Let
$$\begin{aligned} {\varvec{x}}= & {} (x_{11}, x_{12}, \dots , x_{1r}, x_{21}, x_{22},\dots , x_{2r}, \dots , x_{r1}, x_{r2}, \dots , x_{rr})^{\prime }; {\ \mathrm an} {\ r^{2} \times 1} {\ \mathrm vector},\\ {\varvec{p}}= & {} (p_{11}, p_{12}, \dots , p_{1r}, p_{21}, p_{22},\dots , p_{2r}, \dots , p_{r1}, p_{r2}, \dots , p_{rr})^{\prime }; {\ \mathrm an} {\ r^{2} \times 1} {\ \mathrm vector}. \end{aligned}$$
We assume that
\({\varvec{x}}\) has a multinomial distribution Multi
\((n; {\varvec{p}})\) with sample size
n and probability vector
\({\varvec{p}}\). Then
\(\sqrt{n}(\hat{{\varvec{p}}}-{\varvec{p}})\) has asymptotically a normal distribution with zero mean and covariance matrix
\(\mathbf{Diag}({\varvec{p}})-{\varvec{pp}}^{\prime }\), where
\(\hat{{\varvec{p}}}={\varvec{x}}/n\) and
\(\mathbf{Diag}({\varvec{p}})\) is diagonal matrix with the elements of
\({\varvec{p}}\) on the main diagonal (see, e.g., Agresti
2013, p. 590). In order to estimate the indexes,
\(\hat{\varPhi }_{S}\) and
\(\hat{\varPhi }_{PS}\) are given by
\(\varPhi _{S}\) and
\(\varPhi _{PS}\) with
\(\{p_{ij}\}\) replaced by
\(\{\hat{p}_{ij}\}\), respectively. Therefore, the sample version of
\({\varvec{\varPsi }}\), i.e.,
\(\widehat{{\varvec{\varPsi }}}\), is given by
\({\varvec{\varPsi }}\) with
\(\varPhi _{S}\) and
\(\varPhi _{PS}\) replaced by
\(\hat{\varPhi }_{S}\) and
\(\hat{\varPhi }_{PS}\), respectively. Let
\((\partial {\varvec{\varPsi }}/\partial {\varvec{p}}^{\prime })\) denote the
\(2\times r^{2}\) matrix for which the entry in row
k and column
l is
\(\partial \varPsi _{k}({\varvec{p}})/\partial p_{l}\), where
\(\varPsi _{1}\) and
\(\varPsi _{2}\) denote
\(\varPhi _{S}\) and
\(\varPhi _{PS}\), respectively, and
\(p_{l}\) denotes the
lth element of
\({\varvec{p}}\). For
n approaching infinity, the estimated index vector can be approximated by
$$\begin{aligned} \widehat{{\varvec{\varPsi }}}={\varvec{\varPsi }}+\bigg (\frac{\partial {\varvec{\varPsi }}}{\partial {\varvec{p}}^{\prime }}\bigg )(\hat{{\varvec{p}}}-{\varvec{p}})+o(\parallel \hat{{\varvec{p}}}-{\varvec{p}}\parallel ), \end{aligned}$$
where
\(o(\parallel \hat{{\varvec{p}}}-{\varvec{p}}\parallel )\) tends to
\((0,0)^{\prime }\). Using the delta method (see Agresti
2013, Sect. 16.1),
\(\sqrt{n}(\widehat{{\varvec{\varPsi }}}-{\varvec{\varPsi }})\) has asymptotically a bivariate normal distribution with zero mean and covariance matrix
$$\begin{aligned} \varvec{\Sigma }= & {} \bigg (\frac{\partial {\varvec{\varPsi }}}{\partial {\varvec{p}}^{\prime }}\bigg )(\mathbf{Diag}({\varvec{p}})-{\varvec{pp}}^{\prime })\bigg (\frac{\partial {\varvec{\varPsi }}}{\partial {\varvec{p}}^{\prime }}\bigg )^{\prime }\\= & {} \left( \begin{array}{cc} \sigma _{11} &{}\quad \sigma _{12} \\ \sigma _{21} &{}\quad \sigma _{22}\\ \end{array} \right) , \end{aligned}$$
with
\(\sigma _{12}=\sigma _{21}\). The elements
\(\sigma _{11}\),
\(\sigma _{12}\) and
\(\sigma _{22}\) are expressed as follows:
$$\begin{aligned} \sigma _{11}= & {} \bigg (\frac{\partial \varPhi _{S}}{\partial {\varvec{p}}^{\prime }}\bigg )(\mathbf{Diag}({\varvec{p}})-{\varvec{pp}}^{\prime })\bigg (\frac{\partial \varPhi _{S}}{\partial {\varvec{p}}^{\prime }}\bigg )^{\prime }\\= & {} \frac{1}{\delta ^{2}}\bigg [\underset{i\ne j}{\sum \sum }p_{ij}\big (\varOmega _{ij}\big )^{2}-\delta \big (\varPhi _{S}\big )^{2}\bigg ],\\ \sigma _{12}= & {} \bigg (\frac{\partial \varPhi _{S}}{\partial {\varvec{p}}^{\prime }}\bigg )(\mathbf{Diag}({\varvec{p}})-{\varvec{pp}}^{\prime })\bigg (\frac{\partial \varPhi _{PS}}{\partial {\varvec{p}}^{\prime }}\bigg )^{\prime }\\= & {} \frac{1}{\delta \varDelta }\bigg [\underset{i \ne j}{\sum \sum }p_{ij}\big (\varOmega _{ij}-\varPhi _{S}\big )\big ( W_{ij}-\varPhi _{PS}\big )\bigg ],\\ \sigma _{22}= & {} \bigg (\frac{\partial \varPhi _{PS}}{\partial {\varvec{p}}^{\prime }}\bigg )(\mathbf{Diag}({\varvec{p}})-{\varvec{pp}}^{\prime })\bigg (\frac{\partial \varPhi _{PS}}{\partial {\varvec{p}}^{\prime }}\bigg )^{\prime }\\= & {} \frac{1}{\varDelta ^{2}}\bigg [\underset{(i, j)\in D}{\sum \sum }p_{ij}\big (W_{ij}\big )^{2}-\varDelta \big (\varPhi _{PS}\big )^{2}\bigg ], \end{aligned}$$
where
$$\begin{aligned} \varOmega _{ij}=\frac{1}{\log 2}\log \bigg (\frac{2p_{ij}}{p_{ij}+p_{ji}}\bigg ), \quad W_{ij}=\frac{1}{\log 2}\log \bigg (\frac{2p_{ij}}{p_{ij}+p_{r+1-i,r+1-j}}\bigg ). \end{aligned}$$
Note that the previous formulas for the asymptotic variances
\(\sigma _{11}\) and
\(\sigma _{22}\) of
\(\varPhi _{S}\) and
\(\varPhi _{PS}\), respectively, have been derived by Tomizawa (
1994) and Tomizawa et al. (
2007).
Therefore, an approximate bivariate
\(100(1-\alpha )\%\) confidence region for the index vector
\({\varvec{\varPsi }}\) is given by
$$\begin{aligned} n\big (\widehat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\big )^{\prime }\widehat{\varvec{\Sigma }}^{-1}\big (\widehat{{\varvec{\varPsi }}}-{\varvec{\varPsi }}\big )\le \chi ^{2}_{(1-\alpha ; 2)}, \end{aligned}$$
where
\(\chi ^{2}_{(1-\alpha ; 2)}\) is the
\(1-\alpha \) quantile of the chi-square distribution with two degrees of freedom and
\(\widehat{\varvec{\Sigma }}\) is given by
\(\Sigma \) with
\(\{p_{ij}\}\) replaced by
\(\{\hat{p}_{ij}\}\).