A Unified Treatment of Agreement Coefficients and their Asymptotic Results: the Formula of the Weighted Mean of Weighted Ratios

Abstract

A unified treatment of agreement coefficients for multiple raters is shown, where the chance-expected proportions of the Bennett et al.-type, Scott-type, its new variation, and Cohen-type are dealt with using full or lower-order agreement among raters. When only pairwise agreement is used for multiple raters, chance corrections of the Gwet-type and its new variation are also considered. For the unified treatment, Conger’s two formulas of the ratio of means and the mean of ratios are combined into a parent formula of the weighted mean of weighted ratios. The corresponding unified expressions of their new asymptotic results are presented.

Appendix

1.1 The need of unified treatment of the asymptotic results with transparency for agreement coefficients

Fleiss et al. (1979, p.974) started their paper with the statement “Many human endeavors have been cursed before final success is achieved. The scaling of Mount Everest is one example. The discovery of the Northwest Passage is a second. The derivation of a correct standard error for kappa is a third”. Note that “a correct standard error” refers to the ASE of Fleiss’ (1971) coefficient rather than that of the original kappa, whose correct ASE was given by Fleiss et al. (1969) as mentioned earlier. The statement has been quoted by, e.g. Kraemer, Periyakoil and Noda (2002, p.2109) and Berry et al. (2005, p.246).

The above statement suggests the necessity of unified treatment with transparency in the ASEs of the various agreement coefficients. All the ASEs available including that of the Gwet coefficient using linearlization can be seen as those based on the delta method or the Taylor series expansion of a sample coefficient up to the first derivatives. The derivatives for the ASEs are with respect to the sample proportions of the multinomial distribution with k^m cells evaluated at their population values.

While Gwet (2008a, Sections 6 and 7) showed some derivations or intermediate results for the ASEs, Fleiss et al. (1969) and Fleiss (1971) gave only the final expressions of the formulas of the ASEs. The author of the current paper has confirmed that their formulas with Fleiss et al. (1979) are correct ones. The principle of the delta method is quite simple and straightforward since only the first partial derivatives are required. Further, in the case of agreement coefficients, the situation is simpler than that of fitting non-saturated models, e.g. log-linear models (see, e.g. Agresti 2013), where the estimated parameters are generally implicit functions of the sample proportions. Though simple and straightforward, the derivation of the ASEs of the sample agreement coefficients tends to be tedious. Probably, this is one of the reasons of omitting the messy intermediate results in the above literatures yielding some lack of transparency.

To the author’s knowledge, no didactic or expository papers explaining the ASEs are available. It is ironical that the derivation of the ASE in the case of Everitt (1968) is more complicated than those in Fleiss et al. (1969) and Fleiss (1971) while the transparency is kept in that necessary intermediate results or whole derivation was given in Everitt (1968).

1.2 Some relationships among the six agreement coefficients

Lemma 1

When m = 2 and ${w}_{i_1\;{i}_2}={\delta}_{i_1\;{i}_2}\;\left({i}_1,{i}_2=1,\dots, k\right)$, denote ${p}_{\mathrm{o}}^{(w)}$ and ${p}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)}$ by ${p}_{\mathrm{o}}^{\left(w=\delta \right)}$ and ${p}_{\mathrm{e}\;\left(\cdotp \right)}^{\left(w=\delta \right)}$, respectively. Then, (i) for arbitrary k ≥ 2, using ${\overline{p}}_i^2={\left({\overline{p}}_i\right)}^2$(see (2.4)) and $\overline{p_i^2}\kern0.24em \left(i=1,\dots, k\right)$(see (2.9)),

$$ {p}_{\mathrm{e}\left(\mathrm{S}\right)}^{\left(w=\delta \right)}=1-\left(k-1\right){p}_{\mathrm{e}\left(\mathrm{G}\right)}^{\left(w=\delta \right)}=\sum \limits_{i=1}^k{\overline{p}}_i^2\kern0.5em and\kern0.5em {p}_{\mathrm{e}\left(\mathrm{S}\mathrm{O}\right)}^{\left(w=\delta \right)}=1-\left(k-1\right){p}_{\mathrm{e}\left(\mathrm{G}\mathrm{O}\right)}^{\left(w=\delta \right)}=\sum \limits_{i=1}^k\overline{p_i^2}\cdotp $$

(A.1)

(ii) When k = 2, using $\overline{p}$ defined after (2.7) with $\overline{q}\equiv 1-\overline{p}$,

$$ {p}_{\mathrm{e}\;\left(\mathrm{S}\right)}^{\left(w=\delta \right)}=1-{p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{\left(w=\delta \right)}=1-2\overline{p}\;\overline{q}. $$

(A.2)

Proof. (i) The first set of equations of (A.1) is given by the definitions of (2.4) and (2.7) when ${w}_{i_1\;{i}_2}={\delta}_{i_1\;{i}_2}\;\left({i}_1,{i}_2=1,\dots, k\right)$ with ${\sum}_{i=1}^k{\overline{p}}_i=1$. The second set of equations of (A.1) is similarly given by the definitions of (2.5) and (2.9). (ii) When k = 2, it follows that ${p}_{\mathrm{e}\;\left(\mathrm{S}\right)}^{\left(w=\delta \right)}={\overline{p}}^2+{\left(1-\overline{p}\right)}^2={\overline{p}}^2+{\overline{q}}^2=1-2\overline{p}\;\overline{q}=1-{p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{\left(w=\delta \right)}$, where (2.8) is used with $\overline{q}=1-\overline{p}$. Q.E.D.

Lemma 1 shows the compensatory linear relationships between ${p}_{\mathrm{e}\;\left(\mathrm{S}\right)}^{\left(w=\delta \right)}$ and ${p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{\left(w=\delta \right)}$, and between ${p}_{\mathrm{e}\;\left(\mathrm{SO}\right)}^{\left(w=\delta \right)}$ and ${p}_{\mathrm{e}\;\left(\mathrm{GO}\right)}^{\left(w=\delta \right)}$ for arbitrary k ≥ 2 when m = 2. Then, we have the following results.

Theorem 1.

When m = 2,

$$ {\displaystyle \begin{array}{cc}\left(\mathrm{i}\right)& {\hat{K}}_{(s)}={p}_{\mathrm{o}}^{\left(w=\delta \right)}\end{array}}-\frac{{\left(1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}\right)}^2}{p_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}}\le {\hat{K}}_{\left(\mathrm{G}\right)}\kern0.5em for\ k=2. $$

(A.3)

$$ \left(\mathrm{ii}\right)\kern0.5em {\hat{K}}_{(S)}=1-\frac{\left(1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}\right)\left(1-{\hat{K}}_{\left(\mathrm{G}\right)}\right)}{\left(k-1\right)\left({p}_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}\right)}\kern0.5em {\displaystyle \begin{array}{cc}\mathrm{and}& {\hat{K}}_{\left(\mathrm{SO}\right)}=1-\frac{\left(1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}\right)\left(1-{\hat{K}}_{\left(\mathrm{G}\mathrm{O}\right)}\right)}{\left(k-1\right)\left({p}_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\mathrm{O}\right)}\right)}\end{array}} $$

(A.4)

for arbitrary k ≥ 2.

Proof. (i) The definition ${\hat{K}}_{\left(\mathrm{S}\right)}=1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{p}_{\mathrm{e}\left(\mathrm{S}\right)}^{\left(w=\delta \right)}}$ gives ${p}_{\mathrm{e}\left(\mathrm{S}\right)}^{\left(w=\delta \right)}=1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{S}\right)}}$ and similarly ${\hat{K}}_{\left(\mathrm{G}\right)}=1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{p}_{\mathrm{e}\left(\mathrm{G}\right)}^{\left(w=\delta \right)}}$ yields ${p}_{\mathrm{e}\left(\mathrm{G}\right)}^{\left(w=\delta \right)}=1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{G}\right)}}$. Using (A.2) in Lemma 1 and equating these results, we have: $1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{S}\right)}}=\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{G}\right)}}\leftrightarrow \frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{S}\right)}}=1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{G}\right)}}=\frac{p_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}}{1-{\hat{K}}_{\left(\mathrm{G}\right)}}\leftrightarrow \frac{1-{\hat{K}}_{\left(\mathrm{S}\right)}}{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}=\frac{1-{\hat{K}}_{\left(\mathrm{G}\right)}}{p_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}}=1+\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{p_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}}$ giving the equation in (A.3). For the inequality, using (A.2) we have:

$$ {p}_{\mathrm{e}\;\left(\mathrm{S}\right)}^{\left(w=\delta \right)}-{p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{\left(w=\delta \right)}=1-4\overline{p}\;\overline{q}={\left(\overline{p}+\overline{q}\right)}^2-4\overline{p}\;\overline{q}={\left(\overline{p}-\overline{q}\right)}^2\ge 0. $$

(A.5)

Then, the definition ${\hat{K}}_{\left(\cdotp \right)}=1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{p}_{\mathrm{e}\left(\cdotp \right)}^{\left(w=\delta \right)}}$ and (A.5) give the inequality in (A.3) when $1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}>0$. The equality in the inequality is obtained when ${p}_{\mathrm{e}\;\left(\mathrm{S}\right)}^{\left(w=\delta \right)}={p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{\left(w=\delta \right)}$ and $1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}>0$. When $1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}=0$, the definitions of ${\hat{K}}_{\left(\mathrm{S}\right)}$ and ${\hat{K}}_{\left(\mathrm{G}\right)}$ give ${\hat{K}}_{\left(\mathrm{S}\right)}={\hat{K}}_{\left(\mathrm{G}\right)}$, which also satisfies the equality in the inequality.

(ii) For the first equation of (A.4), the definition of ${p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{(w)}$ gives:

${p}_{\mathrm{e}\left(\mathrm{G}\right)}^{\left(w=\delta \right)}=1-\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{G}\right)}}=\frac{p_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}}{1-{\hat{K}}_{\left(\mathrm{G}\right)}}$. This equation and (A.1) give:

$1-{p}_{\mathrm{e}\left(\mathrm{S}\right)}^{\left(w=\delta \right)}=\left(k-1\right)\frac{p_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}}{1-{\hat{K}}_{\left(\mathrm{G}\right)}}$. Since $1-{p}_{\mathrm{e}\left(\mathrm{S}\right)}^{\left(w=\delta \right)}=\frac{1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}}{1-{\hat{K}}_{\left(\mathrm{S}\right)}}$, the last result yields $1-{\hat{K}}_{\left(\mathrm{S}\right)}=\frac{\left(1-{p}_{\mathrm{o}}^{\left(w=\delta \right)}\right)\left(1-{\hat{K}}_{\left(\mathrm{G}\right)}\right)}{\left(k-1\right)\left({p}_{\mathrm{o}}^{\left(w=\delta \right)}-{\hat{K}}_{\left(\mathrm{G}\right)}\right)}$. which gives the first equation of (A.4). The second equation of (A.4) is similarly obtained. Q.E.D.

Two additional inequalities are given.

Theorem 2.

When m = 2 and ${w}_{i_1\;{i}_2}={\delta}_{i_1\;{i}_2}\;\left({i}_1,{i}_2=1,\dots, k\right)$,

$$ \left(\mathrm{i}\right){\hat{K}}_{\left(\mathrm{S}\right)}\ge {\hat{K}}_{\left(\mathrm{S}\mathrm{O}\right)} $$

(A.6)

$$ and\ \left(\mathrm{ii}\right)\ {\hat{K}}_{\left(\mathrm{G}\right)}\le {\hat{K}}_{\left(\mathrm{G}\mathrm{O}\right)}. $$

(A.7)

Proof. (i) Under the conditions of this theorem, ${p}_{\mathrm{e}\;\left(\mathrm{S}\right)}^{(w)}={\sum}_{i=1}^k{\overline{p}}_i^2\le {\sum}_{i=1}^k\overline{p_i^2}=$ ${p}_{\mathrm{e}\;\left(\mathrm{SO}\right)}^{(w)}$ since ${\overline{p}}_i^2\le \overline{p_i^2}\;\left(i=1,\dots, k\right)$. Using the definitions of ${\hat{K}}_{\left(\mathrm{S}\right)}$ and ${\hat{K}}_{\left(\mathrm{SO}\right)}$ based on (2.1), (A.6) follows. (ii) As in (i) ${p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{(w)}=\left(1-{\sum}_{i=1}^k{\overline{p}}_i^2\right)/\left(k-1\right)\ge$ $\left(1-{\sum}_{i=1}^k\overline{p_i^2}\right)/\left(k-1\right)={p}_{\mathrm{e}\;\left(\mathrm{GO}\right)}^{(w)}$ follows. Then, (2.1) gives (A.7). Q.E.D.

1.3 The definitions of the chance-expected proportions for disagreement in the Gwet coefficient and its variation

In the cases of ${\hat{K}}_{\left(\mathrm{G}\right)}$ and ${\hat{K}}_{\left(\mathrm{GO}\right)}$, pairwise agreement is considered with ${w}_{i_1\;{i}_2}={\delta}_{i_1\;{i}_2}\;\left({i}_1,{i}_2=1,\dots, k\right)$. So it is natural to use w_ii (=1)  (i = 1, …, k) rather than v_ii  = 1 − w_ii( = 0)  (i = 1, …, k). However, when v_ij = 1 − w_ij (i, j = 1, …, k) are used for ${\hat{K}}_{\left(\mathrm{G}\right)}$, we require p_{e (G)  ij}(i, j = 1, …, k), which are not defined by Gwet (2008a) except for ${p}_{\mathrm{e}\;\left(\mathrm{G}\right)\kern0.24em ii}={\overline{p}}_i\left(1-{\overline{p}}_i\right)/\left(k-1\right)\kern0.24em \left(i=1,\dots, k\right)$. The undefined p_{e (G)  ij} (i, j = 1, …, k; i ≠ j) are defined as follows. Using the complementary relationship between ${p}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)}$ and ${p}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}$ by construction:

$$ {\displaystyle \begin{array}{c}{p}_{\mathrm{e}\left(\mathrm{G}\right)}^{(v)}=1-{p}_{\mathrm{e}\left(\mathrm{G}\right)}^{(v)}=1-\frac{1}{k-1}\sum \limits_{i,j=1}^k{w}_{ij}\left({\overline{p}}_i-{\overline{p}}_i^2\right)=1-\frac{1}{k-1}\sum \limits_{i,j=1}^k{\delta}_{ij}\left({\overline{p}}_i-{\overline{p}}_i^2\right)\\ {}=1-\frac{1}{k-1}+\frac{1}{k-1}\sum \limits_{l=1}^k{\overline{p}}_l^2=\frac{1}{k-1}\left(k-2+\sum \limits_{l=1}^k{\overline{p}}_l^2\right)=\sum \limits_{\begin{array}{c}i,j=1\\ {}i\ne j\end{array}}^k\frac{1}{k{\left(k-1\right)}^2}\left(k-2+\sum \limits_{l=1}^k{\overline{p}}_l^2\right)\\ {}=\sum \limits_{l=1}^k\frac{1-{\delta}_{ij}}{k{\left(k-1\right)}^2}\left(k-2+\sum \limits_{l=1}^k{\overline{p}}_l^2\right)=\sum \limits_{i,j=1}^k{v}_{ij}\frac{k-2+{\sum}_{l=1}^k{\overline{p}}_l^2}{k{\left(k-1\right)}^2},\end{array}} $$

which gives:

$$ {p}_{\mathrm{e}\;\left(\mathrm{G}\right)\kern0.24em ij}=\frac{k-2+{\sum}_{l=1}^k{\overline{p}}_l^2}{k{\left(k-1\right)}^2}\kern0.24em \left(i,j=1,\dots, k;i\ne j\right) $$

in addition to ${p}_{\mathrm{e}\;\left(\mathrm{G}\right)\kern0.24em ii}={\overline{p}}_i\left(1-{\overline{p}}_i\right)/\left(k-1\right)\kern0.24em \left(i=1,\dots, k\right)$. It is easily confirmed that ${\sum}_{i,j=1}^k{p}_{\mathrm{e}\;\left(\mathrm{G}\right)\kern0.24em ij}=1$. In the above additional definition, the same value is used for p_{e (G)  ij} (i, j = 1, …, k; i ≠ j). Different values can also be used, if necessary, as long as 0 ≤ p_{e (G)  ij} ≤ 1 (i, j = 1, …, k; i ≠ j) and ${\sum}_{\begin{array}{l}i,j=1\\ {}i\ne j\end{array}}^k{p}_{\mathrm{e}\;\left(\mathrm{G}\right)\kern0.24em ij}=$ $\left(k-2+{\sum}_{l=1}^k{\overline{p}}_l^2\right)/\left(k-1\right)$.

The above definition is introduced in the case of the unified treatment using v_ij (i, j = 1, …, k). However, for ${\hat{K}}_{\left(\mathrm{G}\right)}$, the formulation using w_ij ( = δ_ij ) (i = 1, …, k) may be more natural and simpler than using v_ij (i, j = 1, …, k). Note also that the additional definition is given from the complementary property ${p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{(w)}+{p}_{\mathrm{e}\;\left(\mathrm{G}\right)}^{(v)}=1$ without any theoretical basis using e.g., some random rating model as for p_{e (G)  ii}  (i = 1, …, k).

Similarly, for ${\hat{K}}_{\left(\mathrm{GO}\right)}$ using ${p}_{\mathrm{e}\;\left(\mathrm{GO}\right)}^{(w)}$ in (2.9), the following definition is given:

$$ {p}_{\mathrm{e}\left(\mathrm{GO}\right) ij}=\frac{k-2+{\sum}_{l=1}^k\overline{p_i^2}}{k{\left(k-1\right)}^2}\left(i,j=1,\dots, k;\mathrm{i}\ne j\right) $$

in addition to ${p}_{\mathrm{e}\left(\mathrm{GO}\right) ii}=\left({\overline{p}}_i-\overline{p_i^2}\right)/\left(k-1\right)\left(i=1,\dots, k\right)$.

1.4 Asymptotic results for ${\hat{K}}_{\left(\cdotp \right)}$

Define vec(·) as a vectorizing operator stacking the columns of a matrix sequentially with its first column on top.

Theorem 3.

Let ${\kappa}_{(j)}^{\ast}\left(\cdotp \right)$ be the j-th cumulant of a variable. The (local) asymptotic cumulants up to the fourth order and the added higher-order asymptotic variance of order O(n⁻²) in ${\kappa}_{(2)}^{\ast}\left(\cdotp \right)$ are given as follows:

$$ {\displaystyle \begin{array}{c}{\kappa}_1^{\ast}\left({\hat{K}}_{\left(\cdotp \right)}\right)=\mathrm{E}\left({\hat{K}}_{\left(\cdotp \right)}\right)={K}_{\left(\cdotp \right)}+\frac{n^{-1}}{2}\frac{\partial^2{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\pi} \prime \right)}^{<2>}}\mathrm{vec}\left(\boldsymbol{\Omega} \right)+O\left({n}^{-2}\right)\\ {}\equiv {K}_{\left(\cdotp \right)}+{n}^{-1}{\beta}_1+O\left({n}^{-2}\right),\end{array}} $$

(A.8)

$$ {\displaystyle \begin{array}{c}{\kappa}_2^{\ast}\left({\hat{K}}_{\left(\cdotp \right)}\right)=\mathrm{E}\left[{\left\{{\hat{K}}_{\left(\cdotp \right)}-\mathrm{E}\left({\hat{K}}_{\left(\cdotp \right)}\right)\right\}}^2\right]\\ {}={n}^{-1}{\left(\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\pi} \prime}\right)}^{<2>}\mathrm{vec}\left(\boldsymbol{\Omega} \right)+{n}^{-2}\left[\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\pi}^{\prime }}\otimes \frac{\partial^2{K}_{\left(\cdotp \right)}}{2{\left(\partial \boldsymbol{\pi} \prime \right)}^{<2>}}\mathrm{E}\left\{{\left(\mathbf{p}-\boldsymbol{\pi} \right)}^{<3>}\right\}\right.\\ {}\begin{array}{c}\left.+\frac{1}{2}\frac{\partial^2{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\pi} \prime \right)}^{<2>}}{\boldsymbol{\Omega}}^{<\mathbf{2}>}\frac{\partial^2{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\pi} \right)}^{<2>}}+\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\pi}^{\prime }}\otimes \frac{\partial^3{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\pi} \prime \right)}^{<3>}}{\left\{\mathrm{vec}\left(\boldsymbol{\Omega} \right)\right\}}^{<2>}\right]+O\left({n}^{-3}\right)\\ {}\equiv {n}^{-1}{\beta}_2+{n}^{-2}{\beta}_{\Delta 2}+O\left({n}^{-3}\right),\\ {}\begin{array}{c}{\kappa}_3^{\ast}\left({\hat{K}}_{\left(\cdotp \right)}\right)=\mathrm{E}\left[{\left\{{\hat{K}}_{\left(\cdotp \right)}-\mathrm{E}\left({\hat{K}}_{\left(\cdotp \right)}\right)\right\}}^3\right]\\ {}={n}^{-2}\left\{{\left(\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\pi} \prime}\right)}^{<3>}{n}^2{\boldsymbol{\kappa}}_3\left(\mathbf{p}\right)+3\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\pi}^{\prime }}\boldsymbol{\Omega} \frac{\partial^2{K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\pi} \partial \boldsymbol{\pi}^{\prime }}\boldsymbol{\Omega} \frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\pi}}\right\}+O\left({n}^{-3}\right)\\ {}\equiv {n}^{-2}{\beta}_3+O\left({n}^{-3}\right),\end{array}\end{array}\end{array}} $$

$$ {\displaystyle \begin{array}{c}{\kappa}_4^{\ast}\left({\hat{K}}_{\left(\cdotp \right)}\right)=\mathrm{E}\left[{\left\{{\hat{K}}_{\left(\cdotp \right)}-\mathrm{E}\left({\hat{K}}_{\left(\cdotp \right)}\right)\right\}}^4\right]-3{\left\{\mathrm{E}\left[{\left\{{\hat{K}}_{\left(\cdotp \right)}-\mathrm{E}\left({\hat{K}}_{\left(\cdotp \right)}\right)\right\}}^2\right]\right\}}^2\\ {}\equiv {n}^{-3}\underset{\left(\mathrm{A}\right)}{\left[\begin{array}{c}\\ {}\end{array}\right.}{\left(\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\uppi} \prime}\right)}^{<4>}{n}^3{\boldsymbol{\upkappa}}_4\left(\mathbf{p}\right)+2\frac{\partial^2{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\uppi} \prime \right)}^{<2>}}\otimes {\left(\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\uppi} \prime}\right)}^{<3>}\sum \limits^{10}\mathrm{vec}\left(\boldsymbol{\Omega} \right)\otimes {n}^2{\boldsymbol{\upkappa}}_3\left(\mathbf{p}\right)\\ {}+\left[\frac{3}{2}{\left\{\frac{\partial^2{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\uppi} \prime \right)}^{<2>}}\right\}}^{<2>}+\frac{2}{3}\frac{\partial^3{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\uppi} \prime \right)}^{<3>}}\otimes \frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\uppi}^{\prime }}\right]\otimes {\left(\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\uppi} \prime}\right)}^{<2>}\sum \limits^{15}{\left\{\mathrm{vec}\left(\boldsymbol{\Omega} \right)\right\}}^{<3>}\\ {}-\left(4{\beta}_1{\beta}_3+6{\beta}_2{\beta}_{\Delta 2}+6{\beta}_1^2{\beta}_2\right)\underset{\left(\mathrm{A}\right)}{\left.\begin{array}{c}\\ {}\end{array}\right]}+O\left({n}^{-4}\right)\\ {}\equiv {n}^{-3}{\beta}_4+O\left({n}^{-4}\right),\end{array}} $$

where $\underset{\left(\mathrm{A}\right)}{\Big[}\cdotp \underset{\left(\mathrm{A}\right)}{\Big]}$ is for ease of finding correspondence; $\overset{10}{\Sigma}$ is the sum of 10 similar terms considering combinations with $\overset{15}{\Sigma}$ defined similarly; and κ_i(p) is the k^i m × 1 vector of multivariate i-th cumulants of p (i = 3, 4), respectively.

Proof. The results are given from (4.1) using the definitions of cumulants, which are given by Ogasawara (2010, Section 2; 2020a, Theorem 1 with errata in 2019). Q.E.D.

Theorem 4.

The (local) asymptotic cumulants of t up to order O(n⁻¹) are given by:

$$ {\displaystyle \begin{array}{c}{\kappa}_1^{\ast }(t)={n}^{-1/2}\left({\beta}_2^{-1/2}{\beta}_1-\frac{1}{2}{\beta}_2^{-3/2}\frac{\partial {\beta}_2}{\partial \boldsymbol{\uppi}^{\prime }}\boldsymbol{\Omega} \frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\uppi}}\right)+O\left({n}^{-3/2}\right)\\ {}\equiv {n}^{-1/2}{\beta}_1\hbox{'}+O\left({n}^{-3/2}\right),\\ {}\kern-11.6em {\kappa}_2^{\ast }(t)=1+O\left({n}^{-1}\right)\left({\beta}_2^{\prime }=1\right),\\ {}{\kappa}_3^{\ast }(t)={n}^{-1/2}\left({\beta}_2^{-3/2}{\beta}_3-3{\beta}_2^{-3/2}\frac{\partial {\beta}_2}{\partial \boldsymbol{\uppi}^{\prime }}\boldsymbol{\Omega} \frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\uppi}}\right)+O\left({n}^{-3/2}\right)\\ {}\equiv {n}^{-1/2}{\beta}_3\hbox{'}+O\left({n}^{-3/2}\right).\end{array}}. $$

(A.9)

Proof. See Ogasawara (2020a, Theorem 2). Q.E.D.

For the computation of the asymptotic cumulants in Theorems 3 and 4, the partial derivatives of ${\hat{K}}_{\left(\cdotp \right)}$ with respect to p at π up to the third- and second-order are required, respectively. Let w and v be the k^m × 1 vectors corresponding to the weights ${w}_{i_1\cdots {i}_m}$ and ${v}_{i_1\cdots {i}_m}\left({i}_j=1,\dots, k;j=1,\dots, m\right)$, respectively. Then, we have ${p}_{\mathrm{o}}^{(w)}=\mathbf{w}\hbox{'}\mathbf{p}$ and ${p}_{\mathrm{o}}^{(v)}=\mathbf{v}\hbox{'}\mathbf{p}$, whose population counterparts are ${\pi}_{\mathrm{o}}^{(w)}=\mathbf{w}\hbox{'}\boldsymbol{\uppi}$ and ${\pi}_{\mathrm{o}}^{(v)}=\mathbf{v}\hbox{'}\boldsymbol{\uppi}$, respectively. Firstly, the partial derivatives for the single ratio version corresponding to ${\hat{K}}_{\left(\cdotp \right)}^{\left(R\;m\;\left(2,\dots, m\right)\right)}$ in (3.7) are obtained.

Lemma 3.

The partial derivatives of ${\hat{K}}_{\left(\cdotp \right)}^{\left(R\;m\;\left(2,\dots, m\right)\right)}$, synonymously denoted by ${\hat{K}}_{\left(\cdotp \right)}$ for simplicity, with respect to p evaluated at π up to the third order are:

$$ {\displaystyle \begin{array}{c}\frac{\partial {K}_{\left(\cdotp \right)}}{\partial \boldsymbol{\uppi}}=\frac{\partial }{\partial \boldsymbol{\uppi}}\left(1-{\Delta}_{\left(\cdotp \right)}\right)=\frac{\partial }{\partial \boldsymbol{\uppi}}\left(1-\frac{\pi_{\mathrm{o}}^{(v)}}{\pi_{\mathrm{e}\left(\cdotp \right)}^{(v)}}\right)=-\frac{\mathbf{v}}{\pi_{\mathrm{e}\left(\cdotp \right)}^{(v)}}+\frac{\pi_{\mathrm{o}}^{(v)}}{{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}},\\ {}\frac{\partial^2{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<2>}}=\frac{1}{{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}\left(\mathbf{v}\otimes \frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}+\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\otimes \mathbf{v}\right)-\frac{2{\pi}_{\mathrm{o}}^{(v)}}{{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^3}{\left(\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\right)}^{<2>}\end{array}}+\frac{\pi_{\mathrm{o}}^{(v)}}{{\left({\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}\right)}^2}\frac{\partial^2{\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<2>}}, $$

(A.10)

$$ \frac{\partial^3{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<3>}}=-\frac{2}{{\left({\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}\right)}^3}\left\{\mathbf{v}\otimes {\left(\frac{\partial {\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\right)}^{<2>}+\frac{\partial {\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\otimes \mathbf{v}\otimes \frac{\partial {\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}+{\left(\frac{\partial {\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\right)}^{<2>}\otimes \mathbf{v}\right\}+\frac{1}{{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}\left\{\mathbf{v}\otimes \frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<2>}}+\frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\otimes \mathbf{v}\otimes \frac{1}{\partial \boldsymbol{\uppi}}+\frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<2>}}\otimes \mathbf{v}\right\}+\frac{6{\pi}_{\mathrm{o}}^{(v)}}{{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^4}{\left(\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\right)}^{<3>}-\frac{2{\pi}_{\mathrm{o}}^{(v)}}{{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^3}\left\{\frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<2>}}\otimes \frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}+\frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\otimes \frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\otimes \frac{1}{\partial \boldsymbol{\uppi}}+\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\otimes \frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<2>}}\right\}+\frac{\pi_{\mathrm{o}}^{(v)}}{{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}\frac{\partial^3{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<3>}}, $$

where:

$$ {\displaystyle \begin{array}{c}\frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{{\left(\partial \boldsymbol{\uppi} \right)}_{i_1\cdots {i}_m}}\otimes {\left(\mathbf{v}\right)}_{j_1\cdots {j}_m}\otimes \frac{1}{{\left(\partial \boldsymbol{\uppi} \right)}_{k_1\cdots {k}_m}}=\frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial {\pi}_{i_1\cdots {i}_m}}{v}_{j_1\cdots {j}_m}\frac{1}{\partial {\pi}_{k_1\cdots {k}_m}}\\ {}=\frac{\partial^2{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial {\pi}_{i_1\cdots {i}_m}\partial {\pi}_{k_1\cdots {k}_m}}{v}_{j_1\cdots {j}_m}\left({i}_l,{j}_l,{k}_l=1,\dots, k,l=1,\dots, m\right)\end{array}} $$

(A.11)

using lexicographic or m-fold notations for the elements of vectors.

Proof. Direct derivation with $\partial {\pi}_{\mathrm{o}}^{(v)}/\partial \boldsymbol{\uppi} =\mathbf{v}$ gives (A.10). Q.E.D.

Using Lemma 3, we have the partial derivative of the generic expression of the coefficient given by the general formula of the weighted mean of weighted ratios (see (3.10)) as follows.

Lemma 4.

Let ${\hat{K}}_{\left(\cdotp \right)}$ be ${\hat{K}}_{\left(\cdotp \right)}^{\left( Lm\;\left(2,\dots, m\right)\right)}$ in (3.10). Define ${\hat{K}}_{\left(\cdotp \right)}^{\left(L\;m\;j\right)}=1-{\hat{\Delta}}_{\left(\cdotp \right)}^{\left(L\;m\;j\right)}\kern0.24em \left(j=2,\dots, m\right)$ (see (3.10)). Then, the partial derivatives of ${\hat{K}}_{\left(\cdotp \right)}$ up to the third order with respect to p at π are given by:

$$ \frac{\partial^i{K}_{\left(\cdotp \right)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<i>}}=\sum \limits_{j=2}^m{W}^{\ast (j)}\frac{\partial^i{K}_{\left(\cdotp \right)}^{\left(L\;m\;j\right)}}{{\left(\partial \boldsymbol{\uppi} \right)}^{<i>}}\kern0.24em \left(i=1,\kern0.36em 2,\kern0.36em 3\right). $$

(A.12)

Proof. Using ${\hat{K}}_{\left(\cdotp \right)}=1-\sum \limits_{j=2}^m{W}^{\ast (j)}{\hat{\Delta}}_{\left(\cdotp \right)}^{\left(L\;m\;j\right)}=\sum \limits_{j=2}^m{W}^{\ast (j)}{\hat{K}}_{\left(\cdotp \right)}^{\left(L\;m\;j\right)}$, we have (A.12). Q.E.D.

Among the asymptotic cumulants in Theorem 3, that of primary concern may be the asymptotic variance n⁻¹β₂ of ${\hat{K}}_{\left(\cdotp \right)}$, which is given by (A.10) or (A.12).

Corollary 1.

The asymptotic variances of ${\hat{K}}_{\left(\cdotp \right)}$’s in Lemmas 3 and 4 are given as follows:

$$ {\displaystyle \begin{array}{l}\begin{array}{l}n\mathrm{avar}\left({\hat{K}}_{\left(\cdotp \right)}^{\left( Rm\;\left(2,\dots, m\right)\right)}\right)=\frac{\partial {K}_{\left(\cdotp \right)}^{\left( Rm\;\left(2,\dots, m\right)\right)}}{\partial \boldsymbol{\pi}^{\prime }}n\;\mathrm{acov}\left\{\left({p}_{\mathrm{o}}^{(v)},{p}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}\right)^{\prime}\right\}\frac{\partial {K}_{\left(\cdotp \right)}^{\left( Rm\;\left(2,\dots, m\right)\right)}}{\partial \boldsymbol{\pi}}\\ {}=\frac{1}{{\left({\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}\right)}^4}\left({\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)},-{\pi}_{\mathrm{o}}^{(v)}\right)\left(\begin{array}{c}n \operatorname {var}\left(\mathbf{v}\hbox{'}\mathbf{p}\right)\kern1.32em n\;\mathrm{acov}\left\{\left(\mathbf{v}\hbox{'}\mathbf{p},{p}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}\right)\hbox{'}\right\}\\ {}n\;\mathrm{acov}\left\{\left({p}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)},\mathbf{v}\hbox{'}\mathbf{p}\right)\hbox{'}\right\}\kern0.48em n\;\mathrm{avar}\left({p}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}\right)\;\end{array}\right)\left(\begin{array}{c}{\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}\\ {}-{\pi}_{\mathrm{o}}^{(v)}\end{array}\right)\end{array}\\ {}=\frac{1}{{\left(1-{\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)}\right)}^4}\left\{1-{\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)},-\left(1-{\pi}_{\mathrm{o}}^{(w)}\right)\right\}\\ {}\times \left(\begin{array}{c}n \operatorname {var}\left(\mathbf{w}\hbox{'}\mathbf{p}\right)\kern1.32em n\;\mathrm{acov}\left(\mathbf{w}\hbox{'}\mathbf{p},{p}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)}\right)\\ {}n\;\mathrm{acov}\left({p}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)},\mathbf{w}\hbox{'}\mathbf{p}\right)\kern0.48em n\;\mathrm{avar}\left({p}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)}\right)\;\end{array}\right)\left(\begin{array}{c}1-{\pi}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)}\\ {}-\left(1-{\pi}_{\mathrm{o}}^{(w)}\right)\end{array}\right),\end{array}} $$

(A.13)

where:

$$ {\displaystyle \begin{array}{c}n\operatorname{var}\left(\mathbf{v}^{\prime}\mathbf{p}\right)=n\operatorname{var}\left(\mathbf{w}^{\prime}\mathbf{p}\right)=\mathbf{v}\hbox{'}\left\{\operatorname{diag}\left(\boldsymbol{\uppi} \right)-\boldsymbol{\uppi} \boldsymbol{\uppi}^{\prime}\right\}\mathbf{v}=\mathbf{v}\hbox{'}\operatorname{diag}\left(\boldsymbol{\uppi} \right)\mathbf{v}-{\left(\mathbf{v}\prime \boldsymbol{\uppi} \right)}^2,\\ {}n\mathrm{acov}\left(\mathbf{v}^{\prime}\mathbf{p},{p}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)=n\mathrm{acov}\left(\mathbf{w}^{\prime}\mathbf{p},{p}_{\mathrm{e}\left(\cdotp \right)}^{(w)}\right)=\mathbf{v}\hbox{'}\operatorname{diag}\left(\boldsymbol{\uppi} \right)\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}-\mathbf{v}\hbox{'}\boldsymbol{\uppi} \boldsymbol{\uppi} \hbox{'}\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}},\\ {}n\mathrm{avar}\left({p}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)=n\mathrm{avar}\left({p}_{\mathrm{e}\left(\cdotp \right)}^{(w)}\right)=\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}^{\prime }}\operatorname{diag}\left(\boldsymbol{\uppi} \right)\frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}-{\left(\boldsymbol{\uppi} \prime \frac{\partial {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}{\partial \boldsymbol{\uppi}}\right)}^2\cdotp \end{array}} $$

(A.14)

$$ n\mathrm{avar}\left({\hat{K}}_{\left(\cdotp \right)}^{\left( Lm\;\left(2,\dots, m\right)\right)}\right)=\sum \limits_{i=2}^m{W}^{\ast (i)}\frac{\partial {K}_{\left(\cdotp \right)}^{\left( Lm\;i\right)}}{\partial \boldsymbol{\uppi} \hbox{'}}\boldsymbol{\Omega} \sum \limits_{j=2}^m{W}^{\ast (j)}\frac{\partial {K}_{\left(\cdotp \right)}^{\left( Lm\;j\right)}}{\partial \boldsymbol{\uppi}}\cdotp $$

(A.15)

Proof. Using the exact covariance matrix n cov(p) = diag(π) − ππ', (A.13) and (A.15) follow. The first equation of the three sets of equations of (A.14) are due to v ' p = 1 − w ' p and ${p}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}=1-{p}_{\mathrm{e}\;\left(\cdotp \right)}^{(w)}$ (see Lemma 2). Q.E.D.

Theorem 5.

Denote two sets of _mC_m' values of ${p}_{\mathrm{o}}^{\left(v,m\hbox{'}\right)}\left({r}_1,\dots, {r}_{m\hbox{'}}\right)$ and ${p}_{\mathrm{e}\left(\cdotp \right)}^{\left(v,m\hbox{'}\right)}\left({r}_1,\dots, {r}_{m\hbox{'}}\right)$ in Definition 1 by ${p}_{i\;\mathrm{o}}^{(v)}$ and ${p}_{i\;\mathrm{e}\left(\cdotp \right)}^{(v)}\kern0.24em \left(i=1,\dots, G\right)$ with G =_mC_m'. Assume that the corresponding population values are the same over the G cases i.e., $\mathrm{E}\left({p}_{i\;\mathrm{o}}^{(v)}\right)={\pi}_{i\;\mathrm{o}}^{(v)}\equiv {\pi}_{\mathrm{o}}^{(v)}$ and ${\pi}_{i\;\mathrm{e}\left(\cdotp \right)}^{(v)}\equiv {\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\kern0.24em \left(i=1,\dots, G\right)$. Then, $\mathrm{avar}\left({\hat{K}}_{\left(\cdotp \right)}^{\left(R\;m\;m\hbox{'}\right)}\right)=\mathrm{avar}\left({\hat{K}}_{\left(\cdotp \right)}^{\left(M\;m\;m\hbox{'}\right)}\right)$.

Proof. Let ${\boldsymbol{\uppi}}_{\mathrm{o}}^{(v)}=\left({\pi}_{1\;\mathrm{o}}^{(v)},\dots, {\pi}_{G\;\mathrm{o}}^{(v)}\right)\hbox{'}$ and ${\boldsymbol{\uppi}}_{\mathrm{e}\;\left(\cdotp \right)}^{(v)}=\left({\pi}_{1\;\mathrm{e}\;\left(\cdotp \right)}^{(v)},\dots, {\pi}_{G\;\mathrm{e}\;\left(\cdotp \right)}^{(v)}\right)\hbox{'}$. Noting that:

$$ {\hat{K}}_{\left(\cdotp \right)}^{\left(R\;m\;m\hbox{'}\right)}=1-\frac{\sum_{i=1}^G{p}_{i\mathrm{o}}^{(v)}}{\sum_{i=1}^G{p}_{i\;\mathrm{e}\left(\cdotp \right)}^{(v)}}\kern1.1em \mathrm{and}\kern1.1em {\hat{K}}_{\left(\cdotp \right)}^{\left(M\;m\;m\hbox{'}\right)}=1-\frac{1}{G}\sum \limits_{i=1}^G\frac{p_{i\mathrm{o}}^{(v)}}{p_{i\;\mathrm{e}\left(\cdotp \right)}^{(v)}} $$

(A.16)

the following partial derivatives are obtained:

$$ \frac{\partial {K}_{\left(\cdotp \right)}^{\left( Rm m\prime \right)}}{\partial {\boldsymbol{\uppi}}_{\mathrm{o}}^{(v)}}=-\frac{{\mathbf{1}}_{(G)}}{\sum_{i=1}^G{\pi}_{i\mathrm{e}\left(\cdotp \right)}^{(v)}}=-\frac{{\mathbf{1}}_{(G)}}{G{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}},\frac{\partial {K}_{\left(\cdotp \right)}^{\left( Rm{m}^{\prime}\right)}}{\partial {\boldsymbol{\uppi}}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}=\frac{\sum_{i=1}^G{\pi}_{i\mathrm{o}}^{(v)}{\mathbf{1}}_{(G)}}{{\left({\sum}_{i=1}^G{\pi}_{i\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}=\frac{\pi_{\mathrm{o}}^{(v)}{\mathbf{1}}_{(G)}}{G{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}, $$

(A.16)

$$ {\displaystyle \begin{array}{c}\frac{\partial {K}_{\left(\cdotp \right)}^{\left( Mmm\prime \right)}}{\partial {\boldsymbol{\uppi}}_{\mathrm{o}}^{(v)}}=-\frac{1}{G}{\left(\frac{1}{\pi_{1\mathrm{e}\left(\cdotp \right)}^{(v)}},\dots, \frac{1}{\pi_{G\mathrm{e}\left(\cdotp \right)}^{(v)}}\right)}^{\prime }=-\frac{{\mathbf{1}}_{(G)}}{G{\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}},\\ {}\frac{\partial {K}_{\left(\cdotp \right)}^{\left( Mmm\prime \right)}}{\partial {\boldsymbol{\uppi}}_{\mathrm{e}\left(\cdotp \right)}^{(v)}}=\frac{1}{G}{\left\{\frac{\pi_{1\mathrm{o}}^{(v)}}{{\left({\pi}_{1\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2},\dots, \frac{\pi_{G\mathrm{o}}^{(v)}}{{\left({\pi}_{G\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}\right\}}^{\prime }=\frac{\pi_{\mathrm{o}}^{(v)}{\mathbf{1}}_{(G)}}{G{\left({\pi}_{\mathrm{e}\left(\cdotp \right)}^{(v)}\right)}^2}\cdotp \end{array}} $$

(A.17)

Since the above partial derivatives are the same over ${K}_{\left(\cdotp \right)}^{\left(R\;m\;m\hbox{'}\right)}$ and ${K}_{\left(\cdotp \right)}^{\left(M\;m\;m\hbox{'}\right)}$, the required identity follows. Q.E.D.

The asymptotic variance of ${\hat{K}}_{\left(\mathrm{C}\right)}$ when m ≥ 2 and its special case given by Fleiss et al. (1969) when m = 2.

In this subsection for ${\hat{K}}_{\left(\mathrm{C}\right)}$ with m ≥ 2, we use w rather than v without loss of generality for comparison to the special case with m = 2 given by Fleiss et al. (1969). The asymptotic variance of ${\hat{K}}_{\left(\mathrm{C}\right)}$ with m ≥ 2 is obtained from Corollary 1 and $\boldsymbol{\uppi} \hbox{'}\partial {\pi}_{\mathrm{e}\;\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\uppi} =m\;{\pi}_{\mathrm{e}\;\left(\mathrm{C}\right)}^{(w)}$ given by Ogasawara (2020b, Lemma S2):

$$ {\displaystyle \begin{array}{c}n\mathrm{avar}\left({\hat{K}}_{\left(\mathrm{C}\right)}\right)=\left(\frac{1}{1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}},-\frac{1-{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right)\left(\begin{array}{c}\mathbf{w}\hbox{'}\\ {}\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\uppi} \hbox{'}\end{array}\right)\left\{\mathit{\operatorname{diag}}\left(\boldsymbol{\uppi} \right)-\boldsymbol{\uppi} \boldsymbol{\uppi}^{\prime}\right\}\\ {}\times \left(\mathbf{w},\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\uppi} \right){\left(\frac{1}{1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}},-\frac{1-{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right)}^{\prime }=\left(\frac{1}{1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}},-\frac{1-{\pi}_{\mathrm{o}}^{(w)}}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right)\\ {}\times \left\{\left(\begin{array}{cc}\mathbf{w}^{\prime}\operatorname{diag}\left(\boldsymbol{\pi} \right)\mathbf{w}& \mathbf{w}\hbox{'}\operatorname{diag}\left(\boldsymbol{\pi} \right)\left(\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\pi} \right)\\ {}\left(\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial {\boldsymbol{\pi}}^{\prime}\right)\operatorname{diag}\left(\boldsymbol{\pi} \right){\mathbf{w}}^{\prime }& \left(\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\pi}^{\prime}\right) \operatorname {diag}\left(\boldsymbol{\pi} \right)\left(\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\pi} \right)\end{array}\right)\right.\\ {}-\left(\begin{array}{c}\mathbf{w}\hbox{'}\boldsymbol{\pi} \\ {}\left(\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\pi}^{\prime}\right)\boldsymbol{\pi} \end{array}\right)\left.\left(\mathbf{w}^{\prime}\boldsymbol{\pi}, \left(\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\pi}^{\prime}\right)\boldsymbol{\pi} \right)\right\}{\left(\frac{1}{1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}},-\frac{1-{\pi}_{\mathrm{o}}^{(w)}}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right)}^{\prime}\\ {}=\frac{1}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\left\{\mathbf{w}^{\prime}\operatorname{diag}\left(\boldsymbol{\pi} \right)\mathbf{w}-{\left({\pi}_{\mathrm{o}}^{(w)}\right)}^2\right\}\\ {}-2\frac{1-{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^3}\left\{\mathbf{w}^{\prime}\operatorname{diag}\left(\boldsymbol{\pi} \right)\frac{\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial \boldsymbol{\pi}}-m{\pi}_{\mathrm{o}}^{(w)}{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right\}\\ {}+\frac{{\left(1-{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}\right)}^2}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^4}\left\{\frac{\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial \boldsymbol{\pi}^{\prime }}\operatorname{diag}\left(\boldsymbol{\pi} \right)\frac{\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial \boldsymbol{\pi}}-{\left(m{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2\right\}\\ {}={\left\{\frac{\mathbf{w}}{1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}-\frac{\left(1-{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}\right)\left(\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\pi} \right)}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right\}}^{\prime}\operatorname{diag}\left(\boldsymbol{\pi} \right)\left\{\frac{\mathbf{w}}{1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}-\frac{\left(1-{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}\right)\left(\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\pi} \right)}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right\}\\ {}-\frac{1}{{\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^4}{\left\{{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}\left(1-{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)-m\left(1-{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}\right){\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right\}}^2\\ {}=\frac{1}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}{\left\{\mathbf{w}-\left(1-{K}_{\left(\mathrm{C}\right)}\right)\frac{\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial \boldsymbol{\pi}}\right\}}^{\prime}\operatorname{diag}\left(\pi \right)\left\{\mathbf{w}-\left(1-{K}_{\left(\mathrm{C}\right)}\right)\frac{\partial {\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial \boldsymbol{\pi}}\right\}\\ {}-\frac{1}{\left(1-{{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}^4\right)}{\left\{\left(m-1\right){\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}-m{\boldsymbol{\pi}}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}+{\boldsymbol{\pi}}_{\mathrm{o}}^{(w)}\right\}}^2\cdotp \end{array}} $$

(A.18)

On the other hand, when m = 2, Fleiss et al. (1969, Equation (8)) gave:

$$ {\displaystyle \begin{array}{c}n\mathrm{avar}\left({\hat{K}}_{\left(\mathrm{C}\right)}\right)=\frac{1}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^4}\left[\sum \limits_{i=1}^k\sum \limits_{j=1}^k{\pi}_{ij}{\left\{{w}_{ij}\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)-\left({\overline{w}}_{i\cdotp }+{\overline{w}}_{\cdotp j}\right)\left(1-{\pi}_{\mathrm{o}}^{(w)}\right)\right\}}^2\right.\\ {}\left.-{\left({\pi}_{\mathrm{o}}^{(w)}{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}-2{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}+{\pi}_{\mathrm{o}}^{(w)}\right)}^2\begin{array}{c}\\ {}\end{array}\right],\end{array}} $$

where their expressions of marginal weights are equal to the following ones using the notations of the current paper: ${\overline{w}}_{i\cdotp }=\sum \limits_{j=1}^k{w}_{ij}{\pi}_{\cdotp j}=\sum \limits_{j=1}^k{w}_{ij}{\pi}_j^{(2)}$ and ${\overline{w}}_{\cdotp j}=\sum \limits_{i=1}^k{w}_{ij}{\pi}_{i\cdotp }=\sum \limits_{i=1}^k{w}_{ij}{\pi}_i^{(1)}$.

In the above result, using ${\overline{w}}_{i\cdotp }+{\overline{w}}_{\cdotp j}=\partial \sum \limits_{a,b=1}^k{w}_{ab}{\pi}_{a\cdotp }{\pi}_{\cdotp b}/\partial {\pi}_{ij}=\partial {\pi}_{\mathrm{e}\;\left(\mathrm{C}\right)}^{(w)}/\partial {\pi}_{ij}$, it is found that their summarized expression becomes:

$$ {\displaystyle \begin{array}{c}\frac{1}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\sum \limits_{i=1}^k\sum \limits_{j=1}^k{\pi}_{ij}{w}_{ij}^2-2\frac{1-{\pi}_{\mathrm{o}}^{(w)}}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^3}\sum \limits_{i=1}^k\sum \limits_{j=1}^k{\pi}_{ij}{w}_{ij}\frac{\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial {\pi}_{ij}}\\ {}+\frac{{\left(1-{\pi}_{\mathrm{o}}^{(w)}\right)}^2}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^4}\sum \limits_{i=1}^k\sum \limits_{j=1}^k{\pi}_{ij}{\left(\frac{\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial {\pi}_{ij}}\right)}^2-\frac{{\left({\pi}_{\mathrm{o}}^{(w)}{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}-2{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}+{\pi}_{\mathrm{o}}^{(w)}\right)}^2}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^4}\\ {}={\left\{\frac{\mathbf{w}}{1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}-\frac{\left(1-{\pi}_{\mathrm{o}}^{(w)}\right)\left(\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\uppi} \right)}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right\}}^{\prime}\operatorname{diag}\left(\boldsymbol{\uppi} \right)\left\{\frac{\mathbf{w}}{1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}-\frac{\left(1-{\pi}_{\mathrm{o}}^{(w)}\right)\left(\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}/\partial \boldsymbol{\uppi} \right)}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}\right\}\\ {}-\frac{1}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^4}{\left({\pi}_{\mathrm{o}}^{(w)}{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}-2{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}+{\pi}_{\mathrm{o}}^{(w)}\right)}^2\\ {}=\frac{1}{{\left(1-{\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}\right)}^2}{\left\{\mathbf{w}-\left(1-{K}_{\left(\mathrm{C}\right)}\right)\frac{\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial \boldsymbol{\uppi}}\right\}}^{\prime}\operatorname{diag}\left(\boldsymbol{\uppi} \right)\left\{\mathbf{w}-\left(1-{K}_{\left(\mathrm{C}\right)}\right)\frac{\partial {\pi}_{\mathrm{e}\left(\mathrm{C}\right)}^{(w)}}{\partial \boldsymbol{\uppi}}\right\}\\ {}-\frac{1}{{\left(1-{\pi}_{\mathrm{e}\;\left(\mathrm{C}\right)}^{(w)}\right)}^4}{\left({\pi}_{\mathrm{o}}^{(w)}{\pi}_{\mathrm{e}\;\left(\mathrm{C}\right)}^{(w)}-2{\pi}_{\mathrm{e}\;\left(\mathrm{C}\right)}^{(w)}+{\pi}_{\mathrm{o}}^{(w)}\right)}^2,\end{array}} $$

which is found to be equal to the last expression of (A.18) when m = 2.