left(j^7\right)=k_2^{jjj}{k}_3^{j^4}/144,P_3(1^{6}2)=k_2^{111}{k}_3^{1112}/36,\hfill \\ & P_3\left(1^4{2}^3\right)=k_2^{222}{k}_3^{1^4}/144,P_3\left(1^3{2}^4\right)=(k_2^{111}{k}_3^{2^4}+k_2^{222}{k}_3^{1^{3}2})/144,\hfill \\ & P_3\left(j^9\right)={\left(k_2^{jjj}\right)}^3/6^3,P_3\left(1^6{2}^3\right)=3{\left(k_2^{111}\right)}^2{k}_2^{222}/6^3,P_3\left(1^3{2}^6\right)=3k_2^{111}{\left(k_2^{222}\right)}^2/6^3.\hfill \\ & So,{\tilde{p}}_{11}=0,{\tilde{p}}_{13}=P_1\left(1^3\right)H_{30}+P_1\left(2^3\right)H_{03},,{\tilde{p}}_{22}=P_2\left(2^2\right)H_{02},\hfill \\ & {\tilde{p}}_{24}=P_2\left(1^4\right)H_{40}+P_2\left(2^4\right)H_{04}+P_2(1^{3}2)H_{31},{\tilde{p}}_{26}=P_2\left(1^6\right)H_{60}+P_2\left(2^6\right)H_{06}\hfill \\ & +P_2\left(1^3{2}^3\right)H_{33},{\tilde{p}}_{31}=0,{\tilde{p}}_{33}=P_3\left(1^3\right)H_{30}+P_3\left(2^3\right)H_{03},{\tilde{p}}_{35}=P_3\left(2^5\right)H_{05},\hfill \\ & {\tilde{p}}_{37}=P_3\left(1^7\right)H_{70}+P_3\left(2^7\right)H_{07}+P_3(1^{6}2)H_{61}+P_3\left(1^4{2}^3\right)H_{43}+P_3\left(1^3{2}^4\right)H_{34}\hfill \\ & +P_3\left(1^2{2}^5\right)H_{25}+P_3(12^6)H_{16}+P_3\left(2^7\right)H_{07},\hfill \\ & {\tilde{p}}_{39}=P_3\left(1^9\right)H_{90}+P_3\left(1^6{2}^3\right)H_{63}+P_3\left(1^3{2}^6\right)H_{36}+P_3\left(2^9\right)H_{09}.\hfill \end{array}$$

(24)–(26) now give ${\tilde{p}}_r\left(x\right)$ and $p_r{\left(x\right)}^{*}$ for $r\le 3$. By (18) and (47), this gives the conditional density $p_{1\cdot 2}\left({\mathbf{x}}_1\right)$ to $O\left(n^{-2}\right)$. (67) gives $g_r$ needed for the conditional distribution $P_{1\cdot 2}\left({\mathbf{x}}_1\right)$ to $O\left(n^{-2}\right)$ in terms of $(A_r,B_r)$ of (68). So

$$\begin{array}{cc}& B_1=b_{13}=P_1\left(1^3\right)H_{20},B_2=\sum _{k=2,4,6}{b}_{2k}where{b}_{22}=P_2(11)H_{10}+P_2(12)H_{01},\hfill \\ & b_{24}=P_2\left(1^4\right)H_{30}+P_2(1^{3}2)H_{21},b_{26}=P_2\left(1^6\right)H_{50}+P_2\left(1^3{2}^3\right)H_{23},\hfill \\ & B_3=\sum _{k=1,3,5,7,9}{b}_{3k}where{b}_{31}=0,b_{33}=P_3\left(1^3\right)H_{20},b_{35}=0,\hfill \\ & b_{37}=P_3\left(1^7\right)H_{60}+P_3(1^{6}2)H_{51}+P_3\left(1^4{2}^3\right)H_{33}+P_3\left(1^3{2}^4\right)H_{24},\hfill \end{array}$$

$$\begin{array}{cc}& b_{39}=P_3\left(1^9\right)H_{80}+P_3\left(1^6{2}^3\right)H_{53}+P_3\left(1^3{2}^6\right)H_{26}.\hfill \end{array}$$

6. Conditional Cornish-Fisher Expansions

Suppose that $q_1=1$. Here we invert the conditional distribution (56), to obtain its its extended Cornish-Fisher expansions similar to (4). For any function $f\left(u\right)$ with finite derivatives, set $f_{\cdot j}={(d/du)}^{j}f\left(u\right).$

Lemma 1. 

Suppose that $G_0\left(u\right):R\to (0,1)$ is 1 to 1 increasing with jth derivative $G_{0\cdot j}$, and for some $t\in R$,

$$\begin{array}{cc}& P\left(u\right)=G_0\left(u\right)+S_{1}where{S}_1\approx \sum _{r=1}^{\infty }{t}^r{G}_r\left(u\right).\hfill \\ & Set{G}_r=G_r\left(u\right),Q\left(u\right)=G_0^{-1}\left(P\left(u\right)\right).\hfill \end{array}$$

(69)

(70)

(71)

(72)

(73)

PROOF Set $B_{rj}={\tilde{B}}_{rj}\left(J\right).$

$$\begin{array}{cc}& Forr\ge 1,S_2^j\approx \sum _{r=j}^{\infty }{t}^r{B}_{rj}.P\left(u\right)=G_0(u+S_2)\approx \sum _{j=0}^{\infty }{G}_{0\cdot j}{S}_2^j.\hfill \\ & Soby\left(8\right),G_r\left(u\right)=\sum _{j=1}^r{G}_{0\cdot j}{B}_{rj}/j!=G_{0\cdot 1}{J}_r+\sum _{j=2}^r{G}_{0\cdot j}{B}_{rj}/j!.\hfill \end{array}$$

$$\begin{array}{cc}& J_2=G_{0\cdot 1}^{-1}[G_2-G_{0\cdot 2}{J}_1^2/2],J_3=G_{0\cdot 1}^{-1}[G_3-(G_{0\cdot 2}{J}_1{J}_2-G_{0\cdot 3}{J}_1^3/6].\hfill \\ & u=Q(u+S_3)\approx \sum _{j=0}^{\infty }{S}_3^j{Q}_{\cdot j}/j!where{Q}_{\cdot j}\approx \sum _{r=0}^{\infty }{t}^r{J}_{r\cdot j}.\hfill \\ & S_3^j\approx \sum _{r=j}^{\infty }{t}^r{B}_{rj}^{\prime }.Sou=\sum _{r=0}^{\infty }{t}^r\sum _{j=0}^r{Q}_j{B}_{rj}^{\prime }/j!=\sum _{k=0}^{\infty }{t}^k{A}_{k}where\hfill \\ & A_k=\sum _{r+s=k}{C}_{rs}.Also{C}_{0s}=J_s,C_{r0}=K_{r}forr\ge 1.\hfill \\ & So0=A_k=K_k+\sum _{s=1}^k{C}_{k-s,s}:A_0=C_{00}=J_0,A_1=C_{01}+C_{10}.\hfill \\ & 0=A_1=J_1+K_1,K_1=-C_{01}=-J_1.\hfill \\ & 0=A_2=C_{02}+C_{11}+C_{20}where{C}_{11}=\sum _{j=0}^1{J}_{1\cdot j}{B}_{rj}^{\prime }=K_1{J}_{1\cdot 1}.\hfill \\ & So{K}_2=-J_2-K_1{J}_{1\cdot 1}=J_1{J}_{1\cdot 1}-J_2.\hfill \end{array}$$

One obtains $K_3$ similarly. $\square$ A different form for (73) was given in Theorem A2 of Withers (1983). So

$$\begin{array}{cc}& K_4=-J_4-\sum _{j=1}^3{J}_{4-j\cdot 1}{K}_j-J_{2\cdot 2}{J}_1^2/2+J_{1\cdot 2}{J}_1{K}_2+J_{1\cdot 3}{J}_1^3/6,\hfill \\ & K_5=-J_5-\sum _{j=1}^4{J}_{5-j\cdot 1}{K}_j-J_{3\cdot 2}{J}_1^2/2+J_{2\cdot 2}{J}_1{K}_2-J_{1\cdot 2}(K_2^2/2+K_1{K}_3)\hfill \\ & +J_{2\cdot 3}{J}_1^3/6-J_{1\cdot 3}{J}_1^2{K}_2/2-J_{1\cdot 4}{J}_1^4/24.\hfill \end{array}$$

We can now give the quantiles of the conditional distribution (56).

Theorem 2. 

$$\begin{array}{cc}& Supposethat{P}_n\left(u\right)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{G}_r\left(u\right).Set{G}_r=G_r\left(u\right).\hfill \\ & Then{\Phi }^{-1}\left(P_n\left(u\right)\right)\approx \sum _{r=0}^{\infty }{n}^{-r/2}{J}_r,and{P}_n^{-1}(\Phi \left(u\right))\approx \sum _{r=0}^{\infty }{n}^{-r/2}{K}_r,\hfill \\ & where{J}_k=\sum _{r=1}^k\varphi {\left(u\right)}^{-r}{J}_{kr},K_k=\sum _{r=1}^k\varphi {\left(u\right)}^{-r}{K}_{kr},\hfill \end{array}$$

$$\begin{array}{cc}& J_{k1}=-K_{k1}=G_k,J_{22}=u{G}_1^2/2,J_{32}=u{G}_1{G}_2,J_{33}=(2u^2+1)G_1^2/6,\hfill \\ & J_{11,1}=u{G}_1+G_{1\cdot 1},J_{12,1}=(u^2+1)G_1+(u+1)G_{1\cdot 1}+G_{1\cdot 2},\hfill \\ & K_{22}=u{G}_1/2+G_{1\cdot 1},K_{32}=G_{1\cdot 1}(G_2+G_{2\cdot 1})+u{G}_1{G}_{2\cdot 1},\hfill \\ & K_{33}=\sum _{j=0}^3{d}_j{G}_1^{j}for{d}_0=u{G}_2{G}_{1\cdot 1},d_1=u^2{G}_2+(u-1)G_{1\cdot 1}^2,\hfill \\ & d_2=-u^2/2+2(u^2-u)G_{1\cdot 1}-G_{1\cdot 2}/2,d_3=(6u^3-5u^2+3u-4)/6.\hfill \end{array}$$

A simpler formula for $J_k$ is

$$\begin{array}{cc}& J_k=-\sum _{j=1}^k{c}_j\left(u\right){\tilde{B}}_{kj}\left(\lambda \right)/j!where{\lambda }_r=-\varphi {\left(u\right)}^{-1}{G}_r\left(u\right),\hfill \end{array}$$

(74)

PROOF Apply Lemma 6.1 to (56) with $t=n^{-1/2}$. Take $G_0\left(u\right)=\Phi \left(u\right)$ and $G_r=C_r\otimes g_r$ for $g_r$ given by (52) in terms of $G_{rk}$ of (59). So

$$\begin{array}{cc}& G_{0\cdot 1}=\varphi \left(u\right)andforj\ge 1,G_{0\cdot j}={(-1)}^{j-1}{H}_{j-1}\left(u\right)\varphi \left(u\right),\hfill \end{array}$$

and for $P\left(u\right)=P_n\left(u\right)$ of (56), ${\Phi }^{-1}\left(P_n\left(u\right)\right)$ and $P_n^{-1}(\Phi \left(u\right))$ are given by (70) and (72) in terms of $J_k,K_k$ and their derivatives. These are given by

$$\begin{array}{cc}& J_{k\cdot i}=\sum _{r=1}^k\varphi {\left(u\right)}^{-r}{J}_{ki,r},K_{k\cdot i}=\sum _{r=1}^k\varphi {\left(u\right)}^{-r}{K}_{ki,r},\hfill \\ & where{J}_{21,1}=G_{2\cdot 1},J_{21,2}=u{G}_2+(u^2+1/2)G_1^2+u{G}_1{G}_{1\cdot 1},\hfill \\ & K_{32}=-J_{32}+J_{11,1}(J_{21}+J_{21,1}),\hfill \\ & K_{33}=-J_{33}+J_{11,1}(J_{22}+J_{22,1})-J_{11}{J}_{11,1}^2-J_{11}^2{J}_{12,1}/2.\hfill \end{array}$$

(74) follows from (3.2) of Withers (1984). □

We had hoped to read the conditional $a_{ri}$ off the conditional density. But the expansion (47) cannot be put into the form (3) if ${\overline{k}}_2^{1-3}\ne 0$, as the coefficient of $x^2$ in ${\overline{h}}_1\left(x\right)$ of (5) is 0. So the conditional estimate is generally not a standard estimate. (An exception is when $\widehat{w}\sim {\mathcal{N}}_q(w,n^{-1}V)$ since then ${\overline{k}}_2^{1-3}=0$ and by (50), $P_{1\cdot 2}\left({\mathbf{x}}_1\right)={\Phi }_{1\cdot 2}\left({\mathbf{x}}_1\right)$ of (39). We have yet to see what exponential families this extends to.) It might be possible to remedy this by extending the results here to Type B estimates. But there seems little point in doing so.

7. Conclusions

§2 and the Appendix A give the density and distribution of $X_n=n^{1/2}(\widehat{w}-w)$ to $O\left(n^{-2}\right)$, for $\widehat{w}$ any standard estimate, in terms of certain functions of the cumulants coefficients ${\overline{k}}_j^{1-r}$ of (11), the coefficients ${\overline{P}}_r^{1-k}$ of (14)–(17). Most estimates of interest are standard estimates, including functions of sample moments, like the sample correlation, and any multivariate function of k-statistics,. §3 gave the density and distribution of $X_n=n^{1/2}(\widehat{w}-w)$ in more detail when $q=2$ using the dual notation $P_r\left(1^a{2}^b\right)$. §4 gave the conditional density and distribution of ${\mathbf{X}}_{n1}$ given ${\mathbf{X}}_{n2}$ to $O\left(n^{-2}\right)$ where $\left(\genfrac{}{}{0pt}{}{{\mathbf{X}}_{n1}}{{\mathbf{X}}_{n2}}\right)$ is any partition of $X_n$. The expansion (47) gives the conditional density of a standard estimate in terms of $D_r$ of (47). The conditional distribution (50) to $O\left(n^{-2}\right)$ requires the function ${\overline{I}}_{1-k}$ of (54), or its expansion (59) or (65). §6 gave the extended Cornish-Fisher expansions for the quantiles of the conditional distribution when $q_1=1$.

8. Discussion

A good approximation for the distribution of an estimate, is vital for statistical inference. It enables one to explore the distribution’s dependence on underlying parameters, such as correlation. Our analytic method avoids the need for simulation or jack-knife or bootstrap methods while providing greater accuracy than them. Hall (1992) uses the Edgeworth expansion to show that the bootstrap gives accuracy to $O\left(n^{-1}\right)$. Hall (1988) says that “2nd order correctness usually cannot be bettered”. Fortunately this is not true for our analytic method. Simulation, while popular, can at best shine a light on behaviour when there is only a small number of parameters.

Estimates based on a sample of independent but not identically distributed random vectors, are also generally standard estimates. For example for a univariate sample mean $\overline{w}=n^{-1}{\sum }_{j=1}^n{X}_{jn}$ where $X_{jn}$ has rth cumulant ${\kappa }_{rjn}$, then ${\kappa }_r\left(\overline{w}\right)=n^{1-r}{\kappa }_r$ where ${\kappa }_r=n^{-1}{\sum }_{j=1}^n{\kappa }_{rjn}$ is the average rth cumulant. For some examples, see Skovgaard (1981a, 1981b) and Withers and Nadarajah (2010a, 2020b). The last is for a function of a weighted mean of complex random matrices.

A promising approach is the use of conditional cumulants. §6.2 of McCullagh (1984) uses conditional cumulants to give the conditional density of a sample mean to $O\left(n^{-3/2}\right)$. §5.6 of McCullagh (1987) gave formulas for the 1st 4 cumulants conditional on ${\mathbf{X}}_2={\mathbf{x}}_2$ when ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$ are uncorrelated. He says that assumption can be removed but gives no details how. That might give an alternative to our approach, but seems unlikely as the conditional estimate is generally not a standard estimate.

(7.5) of Barndoff-Nielsen and Cox (1989) gave the 3rd order expansion for the conditional density of a sample mean to $O\left(n^{-3/2}\right)$, but did not attempt to integrate it.

Here we have only considered expansions about the normal. However expansions about other distributions can greatly reduce the number of terms by matching the leading bias coefficient. The framework for this is Withers and Nadarajah (2010a). For expansions about a matching gamma, see Withers and Nadarajah (2011, 2014).

The results here can be extended to tilted (saddlepoint) expansions by applying the results of Withers and Nadarajah (2010a). Tilting was 1st used in statistics by Daniels (1954). He gave an approximation to the density of a sample mean. A conditional distribution by tilting was first given by Skovgaard (1987) up to $O\left(n^{-1}\right)$ for the distribution of a sample mean conditional on correlated sample means. For some examples, see Barndoff-Nielsen and Cox (1989). For other some results on conditional distributions, see Pfanzagl (1979), Booth et al. (1992), DiCiccio et al. (1993), Hansen (1994), Moreira (2003), Chapter 4 of Butler (2007), and Kluppelberg and Seifert (2020). The results given here form the basis for constructing confidence intervals and confidence regions. See Withers (1989).