2.1.1. Threshold Estimation

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{ij}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} be the observed frequency and proportion, respectively, of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=i$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=j$$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,m_{U}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,\ldots ,m_{V}$$\end{document} . If the true underlying distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {F}$$\end{document} is different from the assumed distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {H}$$\end{document} , the MLEs of thresholds will be inconsistent estimators of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }_{0}=(\tau _{1,0},\ldots ,\tau _{m_{U}-1,0})'$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\xi }_{0}=(\xi _{1,0},\ldots ,\xi _{m_{V}-1,0})'$$\end{document} , where the subscript 0 indicates true values. Consider the ordinal variable U first. Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{n}_{U}=(n_{1\cdot },\ldots ,n_{m_{U}\cdot })'$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{i \cdot }=\sum _{j=1}^{m_V}n_{ij}$$\end{document} is the marginal total for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2,\ldots ,m_U$$\end{document} . The corresponding marginal proportion is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p}_{U}=(p_{1\cdot },\ldots ,p_{m_{U}\cdot })'$$\end{document} . The pseudo-maximum likelihood estimator (PMLE) of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }$$\end{document} , denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}=(\hat{\tau }_{1},\ldots ,\hat{\tau }_{m_{U}-1})'$$\end{document} , is obtained by maximizing

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textit{Q}(\varvec{\tau }) = \sum _{i=1}^{m_{U}} n_{i\cdot } \log \intop _{\tau _{i-1}}^{\tau _{i}}\text {h}_1(x)\mathrm{d}x \,. \end{aligned}$$\end{document}

It is easy to see that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}$$\end{document} is a consistent estimator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }^{*}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {H}_1 \left( \varvec{\tau }^{*} \right) =\text {F}_{1}(\varvec{\tau }_{0})$$\end{document} , because the observed cell probabilities are consistent estimators of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {F}_{1}(\varvec{\tau }_{0})$$\end{document} . Similarly, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\xi }}$$\end{document} is a consistent estimator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\xi }^{*}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {H}_2 \left( \varvec{\xi }^{*} \right) =\text {F}_{2}(\varvec{\xi }_{0})$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{P}$$\end{document} be an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{U}\times m_{V}$$\end{document} matrix with (i, j)-th entry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} . Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \text {Q} \left( \varvec{\tau } \right) }{\partial \varvec{\tau }}&= n\varvec{B}_U \left( \varvec{\tau } \right) '\varvec{D}_U^{-1} \left( \varvec{\tau } \right) \varvec{p}_{U}, \\ \frac{\partial ^2 \text {Q} \left( \varvec{\tau } \right) }{\partial \varvec{\tau }\partial \varvec{\tau }'}&= -n \varvec{B}_U \left( \varvec{\tau } \right) '\varvec{D}_U^{-1} \left( \varvec{\tau } \right) \varvec{D}_{\varvec{p}}\varvec{D}_U^{-1} \left( \varvec{\tau } \right) \varvec{B}_U \left( \varvec{\tau } \right) + n \varvec{S}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=\sum _{i=1}^{m_U}\sum _{j=1}^{m_V}n_{ij}$$\end{document} is the total number of observations,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{B}_U(\varvec{\tau })&=\left( \begin{array}{cccc} \text {h}_1(\tau _{1}) &{} 0 &{} \cdots &{} 0\\ -\text {h}_1(\tau _{1}) &{} \text {h}_1(\tau _{2}) &{} \cdots &{} 0\\ 0 &{} -\text {h}_1(\tau _{2}) &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} \text {h}_1(\tau _{m_{U}-1})\\ 0 &{} 0 &{} \cdots &{} -\text {h}_1(\tau _{m_{U}-1}) \end{array}\right) , \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{D}_U(\varvec{\tau })=\text {Diag}\left( \intop _{\tau _{0}}^{\tau _{1}}\text {h}_1(\textit{x})\mathrm{d}{} \textit{x},\ldots ,\intop _{\tau _{m_{U}-1}}^{\tau _{m_{U}}}\text {h}_1(\textit{x})\mathrm{d}{} \textit{x}\right) , \varvec{D}_{\varvec{p}}=\text {Diag}(p_{1\cdot },\ldots ,p_{m_{U}\cdot }), \varvec{p}_{U}=\varvec{P}\varvec{1}_{m_{V}}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{1}_{m_{V}}$$\end{document} being an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_V \times 1$$\end{document} vector of 1’s, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S}$$\end{document} is a diagonal matrix with i-th element

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \frac{p_{i,\cdot }}{\intop _{\tau _{i-1}}^{\tau _{i}}\text {h}_1(x)\mathrm{d}x}-\frac{p_{i+1,\cdot }}{\intop _{\tau _{i}}^{\tau _{i+1}}\text {h}_1(x)\mathrm{d}x}\right) \frac{\partial \text {h}_{1}(\tau _{i})}{\partial \tau _{i}} \,, \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \,=\, 1,\ldots ,m_{U}-1$$\end{document} . The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Diag}(\cdot )$$\end{document} constructs a diagonal matrix using the enclosed vector as diagonal elements. The Taylor expansion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{-1/2} \partial \textit{Q}(\hat{\varvec{\tau }})/\partial \varvec{\tau }$$\end{document} around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }^*$$\end{document} is

(1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0 \,=\,&n^{-1/2} \frac{\partial \textit{Q} \left( \hat{\varvec{\tau }} \right) }{\partial \varvec{\tau }} = n^{-1/2}\frac{\partial \textit{Q} \left( \varvec{\tau }^* \right) }{\partial \varvec{\tau }} + n^{-1/2} \frac{\partial ^2 \textit{Q} \left( \tilde{\varvec{\tau }} \right) }{\partial \varvec{\tau } \partial \varvec{\tau }' } \left( \hat{\varvec{\tau }}-\varvec{\tau }^{*} \right) +o_{p}( 1), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{\tau }}$$\end{document} lies between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }^*$$\end{document} . Because both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{\tau }}$$\end{document} are consistent estimators of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }^*, n^{-1} \partial ^2 \textit{Q}(\tilde{\varvec{\tau }})/\partial \varvec{\tau } \partial \varvec{\tau }'$$\end{document} is consistent for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\varvec{B}_U \left( \varvec{\tau }^{*} \right) '\varvec{D}_U^{-1} \left( \varvec{\tau }^{*} \right) \varvec{B}_U \left( \varvec{\tau }^{*} \right) $$\end{document} . So, Eq. (1) implies

(2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2} \left( \hat{\varvec{\tau }}-\varvec{\tau }^{*} \right) \,=\,&n^{1/2} \left[ \varvec{B}_U \left( \varvec{\tau }^{*} \right) '\varvec{D}_U^{-1} \left( \varvec{\tau }^{*} \right) \varvec{B}_U \left( \varvec{\tau }^{*} \right) \right] ^{-1} \varvec{B}_U \left( \varvec{\tau }^{*} \right) '\varvec{D}_U^{-1} \left( \varvec{\tau }^{*} \right) \varvec{p}_U+o_{p}(1). \end{aligned}$$\end{document}

Similar arguments applying to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\xi }$$\end{document} yield

(3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2} \left( \hat{\varvec{\xi }}-\varvec{\xi }^{*} \right) =&n^{1/2} \left[ \varvec{B}_V \left( \varvec{\xi }^{*} \right) '\varvec{D}_V^{-1} \left( \varvec{\xi }^{*} \right) \varvec{B}_V \left( \varvec{\xi }^{*} \right) \right] ^{-1} \varvec{B}_V \left( \varvec{\xi }^{*} \right) '\varvec{D}_V^{-1} \left( \varvec{\xi }^{*} \right) \varvec{p}_V+o_{p}(1), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p}_V=\varvec{P}'\varvec{1}_{m_{U}}$$\end{document} . Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B}_V$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{D}_V$$\end{document} are defined by substituting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {h}_1$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {h}_2$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B}_U$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{D}_U$$\end{document} . The PMLEs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\xi }}$$\end{document} are inconsistent in the sense that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }^{*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\xi }^{*}$$\end{document} are different from the true values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }_{0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\xi }_{0}$$\end{document} .

2.1.2. Polychoric Correlation Coefficient Estimation

Under the distributional assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {H}$$\end{document} , the assumed cell probability is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi _{ij,(\text {H})}={\int }_{\tau _{i-1}}^{\tau _{i}}{\int }_{\xi _{j-1}}^{\xi _{j}}\text {h}(x,y)\mathrm{d}y\mathrm{d}x, \end{aligned}$$\end{document}

while the true cell probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,(\text {F})}$$\end{document} is obtained by substituting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {h}(x,y)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {f}(x,y)$$\end{document} . Conditionally on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\xi }}$$\end{document} , the polychoric correlation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} is estimated by maximizing

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {L} \left( \rho ,\hat{\varvec{\tau }},\hat{\varvec{\xi }} \right) =\sum _{i=1}^{m_{U}}\sum _{j=1}^{m_{V}}p_{ij}\log \,\pi _{ij,(\text {H})}. \end{aligned}$$\end{document}

Theorem 2.2 in White (Reference White1982) shows that the PMLE is a consistent estimator that minimizes the Kullback–Leibler information (Kullback & Leibler, Reference Kullback and Leibler1951) under some regularity conditions, one of which is that the absolute value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log \pi _{ij,(\text {H})} $$\end{document} is dominated by a variable with finite expectation. Such a regularity condition is satisfied if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,(\text {H})}=0$$\end{document} implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,(\text {F})}=0$$\end{document} for all (i, j). Consequently, Theorem 2.2 in White (Reference White1982) shows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }$$\end{document} converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^{*}$$\end{document} that minimizes the Kullback–Leibler information

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{i=1}^{m_{U}}\sum _{j=1}^{m_{V}} \pi _{ij,(\text {F})} \left( \log \,\pi _{ij,(\text {F})} - \log \,\pi _{ij,(\text {H})} \right) . \end{aligned}$$\end{document}

Theorem 1

Assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {g} \left( \rho ,\varvec{\tau }^{*},\varvec{\xi }^{*} \right) = \sum _{i=1}^{m_{U}}\sum _{j=1}^{m_{V}} \pi _{ij,(\text {F})}\log \,\pi _{ij,(\text {H})}$$\end{document} , as a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} , has a unique maximum at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^*$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,(\text {H})}=0$$\end{document} implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,(\text {F})}=0$$\end{document} for all (i, j), then there exists a root \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }$$\end{document} of the equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial }{\partial \rho } \sum _{i=1}^{m_{U}}\sum _{j=1}^{m_{V}}p_{ij}\log \,\pi _{ij,(\text {H})}=0 \end{aligned}$$\end{document}

such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }$$\end{document} is a consistent estimator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^{*}$$\end{document} .

That is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }$$\end{document} is a consistent estimator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^{*}$$\end{document} that minimizes the probabilistic divergence between H and F (Kullback, Reference Kullback1959) in the sense of the Kullback–Leibler information. This minimized divergence implies similarities of H and F in terms of cell probabilities.

The assumption in Theorem 1 requires uniqueness of the maximum. In so doing, we rule out all cases with local maxima. If we have several stationary points, we can then only conclude that one of the stationary points minimizes the Kullback–Leibler information.

2.1.3. Asymptotic Variance of Polychoric Correlations

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {L}_{\rho } \left( \rho ,\varvec{\tau },\varvec{\xi } \right) $$\end{document} denote the first order partial derivative of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {L} \left( \rho ,\varvec{\tau },\varvec{\xi } \right) $$\end{document} with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} . Similar symbols are used to represent other partial derivatives and higher order partial derivatives. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {L}_{\rho } \left( \hat{\rho },\hat{\varvec{\tau }},\hat{\varvec{\xi }} \right) $$\end{document} can be expanded around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^{*}$$\end{document} for a sufficiently large n,

(4) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0= n^{1/2} \text {L}_{\rho }\left( \hat{\rho },\hat{\varvec{\tau }},\hat{\varvec{\xi }} \right) = \underset{\left( i\right) }{\underbrace{n^{1/2} \text {L}_{\rho }\left( \rho ^{*},\hat{\varvec{\tau }},\hat{\varvec{\xi }} \right) }}+\underset{\left( ii\right) }{\underbrace{ n^{1/2} \left( \hat{\rho }-\rho ^{*}\right) \text {L}_{\rho \rho }\left( \tilde{\rho },\hat{\varvec{\tau }},\hat{\varvec{\xi }} \right) }}+o_{p}(1), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\rho }$$\end{document} lies between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^*$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }$$\end{document} . Term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( i\right) $$\end{document} in Eq. (4) is equivalent to

(5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2}\text {L}_{\rho }\left( \rho ^{*},\hat{\varvec{\tau }},\hat{\varvec{\xi }}\right) \,=\,&n^{1/2}\text {L}_{\rho }\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) + n^{1/2} \text {L}_{\rho \varvec{\tau }} \left( \rho ^{*},\tilde{\varvec{\tau }},\tilde{\varvec{\xi }}\right) '\left( \hat{\varvec{\tau }}-\varvec{\tau }^{*}\right) \nonumber \\&+n^{1/2} \text {L}_{\rho \varvec{\xi }} \left( \rho ^{*},\tilde{\varvec{\tau }},\tilde{\varvec{\xi }}\right) '\left( \hat{\varvec{\xi }}-\varvec{\xi }^{*}\right) +o_{p}(1), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{\tau }}$$\end{document} lies between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\tau }^{*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{\xi }}$$\end{document} lies between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\xi }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\xi }^{*}$$\end{document} . Hence, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,\left( \text {H}\right) }=0$$\end{document} implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,\left( \text {F}\right) }=0$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( i,j\right) $$\end{document} in a neighbourhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varvec{\tau }^{*},\varvec{\xi }^{*}) $$\end{document} given the correlation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^{*}, \text {L}_{\rho \varvec{\tau }}\left( \rho ^{*},\tilde{\varvec{\tau }},\tilde{\varvec{\xi }}\right) $$\end{document} is consistent for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {g}_{\rho \varvec{\tau }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {L}_{\rho \varvec{\xi }}\left( \rho ^{*},\tilde{\varvec{\tau }},\tilde{\varvec{\xi }}\right) $$\end{document} is consistent for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {g}_{\rho \varvec{\xi }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) $$\end{document} . Thus, Eq. (5) is equivalent to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2}\text {L}_{\rho }\left( \rho ^{*},\hat{\varvec{\tau }},\hat{\varvec{\xi }}\right)&=n^{1/2}\text {L}_{\rho }\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) +n^{1/2} \text {g}_{\rho \varvec{\tau }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) '\left( \hat{\varvec{\tau }}-\varvec{\tau }^{*}\right) \nonumber \\&+n^{1/2} \text {g}_{\rho \varvec{\xi }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) '\left( \hat{\varvec{\xi }}-\varvec{\xi }^{*}\right) +o_{p}(1). \end{aligned}$$\end{document}

Likewise, Term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( ii\right) $$\end{document} can be written as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2}\left( \hat{\rho }-\rho ^{*}\right) \text {L}_{\rho \rho }\left( \tilde{\rho },\hat{\varvec{\tau }},\hat{\varvec{\xi }}\right) = n^{1/2}\left( \hat{\rho }-\rho ^{*}\right) \text {g}_{\rho \rho }\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) +o_{p}(1), \end{aligned}$$\end{document}

provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,\left( \text {H}\right) }=0$$\end{document} implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,\left( \text {F}\right) }=0$$\end{document} in a neighbourhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \rho ^*,\varvec{\tau }^{*},\varvec{\xi }^{*}\right) $$\end{document} . Hence, combining with Eqs. (2) and (3), (4) is equivalent to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2}\left( \hat{\rho }-\rho ^{*}\right) =&-\frac{n^{1/2}}{ \text {g}_{\rho \rho }\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) } \left\{ \text {L}_{\rho }\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) +\, \text {g}_{\rho \varvec{\tau }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) ' \left( \hat{\varvec{\tau }}-\varvec{\tau }^{*}\right) \right. \nonumber \\&\quad \left. +\, \text {g}_{\rho \varvec{\xi }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) ' \left( \hat{\varvec{\xi }}-\varvec{\xi }^{*}\right) \right\} +o_{p}\left( 1\right) \nonumber \\ =&- \frac{n^{1/2}}{ \text {g}_{\rho \rho }\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) } \text {tr} \left( \varvec{\Lambda }'\varvec{P} \right) +o_{p}(1), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Lambda }=\varvec{A}+\varvec{E}_{\tau }\text {g}_{\rho \varvec{\tau }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) \varvec{1}_{m_{V}}'+\varvec{1}{}_{m_{U}}\text {g}_{\rho \varvec{\xi }}\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) '\varvec{E}'_{\xi }$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{A}$$\end{document} being an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_U\times m_V$$\end{document} matrix with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( i,j\right) $$\end{document} -th element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \partial \pi _{ij}/\partial \rho \right) /\pi _{ij}, \varvec{E}_{\tau } =\varvec{D}_{\tau }^{-1}\varvec{B}_{\tau }\left( \varvec{B}_{\tau }'\varvec{D}_{\tau }^{-1}\varvec{B}_{\tau }\right) ^{-1}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E}_{\xi } =\varvec{D}_{\xi }^{-1}\varvec{B}_{\xi }\left( \varvec{B}_{\xi }'\varvec{D}_{\xi }^{-1}\varvec{B}_{\xi }\right) ^{-1}$$\end{document} . Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {tr}\left( \varvec{\Lambda }'\varvec{P} \right) = \mathrm {vec}{ (\varvec{\Lambda })}' \mathrm {vec}{(\varvec{P})}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {vec}(\cdot )$$\end{document} stacks the columns of the enclosed matrix and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sqrt{n} \left( \mathrm {vec}{(\varvec{P})} - \mathrm {vec}(\varvec{\pi }_{\left( \text {F}\right) }) \right) \overset{d}{\rightarrow } N \left( \varvec{0}, \text {Diag}(\varvec{P}) - \mathrm {vec}{(\varvec{P})} \mathrm {vec}{(\varvec{P})}' \right) \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\pi }_{\left( \text {F}\right) }$$\end{document} being an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{U}\times m_{V}$$\end{document} matrix with (i, j)-th entry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,\left( \text {F}\right) }$$\end{document} . The arguments above establish the following theorem.

Theorem 2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }$$\end{document} be the consistent root of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {L}_{\rho }\left( \rho ,\hat{\varvec{\tau }},\hat{\varvec{\xi }}\right) =0$$\end{document} given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\tau }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\xi }}$$\end{document} . Assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,\left( \text {H}\right) }=0$$\end{document} implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{ij,\left( \text {F}\right) }=0$$\end{document} in a neighbourhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \rho ^*,\varvec{\tau }^{*},\varvec{\xi }^{*}\right) $$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2}\left( \hat{\rho }-\rho ^{*}\right)&\overset{d}{\rightarrow }N\left( 0,\sigma ^{2}\right) , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^{2}=\left[ \text {tr} \left( \left( \varvec{\Lambda }\odot \varvec{\Lambda } \right) '\varvec{\pi }_{\left( \text {F}\right) } \right) -\left( \text {tr}\left( \varvec{\Lambda }'\varvec{\pi }_{\left( \text {F}\right) } \right) \right) ^{2} \right] / \left( \text {g}_{\rho \rho } \left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*} \right) \right) ^{2}$$\end{document} . Here, matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Lambda }$$\end{document} is evaluated under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*}\right) $$\end{document} . The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\odot $$\end{document} implies element-wise multiplication.

2.1.4. Estimating the Asymptotic Covariance Matrix

Following Theorem 2, the asymptotic variance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }$$\end{document} can be consistently estimated by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{\left[ \text {L}_{\rho \rho }\left( \hat{\rho },\hat{\varvec{\tau }},\hat{\varvec{\xi }}\right) \right] ^{2} } \left[ \sum _{i=1}^{m_U}\sum _{j=1}^{m_V} \hat{\lambda }_{ij}^2 p_{ij} - \left( \sum _{i=1}^{m_U}\sum _{j=1}^{m_V} \hat{\lambda }_{ij} p_{ij} \right) ^2 \right] , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\lambda }_{ij}$$\end{document} is the (i, j)-th element in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\Lambda }}$$\end{document} . The polychoric correlation between variables U and V satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2}\left( \hat{\rho }^{(UV)}-\rho ^{*(UV)}\right)&=\frac{-n^{1/2}}{ \text {g}_{\rho \rho }^{(UV)} \left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*} \right) }\text {tr} \left( \varvec{\Lambda }^{\left( UV\right) '}\varvec{P}^{\left( UV\right) } \right) +o_{p}\left( 1\right) , \end{aligned}$$\end{document}

where the superscript (UV) emphasizes that all quantities are evaluated under the distributional assumption for U and V. Similarly, the polychoric correlation between variables K and Z satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n^{1/2}\left( \hat{\rho }^{(KZ)}-\rho ^{*(KZ)}\right)&=\frac{-n^{1/2}}{ \text {g}_{\rho \rho }^{(KZ)} \left( \rho ^{*},\varvec{\tau }^{*},\varvec{\xi }^{*} \right) }\text {tr} \left( \varvec{\Lambda }^{\left( KZ\right) '}\varvec{P}^{\left( KZ\right) } \right) +o_{p}\left( 1\right) . \end{aligned}$$\end{document}

The underlying distributional assumption for U and V can either be the same as that for K and Z or different. Thus, the asymptotic covariance between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }^{(UV)}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }^{(KZ)}$$\end{document} is consistently estimated by

(6) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\sum _{a=1}^{m_{U}}\sum _{b=1}^{m_{V}}\sum _{c=1}^{m_{K}}\sum _{d=1}^{m_{Z}} \hat{\lambda }_{ab}^{\left( UV\right) } \left( p_{abcd}^{\left( UVKZ\right) }-p_{ab}^{\left( UV\right) }p_{cd}^{\left( KZ\right) } \right) \hat{ \lambda }_{cd}^{\left( KZ\right) }}{\text {L}_{\rho \rho }\left( \hat{\rho }^{(UV)},\hat{\varvec{\tau }}^{\left( UV\right) },\hat{\varvec{\xi }}^{\left( UV\right) }\right) \text {L}_{\rho \rho }\left( \hat{\rho }^{(KZ)},\hat{\varvec{\tau }}^{\left( KZ\right) },\hat{\varvec{\xi }}^{\left( KZ\right) }\right) } \,, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{abcd}^{(UVKZ)}$$\end{document} is the sample proportion of observing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=a, V=b, K=c$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z=d$$\end{document} . Under the assumption that the underlying distribution is normal and correctly specified, Eq. (6) reduces to the estimator in Jöreskog (Reference Jöreskog1994).

3.1.1. Experiment 1: Elliptical Distribution

In probability and statistics, an elliptical distribution belongs to a broad family of probability distributions. The bivariate joint density function of an elliptical distribution is of the form

(11) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{2\pi \sigma _{11}\sigma _{22} \left( 1-\rho ^2 \right) ^{1/2} } \text {q} \left( z \right) , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {q} \left( \cdot \right) $$\end{document} is a univariate function and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} z = \frac{1}{1-\rho ^{2}}\left[ \frac{(x-\mu _{1})^{2}}{\sigma _{11}^2}-\frac{2\rho (x-\mu _{1})(y-\mu _{2})}{ \sigma _{11}\sigma _{22} }+\frac{(y-\mu _{2})^{2}}{\sigma _{22}^2}\right] \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{11}$$\end{document} being the variance of X and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{22}$$\end{document} being the variance of Y. An elliptical distribution generalizes the normal distribution and keeps some properties (e.g. Balakrishnan & Lai, Reference Balakrishnan and Lai2009; Fang, Kotz, & Ng, Reference Fang, Kotz and Ng1990; Kelker, Reference Kelker1970). Some examples of the bivariate elliptical distributions that will be used later are

1. 1. Normal distributions: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\left( z \right) = \exp \left( -z/2 \right) $$\end{document} ;

2. 2. t(v) distributions with degrees of freedom v: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\left( z \right) = \left( 1+z/v \right) ^{-(v+2)/2}$$\end{document} ;

3. 3. Bivariate uniform distributions: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\left( z \right) = 2\text {I}_{ \{z\le 1\} }$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {I}$$\end{document} being an indicator function;

4. 4. Bivariate Logistic distributions: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\left( z \right) = 4\exp \left( -z \right) /\left[ 1+\exp \left( -z \right) \right] ^2$$\end{document} ;

5. 5. Bivariate exponential power distributions: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\left( z \right) = 2\exp \left( -z^{\beta }/2 \right) /\left( 2^{1/\beta } \Gamma (1+1/\beta )\right) $$\end{document} .

The elliptical distribution family plays a very important role in robustness studies (e.g. Kano, Berkane, & Bentler, Reference Kano, Berkane and Bentler1993). In the context of Pearson correlation estimation, Hampel, Ronchetti, Rousseeuw, and Stahel (Reference Hampel, Ronchetti, Rousseeuw and Stahel1986) showed that the PMLE of the covariance matrix is proportional to the MLE under the true distributional assumption, provided that continuous data have been acquired. This result enables us to use any member of the family to estimate the correlation matrix, having the same estimates as if the true distribution were used. Likewise, Berkane, Kano, and Bentler (Reference Berkane, Kano and Bentler1994) claimed that “there is practically no cost in treating the distribution as multivariate t with specified (possibly small) degrees of freedom” (Berkane et al., Reference Berkane, Kano and Bentler1994, p. 266) when the true distribution is normal and continuous data are observed. It only slightly inflates the variance of the resulting estimator. Thus, it is worth investigating the effect of an underlying elliptical distribution. Because we have only categorical data, the mean and variance are not identified. But then only the correlation coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} is the parameter of interest, so we can assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{1}=\mu _{2}=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{11}=\sigma _{22}=1$$\end{document} .

For some members of the elliptical distribution family, the marginal distribution is still elliptical but not of the same type (Gómez, Gómez-villegas, & Marín, Reference Gómez, Gómez-villegas and Marín2003). The bivariate uniform distribution, the logistic distribution, and the exponential power distribution possess such properties. The support of the bivariate uniform distribution is not the whole Cartesian plane, whereas the other distributions have the whole Cartesian plane as their support. The exponential power distribution includes the normal distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta =1)$$\end{document} and the Laplace distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta =1/2)$$\end{document} as special cases.

3.1.2. Experiment 2: Skew-Normal Distribution

An elliptical distribution is symmetric. Qui-roga (Reference Quiroga1992) reported that kurtosis does not have strong effects on the polychoric correlation but that skewness increases the bias. The above elliptical distributions examine various values of kurtosis. The following distributions introduce nonzero values of skewness.

A natural generalization of a standard normal distribution is the univariate skew-normal distribution proposed by Azzalini (Reference Azzalini1985) and extended by Azzalini and Valle (Reference Azzalini and Valle1996) to a multivariate skew-normal distribution. The bivariate density function, covariance matrix and marginal density function are shown in Eqs. (8), (9), and (10), respectively. The ranges of the skewness and excess kurtosis are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-0.9953,0.9953)$$\end{document} and [0, 0.8692), respectively (Azzalini & Capitanio, Reference Azzalini and Capitanio2014, p. 32). This range is close to the low skewness and low kurtosis case in Flora and Curran (Reference Flora and Curran2004). The reader can refer to Azzalini (Reference Azzalini2005) for an overview of the skew-normal distribution and to Azzalini and Capitanio (Reference Azzalini and Capitanio2014) for the expressions of skewness and excess kurtosis. Note that the bivariate skew-normal distribution proposed by Azzalini and Valle (Reference Azzalini and Valle1996) is different from the skew-normal distribution in Quiroga (Reference Quiroga1992). The specification in Azzalini and Valle (Reference Azzalini and Valle1996) is used in the present study for its connection with the skew-t(v) distribution in the next experiment.

3.1.3. Experiment 3: Skew-t(v) Distribution

Skewness can also be introduced to the t(v) distribution. Azzalini and Capitanio (Reference Azzalini and Capitanio2003) proposed a multivariate skew-t(v) distribution whose bivariate density function is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {f}\left( x,y \right)&= 2 \text {t} \left( x,y; \omega , v \right) \text {T} \left( (\alpha _1x+\alpha _2y) \left( \frac{v+2}{v+ (x^2 - 2\omega xy + y^2)/(1-\rho ^2)} \right) ^{1/2} ; v+2 \right) , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {t}(\cdot ,\cdot ; \omega , v)$$\end{document} is the density function of a standard bivariate t distribution with correlation w and degrees of freedom v and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \text {T}(\cdot ;v+2)$$\end{document} is the distribution function of a univariate t distribution with degrees of freedom \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v+2$$\end{document} . The covariance matrix of X and Y is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{v}{v-2} \varvec{W} -\frac{2}{\pi \left( 1+\alpha _{1}^{2}+2\omega \alpha _{1}\alpha _{2}+\alpha _{2}^{2}\right) }\varvec{R} \,, \end{aligned}$$\end{document}

provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v>2$$\end{document} . Both marginal distributions are univariate skew-t distributions with density function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {f}\left( x \right)&= 2 \text {t} \left( x; v \right) \text {T} \left( \alpha x \left( \frac{v+1}{v+x^2} \right) ^{1/2} ; v+1 \right) . \end{aligned}$$\end{document}

The reader is directed to Azzalini and Capitanio (Reference Azzalini and Capitanio2003) for the expressions of skewness and excess kurtosis.

3.1.4. Experiment 4: Other Distributions

The skew-normal and t(v) distributions are special cases of the skew-t(v) distribution family. There are many distributions that are not members of the skew-elliptical distribution family. In addition, the underlying distribution cannot be truly determined from the observed ordinal data. It is therefore important to investigate the effect of distributional misspecification using the distributions that do not belong to the skew-t distribution family. A Pareto distribution is commonly used to model income (Arnold, Reference Arnold and Chotikapanich2008) and income is commonly used as an indicator of socio-economic status. Mardia (Reference Mardia1962) proposed a multivariate Pareto distribution in which the bivariate density function is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {f}(x,y;a,\theta _1,\theta _2) = (a+1)a(\theta _1 \theta _2)^{a+1}(\theta _2x + \theta _1y - \theta _1 \theta _2)^{-(a+2)}, \end{aligned}$$\end{document}

and the marginal density function is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {f}(x)=a\theta _i^a x^{-(a+1)}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \ge \theta _1>0, y \ge \theta _2>0$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} . The correlation coefficient between X and Y is 1 / a, which is always positive.

3.3.1. Experiment 1

Figures 1 displays the RB and RMSEA values when the true correlation is 0.4. As expected, assuming a wrong underlying distribution generally biases the polychoric correlation. Observe that the skew-normal distribution contains the normal distribution as a special case. Thus, both the normal and skew-normal assumptions consistently estimate the polychoric correlation when the true underlying distribution is normal. When the true underlying distribution is a normal distribution or a t distribution, all distributional assumptions produce a low RB (less than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\,\%$$\end{document} ). When the true underlying distribution is a uniform distribution or logistic distribution, the normal assumption generally produces a low-biased correlation estimate. However, the t assumption may produce a high RB (Figure 1). The normal and skew-normal assumptions can produce moderately biased polychoric correlations when the underlying distribution is the exponential power distribution with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =0.3$$\end{document} that corresponds to a distribution with high kurtosis (Figure 1). As the kurtosis in the exponential power family decreases, the magnitude of RB concomitantly decreases. When the underlying distribution is non-normal, the normal and skew-normal assumptions may produce different correlation estimates. Thus, the skew-normal distribution adjusts the underlying non-normality by introducing some degree of skewness. Consequently, the magnitude of RB may become higher but the RMSEA may become lower (Figure 1), which occurs when the number of categories is three for both ordinal variables. The polychoric correlation based on the underlying normal assumption generally underestimates the true correlation coefficient. The t(4) and t(6) assumptions sometimes outperform the normal assumption in Experiment 1.

Figure 1.

Relative bias (RB) and root-mean-square error of approximation (RMSEA) of correlation estimates when the true underlying distribution belongs to the elliptical distribution family. The true correlation coefficient is 0.4, a RB when both ordinal variables have five categories. b RMSEA when both ordinal variables have five categories. c RB when both ordinal variables have three categories. d RMSEA when both ordinal variables have three categories. Note Nor normal, Uni uniform, Logi logistic, EP \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\cdot )$$\end{document} =exponential power distribution with the enclosed value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta .$$\end{document}

3.3.2. Experiment 2

As expected, the polychoric correlation is consistently estimated when the true and assumed underlying distributions are both skew-normal (Figure 2). The normal assumption produces negatively biased correlation estimates. It can be moderately or strongly biased unless both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2$$\end{document} are small. Recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2$$\end{document} control the skewness and kurtosis of the underlying distribution. Small values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2$$\end{document} only introduce a small departure from the bivariate normal distribution. All the t distribution assumptions produce similar RBs relative to the normal assumption. Under both the normal and t(v) distribution assumptions, three categories in both ordinal variables generally lead to a higher magnitude of the RB value than five categories in both variables. For example, the RB with five-category variables does not exceed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-15$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0=0.4$$\end{document} , whereas the RB with three-category variables frequently exceeds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-25$$\end{document} under the same condition (Figure 2). As the true value of the correlation increases while the other conditions remain the same, the RB generally becomes smaller (see Figures 7, 8, and 9 in the supplementary materials).

In Experiment 2, the RMSEA can be misleading when the number of categories is three in both variables. Consider the normal assumption as an example. The magnitude of RB may exceed 10 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.1, \rho _0=0.4$$\end{document} , and both ordinal variables have three categories (Figure 2), whereas the RMSEA is still below 0.05 (Figure 3). The pattern is more dramatic when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=1$$\end{document} . The RB is almost \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-20$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2=1.5$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0=0.4$$\end{document} , but the RMSEA is slightly below 0.05. Thus, the estimated probabilities can be rather close to the true probabilities but the polychoric correlation can be largely biased. This event occurs because RMSEA only measures the closeness between the estimated and true category probabilities, and is not a direct measure of the correlation estimate.

Figure 2.

Relative bias (RB) of correlation estimates when the true underlying distribution is skew-normal. The true correlation coefficient is 0.4. a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.1$$\end{document} and both ordinal variables have five categories. b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.5$$\end{document} and both ordinal variables have five categories. c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=1$$\end{document} and both ordinal variables have five categories. d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.1$$\end{document} and both ordinal variables have three categories. e \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.5$$\end{document} and both ordinal variables have three categories. f \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=1$$\end{document} and both ordinal variables have three categories.

Figure 3.

Root-mean-square error of approximation (RMSEA) of correlation estimates when the true underlying distribution is skew-normal. The true correlation coefficient is 0.4. Both ordinal variables have three categories. a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.1$$\end{document} . b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.5$$\end{document} . (c) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=1.$$\end{document}

On the other hand, although the skew-normal assumption consistently estimates the polychoric correlation in Experiment 2, the numerical difficulties (such as non-convergence and local maximizer) are encountered in the present study. The fit function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\left( \varvec{\theta } \right) $$\end{document} can be fairly flat (see Figure 13 in the supplementary materials as an illustration). A bad choice of the starting value for the numerical optimization process can lead to the aforementioned issues. Thus, 20 starting values are employed. As a result, the skew-normal assumption is computationally much more intensive than the normal assumption.

3.3.3. Experiment 3

When the true underlying distribution is a skew-t(4) distribution, the normal and t(v) underlying distributional assumptions lead to a largely biased polychoric correlation, except when both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2$$\end{document} are small (Figure 4). A small pair of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _1,\alpha _2)$$\end{document} only introduces a small skewness and kurtosis to the underlying distribution, which is similar to a t(4) distribution. As known from Experiment 1, the normal and t(v) underlying distributional assumptions are only slightly biased when the true underlying distribution is a t distribution. The skew-normal assumption may produce not so biased correlations when both ordinal variables have three categories and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document} is small (Figure 4). In general, the skew-normal assumption is less biased than the normal and t(v) assumptions. As the degrees of freedom of the skew-t(v) distribution increases, all distributional assumptions become less biased, and the skew-normal assumption in particular is often robust (See the figures in the supplementary materials). This effect is expected from the fact that the skew-normal distribution corresponds to the skew-t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\infty )$$\end{document} distribution. Nevertheless, the normal and t(v) assumptions still produce moderately or largely biased polychoric correlations. Similar to the conclusions from the underlying skew-normal distribution, a wrong distributional assumption tends to underestimate the polychoric correlation. Three categories in both ordinal variables generally lead to a higher RB in magnitudes than five categories in both ordinal variables; and a higher value of the true correlation coefficient generally leads to less biased estimates. Similar to the case in Experiment 2, the RMSEA can be misleading as well. A low RMSEA does not necessarily indicate a low RB (e.g. see Figure 26 in the supplementary materials).

Figure 4.

Relative bias (RB) of correlation estimates when the true underlying distribution is skew-t(4). The true correlation coefficient is 0.4. a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.1$$\end{document} and both ordinal variables have five categories. b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.5$$\end{document} and both ordinal variables have five categories. c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=1$$\end{document} and both ordinal variables have five categories. d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.1$$\end{document} and both ordinal variables have three categories. e \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=0.5$$\end{document} and both ordinal variables have three categories. f \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1=1$$\end{document} and both ordinal variables have three categories.

3.3.4. Experiment 4

Table 1 shows that all the underlying distributional assumptions tend to be extremely biased when the true underlying distribution is a Pareto distribution. Similar to Experiments 2 and 3, the polychoric correlation tends to be underestimated across all conditions in Experiment 4. The skew-normal assumption produces a lower RB than the normal and t(v) assumptions, although all assumptions generally produce a large RB. The value of RMSEA tends to be small despite the heavily biased polychoric correlation. In particular, the RMSEA produced by the skew-normal assumption is always low. Note that the Pareto distribution is skewed. Thus, the skew-normal distribution assumption mimics the skewed pattern, although the true correlation coefficient is inconsistently estimated.

Table 1.

Relative bias (RB) and root-mean-squared error of approximation (RMSEA) of polychoric correlations in Experiment 4.