Mixtures of restricted skew-t factor analyzers with common factor loadings

Abstract

Mixtures of common t factor analyzers (MCtFA) have been shown its effectiveness in robustifying mixtures of common factor analyzers (MCFA) when handling model-based clustering of the high-dimensional data with heavy tails. However, the MCtFA model may still suffer from a lack of robustness against observations whose distributions are highly asymmetric. This paper presents a further robust extension of the MCFA and MCtFA models, called the mixture of common restricted skew-t factor analyzers (MCrstFA), by assuming a restricted multivariate skew-t distribution for the common factors. The MCrstFA model can be used to accommodate severely non-normal (skewed and leptokurtic) random phenomena while preserving its parsimony in factor-analytic representation and performing graphical visualization in low-dimensional plots. A computationally feasible expectation conditional maximization either algorithm is developed to carry out maximum likelihood estimation. The numbers of factors and mixture components are simultaneously determined based on common likelihood penalized criteria. The usefulness of our proposed model is illustrated with simulated and real datasets, and experimental results signify its superiority over some existing competitors.

Appendices

Appendix A: Proof of hierarchical representation (8)

It follows from (7) that

$$\begin{aligned}&E\left( {\varvec{Y}}_j{\varvec{U}}_{ij}^{\top }\mid \gamma _j,\tau _j,Z_{ij}=1\right) \nonumber \\&\quad =E\left[ E\left( {\varvec{Y}}_j{\varvec{U}}_{ij}^{\top }\mid {\varvec{U}}_{ij},\gamma _j,\tau _j,Z_{ij}=1\right) \mid \gamma _j,\tau _j,Z_{ij}=1\right] \\&\quad =E\left[ E({\varvec{Y}}_j\mid {\varvec{U}}_{ij},\gamma _j,\tau _j,Z_{ij}=1){\varvec{U}}_{ij}^{\top }\mid \gamma _j,\tau _j,Z_{ij}=1\right] \\&\quad =E\left( {\varvec{A}}{\varvec{U}}_{ij}{\varvec{U}}_{ij}^{\top }\mid \gamma _j,\tau _j,Z_{ij}=1\right) \\&\quad ={\varvec{A}}\left[ \tau _j^{-1}{\varvec{\varOmega }}_i+({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)^{\top }\right] , \end{aligned}$$

and

$$\begin{aligned}&\text{ cov }\left( {\varvec{Y}}_j, {\varvec{U}}_{ij}^{\top }\mid \gamma _j,\tau _j,Z_{ij}=1\right) \\&\quad =E\left( {\varvec{Y}}_j{\varvec{U}}_{ij}^{\top }\mid \gamma _j,\tau _j,Z_{ij}=1\right) -E({\varvec{Y}}_j\mid \gamma _j,\tau _j,Z_{ij}=1)E\left( {\varvec{U}}_{ij}^{\top }\mid \gamma _j,\tau _j,Z_{ij}=1\right) \\&\quad ={\varvec{A}}\left[ \tau _j^{-1}{\varvec{\varOmega }}_i+({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)^{\top }\right] -({\varvec{\mu }}_i+{\varvec{\alpha }}_i\gamma _j)({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)^{\top }\\&\quad ={\varvec{A}}\left[ \tau _j^{-1}{\varvec{\varOmega }}_i+({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)^{\top }\right] -{\varvec{A}}({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)^{\top }\\&\quad =\tau _j^{-1}{\varvec{A}}{\varvec{\varOmega }}_i. \end{aligned}$$

This gives rise to the following joint distribution:

$$\begin{aligned}\left[ \begin{array}{c} {\varvec{Y}}_j\\ {\varvec{U}}_{ij} \end{array} \right] \bigg |(\gamma _j,\tau _j,Z_{ij}=1)\sim N_{p+q} \left( \left[ \begin{array}{c} {\varvec{\mu }}_i+{\varvec{\alpha }}_i\gamma _j\\ {\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j \end{array} \right] , \tau _j^{-1}\left[ \begin{array}{cc} {\varvec{\varSigma }}_i &{} {\varvec{A}}{\varvec{\varOmega }}_i\\ {\varvec{\varOmega }}_i^{\top }{\varvec{A}}^{\top } &{} {\varvec{\varOmega }}_i \end{array} \right] \right) . \end{aligned}$$

We then have the following standard results:

$$\begin{aligned}&E({\varvec{U}}_{ij}\mid {\varvec{y}}_j,\gamma _j,\tau _j,Z_{ij}=1)\\&\quad =({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j)+\left( \tau _j^{-1}{\varvec{\varOmega }}_i^{\top }{\varvec{A}}^{\top }\right) (\tau _j^{-1}{\varvec{\varSigma }}_i)^{-1}({\varvec{y}}_j-{\varvec{\mu }}_i-{\varvec{\alpha }}_i\gamma _j)\\&\quad ={\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i-{\varvec{\alpha }}_i\gamma _j), \end{aligned}$$

and

$$\begin{aligned} \text{ cov }({\varvec{U}}_{ij}\mid {\varvec{y}}_j,\gamma _j,\tau _j,Z_{ij}=1)= & {} \tau _j^{-1}{\varvec{\varOmega }}_i-\left( \tau _j^{-1}{\varvec{\varOmega }}_i^{\top }{\varvec{A}}^{\top }\right) (\tau _j^{-1}{\varvec{\varSigma }}_i)^{-1}(\tau _j^{-1}{\varvec{A}}{\varvec{\varOmega }}_i)\\= & {} \tau _j^{-1}\left( {\varvec{I}}_q-{\varvec{\beta }}_i^{\top }{\varvec{A}}\right) {\varvec{\varOmega }}_i, \end{aligned}$$

where ${\varvec{\beta }}_i={\varvec{\varSigma }}_i^{-1}{\varvec{A}}{\varvec{\varOmega }}_i$. Using the characterization of the multivariate normal distribution, we can obtain

$$\begin{aligned}&{\varvec{U}}_{ij}\mid ({\varvec{y}}_j,\gamma _j,\tau _j,Z_{ij}=1) \sim N_q\left( {\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i-{\varvec{\alpha }}_i\gamma _j),\tau _j^{-1}\left( {\varvec{I}}_q\right. \right. \\&\quad \left. -\,{\varvec{\beta }}_i^{\top }\left. {\varvec{A}}\right) {\varvec{\varOmega }}_i\right) . \end{aligned}$$

With similar arguments, we have

$$\begin{aligned}&f\left( {\varvec{y}}_j,\gamma _j,\tau _j\mid z_{ij}=1\right) \\&\quad =f\left( {\varvec{y}}_j\mid \gamma _j,\tau _j,z_{ij}=1\right) f\left( \gamma _j\mid \tau _j,z_{ij}=1\right) f\left( \tau _j\mid z_{ij}=1\right) \\&\quad =\frac{2|{\varvec{\varSigma }}_i|^{-\frac{1}{2}}\left( \frac{\nu _i}{2}\right) ^{\frac{\nu _i}{2}}}{\left( 2\pi \right) ^{\frac{p+1}{2}}{\varGamma }\left( \frac{\nu _i}{2}\right) }\tau _j^{\frac{p+\nu _i+1}{2}-1}\exp \left\{ -\frac{\tau _j}{2}\left[ \frac{\left( \gamma _j-h_i\right) ^2}{\sigma _i^2}+\delta _{ij}+\nu _i\right] \right\} ,\\&f\left( {\varvec{y}}_j,\tau _j\mid z_{ij}=1\right) =\frac{2|{\varvec{\varSigma }}_i|^{-\frac{1}{2}}\sigma _i\left( \frac{\nu _i}{2}\right) ^{\frac{\nu _i}{2}}}{\left( 2\pi \right) ^{\frac{p}{2}}{\varGamma }\left( \frac{\nu _i}{2}\right) }\tau _j^{\frac{p+\nu _i}{2}-1}\nonumber \\&\quad \exp \left\{ -\frac{\tau _j}{2}\left[ \delta _{ij}+\nu _i\right] \right\} {\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) ,\\&f\left( \gamma _j\mid {\varvec{y}}_j,\tau _j,z_{ij}=1\right) =\frac{f\left( {\varvec{y}}_j,\gamma _j,\tau _j\mid z_{ij}=1\right) }{f\left( {\varvec{y}}_j,\tau _j\mid z_{ij}=1\right) } =\frac{\phi \left( \gamma _j;h_i,\tau _j^{-1}\sigma _i^2\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }. \end{aligned}$$

Hence, it is trivial to establish that $\gamma _j\mid ({\varvec{y}}_j,\tau _j,Z_{ij}=1) \sim TN(h_i,\tau _j^{-1}\sigma _i^2;(0,\infty ))$. Furthermore, standard calculation gives

$$\begin{aligned} f(\tau _j\mid {\varvec{y}}_j,Z_{ij}=1)= & {} \frac{f({\varvec{y}}_j,\tau _j\mid Z_{ij}=1)}{f({\varvec{y}}_j\mid Z_{ij}=1)}\\= & {} \frac{\tau _j^{\frac{\nu _i+p}{2}-1}}{{\varGamma }\left( \frac{\nu _i+p}{2}\right) }\left( \frac{\nu _i+\delta _{ij}}{2}\right) ^{\frac{\nu _i+p}{2}}\frac{{\varPhi }(\sqrt{\tau _j}M_{ij})}{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\\&\times \exp \left\{ -\frac{\tau _j}{2}\left[ \delta _{ij}+\nu _i\right] \right\} \\= & {} \frac{{\varPhi }(\sqrt{\tau _j}M_{ij})}{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }g\left( \tau _j;\frac{\nu _i+p}{2},\,\frac{\nu _i+\delta _{ij}}{2}\right) . \end{aligned}$$

Appendix B: Proof of Proposition 1

• (a) Standard calculation of conditional expectation yields

  $$\begin{aligned}&E(\tau _j\mid {\varvec{y}}_j,\,z_{ij}=1)=\int _{0}^{\infty }\tau _j f(\tau _j\mid {\varvec{y}}_j,\,z_{ij}=1)d\tau _j\nonumber \\&\quad =\int _{0}^{\infty }\tau _j\frac{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }g\left( \tau _j;\frac{\nu _i+p}{2},\,\frac{\nu _i+\delta _{ij}}{2}\right) d\tau _j\nonumber \\&\quad =\frac{\left( \frac{\nu _i+p}{\nu _i+\delta _{ij}}\right) }{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\int _{0}^{\infty }{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) g\left( \tau _j;\frac{\nu _i+p+2}{2},\,\frac{\nu _i+\delta _{ij}}{2}\right) d\tau _j\nonumber \\&\quad =\left( \frac{\nu _i+p}{\nu _i+\delta _{ij}}\right) \frac{T\left( M_{ij}\sqrt{\frac{\nu _i+p+2}{\nu _i+\delta _{ij}}};\nu _i+p+2\right) }{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }. \end{aligned}$$

  (B.1)

• (b) Because $\gamma _j\mid ({\varvec{y}}_j,\tau _j,Z_{ij}=1) \sim TN(h_i,\tau _j^{-1}\sigma _i^2;(0,\infty ))$, we obtain

  $$\begin{aligned} E(\gamma _j\mid {\varvec{y}}_j,\,\tau _j,\,z_{ij}=1)=h_{ij}+\frac{\sigma _i}{\sqrt{\tau _j}}\frac{\phi \left( \sqrt{\tau _j}M_{ij}\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }. \end{aligned}$$

  (B.2)

• (c) We first need to show

  $$\begin{aligned}&E\left( \tau _j^{\frac{k}{2}}\frac{\phi \left( \sqrt{\tau _j}M_{ij}\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }\bigg |{\varvec{y}}_j,\,z_{ij}=1\right) \nonumber \\&\quad =\frac{1}{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\int _{0}^{\infty }\tau _j^{\frac{k}{2}}\phi \left( \sqrt{\tau _j}M_{ij}\right) g\left( \tau _j;\frac{\nu _i+p}{2},\,\frac{\nu _i+\delta _{ij}}{2}\right) d\tau _j\nonumber \\&\quad =\frac{1}{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\int _{0}^{\infty }\tau _j^{\frac{k-1}{2}}\phi \big (M_{ij};0,\,\tau _j^{-1}\big )g\left( \tau _j;\frac{\nu _i+p}{2},\,\frac{\nu _i+\delta _{ij}}{2}\right) d\tau _j\nonumber \\&\quad =\frac{{\varGamma }\left( \frac{\nu _i+p+k-1}{2}\right) \int _{0}^{\infty }\phi \left( M_{ij};0,\,\tau _j^{-1}\right) g\left( \tau _j;\frac{\nu _i+p+k-1}{2},\,\frac{\nu _i+\delta _{ij}}{2}\right) d\tau _j}{{\varGamma }\left( \frac{\nu _i+p}{2}\right) \left( \frac{\nu _i+\delta _{ij}}{2}\right) ^{\frac{k-1}{2}}T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\nonumber \\&\quad =\frac{{\varGamma }\left( \frac{\nu _i+p+k-1}{2}\right) \sqrt{\frac{\nu _i+p+k-1}{\nu _i+\delta _{ij}}}\,t\left( M_{ij}\sqrt{\frac{\nu _i+p+k-1}{\nu _i+\delta _{ij}}};\nu _i+p+k-1\right) }{{\varGamma }\left( \frac{\nu _i+p}{2}\right) \left( \frac{\nu _i+\delta _{ij}}{2}\right) ^{\frac{k-1}{2}}T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }. \end{aligned}$$

  (B.3)

  Applying the result in (B.3) with $k=-\,1$ and (B.2) yields

  $$\begin{aligned}&E(\gamma _j\mid {\varvec{y}}_j,\,z_{ij}=1) =E[E(\gamma _j\mid {\varvec{y}}_j,\,\tau _j,\,z_{ij}=1)\mid {\varvec{y}}_j,\,z_{ij}=1]\nonumber \\&\quad =E\left[ h_{ij}+\frac{\sigma _i}{\sqrt{\tau _j}}\frac{\phi \left( \sqrt{\tau _j}M_{ij}\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }\bigg |{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad =h_{ij}+\sigma _iE\left( \frac{1}{\sqrt{\tau _j}}\frac{\phi \left( \sqrt{\tau _j}M_{ij}\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }\bigg |{\varvec{y}}_j,\,z_{ij}=1\right) \nonumber \\&\quad =h_{ij}+\frac{\sigma _i}{\sqrt{\frac{\nu _i+p-2}{\nu _i+\delta _{ij}}}}\frac{t\left( M_{ij}\sqrt{\frac{\nu _i+p-2}{\nu _i+\delta _{ij}}};\nu _i+p-2\right) }{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }. \end{aligned}$$

  (B.4)

• (d) Using (B.1), (B.2) and (B.3) with $k=1$, we have

  $$\begin{aligned}&E(\tau _j\gamma _j|{\varvec{y}}_j,\,z_{ij}=1) =E[E(\tau _j\gamma _j|{\varvec{y}}_j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1]\nonumber \\&\quad =E[\tau _jE(\gamma _j|{\varvec{y}}_j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1]\nonumber \\&\quad =E\left[ \tau _j\left( h_{ij}+\frac{\sigma _i}{\sqrt{\tau _j}}\frac{\phi \left( \sqrt{\tau _j}M_{ij}\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }\right) \bigg |{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad =h_{ij} E(\tau _j|{\varvec{y}}_j,\,z_{ij}=1)+\sigma _i E\left[ \sqrt{\tau _j}\frac{\phi \left( \sqrt{\tau _j}M_{ij}\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }\bigg |{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad =h_{ij}\left[ \frac{\nu _i+p}{\nu _i+\delta _{ij}}\frac{T\left( M_{ij}\sqrt{\frac{\nu _i+p+2}{\nu _i+\delta _{ij}}};\nu _i+p+2\right) }{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\right] \nonumber \\&\qquad +\,\sigma _i\left[ \sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}}\frac{t\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\right] . \end{aligned}$$

  (B.5)

• (e) Using the result of (B.2), the second moment of a truncated normal distribution is given by

  $$\begin{aligned} E\left( \gamma _j^2|{\varvec{y}}_j,\,\tau _j,\,z_{ij}=1\right)= & {} h_{ij}E(\gamma _j|{\varvec{y}}_j,\,\tau _j,\,z_{ij}=1)+\frac{\sigma _i^2}{\tau _j}\nonumber \\= & {} h_{ij}\left( h_{ij}+\frac{\sigma _i}{\sqrt{\tau _j}}\frac{\phi \left( \sqrt{\tau _j}M_{ij}\right) }{{\varPhi }\left( \sqrt{\tau _j}M_{ij}\right) }\right) +\frac{\sigma _i^2}{\tau _j}. \end{aligned}$$

  (B.6)

• (f) Applying the double expectation and using (B.5) and (B.6), we have

  $$\begin{aligned} E\left( \tau _j\gamma _j^2|{\varvec{y}}_j,\,z_{ij}=1\right)= & {} E\left[ E\left( \tau _j\gamma _j^2|{\varvec{y}}_j,\,\tau _j,\,z_{ij}=1\right) |{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\= & {} E\left[ \tau _jE\left( \gamma _j^2|{\varvec{y}}_j,\,\tau _j,\,z_{ij}=1\right) |{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\= & {} E\left\{ \tau _j\left[ h_{ij}E(\gamma _j|{\varvec{y}}_j,\,\tau _j,\,z_{ij}=1)+\tau _j^{-1}\sigma _i^2\right] |{\varvec{y}}_j,\,z_{ij}=1\right\} \nonumber \\= & {} h_{ij} E\left( \tau _j\gamma _j|{\varvec{y}}_j,\,z_{ij}=1\right) +\sigma _i^2. \end{aligned}$$

  (B.7)

• (g) Applying the double expectation and the result of (B.4), we have

  $$\begin{aligned}&E({\varvec{U}}_{ij}|{\varvec{y}}_j,\,z_{ij}=1)=E\left[ E({\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad =E\left[ {\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i-{\varvec{\alpha }}_i\gamma _j)|{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad ={\varvec{\xi }}_i+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i)+({\varvec{\lambda }}_i-{\varvec{\beta }}_i^{\top }{\varvec{\alpha }}_i)E(\gamma _j|{\varvec{y}}_j,\,z_{ij}=1). \end{aligned}$$

  (B.8)

• (h) Applying the double expectation and using (B.1) and (B.5), we have

  $$\begin{aligned}&E(\tau _j{\varvec{U}}_{ij}|{\varvec{y}}_j,\,z_{ij}=1)=E\left[ E(\tau _j{\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad =E\left[ \tau _j E({\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad =E\left\{ \tau _j\left[ {\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i-{\varvec{\alpha }}_i\gamma _j)\right] |{\varvec{y}}_j,\,z_{ij}=1\right\} \nonumber \\&\quad =\left[ {\varvec{\xi }}_i+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i)\right] E(\tau _j|{\varvec{y}}_j,\,z_{ij}=1)\nonumber \\&\qquad +\,\left( {\varvec{\lambda }}_i-{\varvec{\beta }}_i^{\top }{\varvec{\alpha }}_i\right) E(\tau _j\gamma _j|{\varvec{y}}_j,\,z_{ij}=1). \end{aligned}$$

  (B.9)

• (i) Applying the double expectation and using (B.5) and (B.7), we have

  $$\begin{aligned}&E\left( \tau _j\gamma _j{\varvec{U}}_{ij}|{\varvec{y}}_j,\,z_{ij}=1\right) \\&\quad =E\left[ E(\tau _j\gamma _j{\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1\right] \nonumber \\&\quad =E\left[ \tau _j\gamma _j E({\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1\right] \\&\quad =E\left\{ \tau _j\gamma _j[{\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j+{\varvec{\beta }}_i^\top ({\varvec{y}}_j-{\varvec{\mu }}_i-{\varvec{\alpha }}_i\gamma _j)]|{\varvec{y}}_j,\,z_{ij}=1\right\} \nonumber \\&\quad =\left[ {\varvec{\xi }}_i+{\varvec{\beta }}_i^\top ({\varvec{y}}_j-{\varvec{\mu }}_i)\right] E(\tau _j\gamma _j|{\varvec{y}}_j,\,z_{ij}=1)\\&\qquad +\,({\varvec{\lambda }}_i-{\varvec{\beta }}_i^\top {\varvec{\alpha }}_i)E(\tau _j\gamma _j^2|{\varvec{y}}_j,\,z_{ij}=1). \end{aligned}$$

• (j) Applying the double expectation and using (B.8) and (B.9), we have

  $$\begin{aligned}&E\left( \tau _j{\varvec{U}}_{ij}{\varvec{U}}_{ij}^{\top }|{\varvec{y}}_j,\,z_{ij}=1\right) =E\left[ E\left( \tau _j{\varvec{U}}_{ij}{\varvec{U}}_{ij}^{\top }|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1\right) |{\varvec{y}}_j,\,z_{ij}=1\right] \\&\quad =E\left[ \tau _jE({\varvec{U}}_{ij}{\varvec{U}}_{ij}^{\top }|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)|{\varvec{y}}_j,\,z_{ij}=1\right] \\&\quad =E\big \{\tau _j[E({\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)E({\varvec{U}}_{ij}^{\top }|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)\\&\qquad +\,\text{ cov }({\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)]|{\varvec{y}}_j,\,z_{ij}=1\big \}\\&\quad =E\big \{\tau _j[E({\varvec{U}}_{ij}|{\varvec{y}}_j,\,\gamma _j,\,\tau _j,\,z_{ij}=1)({\varvec{\xi }}_i+{\varvec{\lambda }}_i\gamma _j+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i-{\varvec{\alpha }}_i\gamma _j))^{\top }\\&\qquad +\,\tau _j^{-1}({\varvec{I}}_q-{\varvec{\beta }}_i^{\top }{\varvec{A}}){\varvec{\varOmega }}_i]|{\varvec{y}}_j,\,z_{ij}=1\big \}\\&\quad =E(\gamma _j\tau _j{\varvec{U}}_{ij}|{\varvec{y}}_j,\,z_{ij}=1)({\varvec{\lambda }}_i-{\varvec{\beta }}_i^{\top }{\varvec{\alpha }}_i)^{\top }\\&\qquad +\,E(\tau _j{\varvec{U}}_{ij}|{\varvec{y}}_j,\,z_{ij}=1)[{\varvec{\xi }}_i+{\varvec{\beta }}_i^{\top }({\varvec{y}}_j-{\varvec{\mu }}_i)]^{\top }+({\varvec{I}}_q-{\varvec{\beta }}_i^{\top }{\varvec{A}}){\varvec{\varOmega }}_i. \end{aligned}$$

• (k) It is known that $\int _0^{\infty }f(\tau _j|{\varvec{y}}_j,\,Z_{ij}=1)d\tau _j=1$, that is,

  $$\begin{aligned} \int _0^{\infty }\frac{{\varPhi }\big (\sqrt{\tau _j}M_{ij}\big )}{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }\frac{{(\frac{\nu _i+\delta _{ij}}{2})}^{(\frac{\nu _i+p}{2})}}{{\varGamma }(\frac{\nu _i+p}{2})}\exp \left\{ -\frac{\nu _i+\delta _{ij}}{2}\tau _j\right\} d\tau _j =1. \end{aligned}$$

  Then

  $$\begin{aligned} \frac{d}{d\nu _i}\int _0^{\infty }b_j{\varPhi }\big (\sqrt{\tau _j}M_{ij}\big )\exp \left\{ -\frac{\nu _i+\delta _{ij}}{2}\tau _j\right\} d\tau _j=0, \end{aligned}$$

  where

  $$\begin{aligned} b_j=\frac{{\left( \frac{\nu _i+\delta _{ij}}{2}\right) }^{\left( \nu _i+p\right) /2}}{{\varGamma }\left( \frac{\nu _i+p}{2}\right) T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }. \end{aligned}$$

  By Leibnitz’s rule, we can obtain

  $$\begin{aligned}&E(\log \tau _j|{\varvec{y}}_j,\,z_{ij}=1)-E(\tau _j|{\varvec{y}}_j,\,z_{ij}=1)+\log \left( \frac{\nu _i+\delta _{ij}}{2}\right) +\left( \frac{\nu _i+p}{\nu _i+\delta _{ij}}\right) \\&\quad -\,\mathrm{DG}\left( \frac{\nu _i+p}{2}\right) -\frac{\int _{-\infty }^{M_{ij}}t\left( x;0,\frac{\nu _i+\delta _{ij}}{\nu _i+p},\nu _i+p\right) f_{\nu _i}(x)dx}{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }=0, \end{aligned}$$

  where

  $$\begin{aligned} f_{\nu _i}(x)= & {} \mathrm{DG}\left( \frac{\nu _i+p+1}{2}\right) -\mathrm{DG}\left( \frac{\nu _i+p}{2}\right) -\frac{1}{\pi (\nu _i+\delta _{ij})}\nonumber \\&-\,\log \left( 1+\frac{x^2}{\nu _i+\delta _{ij}}\right) +\frac{(\nu _i+p+1)x^2}{(\nu _i+\delta _{ij})(x^2+\nu _i+\delta _{ij})}. \end{aligned}$$

  (B.10)

  It follows that

  $$\begin{aligned} E(\log \tau _j|{\varvec{y}}_j,\,z_{ij}=1)= & {} E(\tau _j|{\varvec{y}}_j,\,z_{ij}=1)-\log \left( \frac{\nu _i+\delta _{ij}}{2}\right) -\left( \frac{\nu _i+p}{\nu _i+\delta _{ij}}\right) \\&+\,\mathrm{DG}\left( \frac{\nu _i+p}{2}\right) +\,\frac{\int _{-\infty }^{M_{ij}}t\left( x;0,\frac{\nu _i+\delta _{ij}}{\nu _i+p},\nu _i+p\right) f_{\nu _i}(x)dx}{T\left( M_{ij}\sqrt{\frac{\nu _i+p}{\nu _i+\delta _{ij}}};\nu _i+p\right) }. \end{aligned}$$

Appendix C: Proof of CM-steps

• (a) By the Lagrange multiplier method, we define

  $$\begin{aligned} L(\pi _i,\lambda )=Q({\varvec{\varTheta }}\mid \hat{{\varvec{\varTheta }}}^{(k)})-\lambda \left( \sum _{i=1}^g\pi _i-1\right) , \end{aligned}$$

  and then take partial derivatives, yielding

  $$\begin{aligned} \frac{\partial L(\pi _i,\lambda )}{\partial \pi _i} = \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}\frac{1}{\pi _i}-\lambda =0,\quad \text{ and } \quad \frac{\partial L(\pi _i,\lambda )}{\partial \lambda } = -\left( \sum _{i=1}^g\pi _i-1\right) =0. \end{aligned}$$

  Since $\sum _{i=1}^g\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}=n$, we obtain $\hat{\pi }_i^{(k+1)}=\sum _{j=1}^n{\hat{z}}_{ij}^{(k)}/n$.

• (b) Differentiating $Q({\varvec{\varTheta }}\mid \hat{{\varvec{\varTheta }}}^{(k)})$ with respect to ${\varvec{\xi }}_i$ leads to

  $$\begin{aligned} \frac{\partial Q}{\partial {\varvec{\xi }}_i}= & {} -\frac{1}{2}\frac{\partial }{\partial {\varvec{\xi }}_i}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{\varOmega }}_i^{-1}\mathrm{tr}\bigg [-{\hat{{\varvec{\eta }}}_{ij}}^{(k)}{\varvec{\xi }}_i^\top -{\varvec{\xi }}_i{\hat{{\varvec{\eta }}}_{ij}}^{(k)^\top }\\&+\,{\varvec{\xi }}_i{\hat{\tau }_{ij}}^{(k)}{\varvec{\xi }}_i^\top +{\varvec{\xi }}_i{{\hat{s}}_{1ij}}^{(k)}{\varvec{\lambda }}_i^\top +{\varvec{\lambda }}_i{{\hat{s}}_{1ij}}^{(k)}{\varvec{\xi }}_i^\top \bigg ]\\= & {} \mathrm{tr}\bigg \{{\varvec{\varOmega }}_i^{-1}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}\left[ {\hat{{\varvec{\eta }}}_{ij}}^{(k)}-{\hat{\tau }_{ij}}^{(k)}{\varvec{\xi }}_i-{{\hat{s}}_{1ij}}^{(k)}{\varvec{\lambda }}_i\right] \bigg \}. \end{aligned}$$

  Moreover, the partial derivative of $Q({\varvec{\varTheta }}\mid \hat{{\varvec{\varTheta }}}^{(k)})$ with respect to ${\varvec{\lambda }}_i$ is

  $$\begin{aligned} \frac{\partial Q}{\partial {\varvec{\lambda }}_i}= & {} -\frac{1}{2}\frac{\partial }{\partial {\varvec{\lambda }}_i}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{\varOmega }}_i^{-1}\mathrm{tr}\bigg [-{\hat{{\varvec{\zeta }}}_{ij}}^{(k)}{\varvec{\lambda }}_i^\top +{\varvec{\xi }}_i{{\hat{s}}_{1ij}}^{(k)}{\varvec{\lambda }}_i^\top \\&-\,{\varvec{\lambda }}_i{\hat{{\varvec{\zeta }}}_{ij}}^{(k)^\top }+{\varvec{\lambda }}_i{{\hat{s}}_{1ij}}^{(k)}{\varvec{\xi }}_i^\top +{\varvec{\lambda }}_i{{\hat{s}}_{2ij}}^{(k)}{\varvec{\lambda }}_i^\top \bigg ]\\= & {} \mathrm{tr}\bigg \{{\varvec{\varOmega }}_i^{-1}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}\left[ {\hat{{\varvec{\zeta }}}_{ij}}^{(k)}-{{\hat{s}}_{1ij}}^{(k)}{\varvec{\xi }}_i-{{\hat{s}}_{2ij}}^{(k)}{\varvec{\lambda }}_i\right] \bigg \}. \end{aligned}$$

  Solving the above two equations, we get

  $$\begin{aligned} \frac{\partial Q}{\partial {\varvec{\xi }}_i}= & {} \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{\varOmega }}_i^{-1}({\hat{{\varvec{\eta }}}_{ij}}^{(k)}-{{\hat{s}}_{1ij}}^{(k)}{\varvec{\lambda }}_i)-\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{\varOmega }}_i^{-1}{\hat{\tau }_{ij}}^{(k)}{\varvec{\xi }}_i=\mathbf 0, \end{aligned}$$

  (C.1)
  $$\begin{aligned} \frac{\partial Q}{\partial {\varvec{\lambda }}_i}= & {} \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{\varOmega }}_i^{-1}({\hat{{\varvec{\zeta }}}_{ij}}^{(k)}-{{\hat{s}}_{1ij}}^{(k)}{\varvec{\xi }}_i)-\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{\varOmega }}_i^{-1}{{\hat{s}}_{2ij}}^{(k)}{\varvec{\lambda }}_i=\mathbf 0. \end{aligned}$$

  (C.2)

  After rearrangement, (C.1) and (C.2) can be rewritten as

  $$\begin{aligned} \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\hat{\tau }_{ij}}^{(k)}{\varvec{\xi }}_i+\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{{\hat{s}}_{1ij}}^{(k)}{\varvec{\lambda }}_i= & {} \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\hat{{\varvec{\eta }}}_{ij}}^{(k)},\\ \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{{\hat{s}}_{1ij}}^{(k)}{\varvec{\xi }}_i+\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{{\hat{s}}_{2ij}}^{(k)}{\varvec{\lambda }}_i= & {} \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\hat{{\varvec{\zeta }}}_{ij}}^{(k)}. \end{aligned}$$

  Using Cramer’s law, the solutions of the two linear equations are

  $$\begin{aligned} \hat{{\varvec{\xi }}}_i^{\left( k+1\right) }= \frac{ \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{\hat{{\varvec{\eta }}}_{ij}}^{\left( k\right) }\right) \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{{\hat{s}}_{2ij}}^{\left( k\right) }\right) - \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{\hat{{\varvec{\zeta }}}_{ij}}^{\left( k\right) }\right) \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{{\hat{s}}_{1ij}}^{\left( k\right) }\right) }{ \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{\hat{\tau }_{ij}}^{\left( k\right) }\right) \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{{\hat{s}}_{2ij}}^{\left( k\right) }\right) - \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{{\hat{s}}_{1ij}}^{\left( k\right) }\right) ^2}, \end{aligned}$$

  and

  $$\begin{aligned} \hat{{\varvec{\lambda }}}_i^{\left( k+1\right) }=\frac{ \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{\hat{\tau }_{ij}}^{\left( k\right) }\right) \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{\hat{{\varvec{\zeta }}}_{ij}}^{\left( k\right) }\right) - \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{{\hat{s}}_{1ij}}^{\left( k\right) }\right) \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{\hat{{\varvec{\eta }}}_{ij}}^{\left( k\right) }\right) }{ \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{\hat{\tau }_{ij}}^{\left( k\right) }\right) \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{{\hat{s}}_{2ij}}^{\left( k\right) }\right) - \left( \sum _{j=1}^n{{\hat{z}}_{ij}}^{\left( k\right) }{{\hat{s}}_{1ij}}^{\left( k\right) }\right) ^2}. \end{aligned}$$

• (c) The partial derivative of $Q({\varvec{\varTheta }}\mid \hat{{\varvec{\varTheta }}}^{(k)})$ with respect to ${\varvec{A}}$ is

  $$\begin{aligned} \frac{\partial Q}{\partial {\varvec{A}}}= & {} -\frac{1}{2}\frac{\partial }{\partial {\varvec{A}}}\sum _{i=1}^g\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}\mathrm{tr}\bigg (-{\varvec{D}}_i^{-1}{\varvec{y}}_j{\hat{{\varvec{\eta }}}_{ij}}^{(k)^\top }{\varvec{A}}^\top \nonumber \\&-\,{\varvec{D}}_i^{-1}{\varvec{A}}{\hat{{\varvec{\eta }}}_{ij}}^{(k)}{\varvec{y}}_j^\top +{\varvec{D}}_i^{-1}{\varvec{A}}{\hat{{\varvec{\varPsi }}}_{ij}}^{(k)}{\varvec{A}}^\top \bigg )\nonumber \\= & {} \mathrm{tr}\left( \sum _{i=1}^g\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{D}}_i^{-1}{\varvec{y}}_j{\hat{{\varvec{\eta }}}_{ij}}^{(k)^\top }-\sum _{i=1}^g\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{D}}_i^{-1}{\hat{{\varvec{\varPsi }}}_{ij}}^{(k)}{\varvec{A}}\right) . \end{aligned}$$

  (C.3)

  Equating (C.3) to the zero matrix, we have

  $$\begin{aligned} \hat{{\varvec{A}}}^{(k+1)}=\left( \sum _{i=1}^g\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\varvec{y}}_j{\hat{{\varvec{\eta }}}_{ij}}^{(k)^\top }\right) \left( \sum _{i=1}^g\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\hat{{\varvec{\varPsi }}}_{ij}}^{(k)}\right) ^{-1}. \end{aligned}$$

• (d) The partial derivative of $Q({\varvec{\varTheta }}\mid \hat{{\varvec{\varTheta }}}^{(k)})$ with respect to ${\varvec{\varOmega }}_i$ is

  $$\begin{aligned} \frac{\partial Q}{\partial {\varvec{\varOmega }}_i^{-1}}= & {} \frac{1}{2}\frac{\partial }{\partial {\varvec{\varOmega }}_i^{-1}}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}\left\{ \log |{\varvec{\varOmega }}_i^{-1}|-\mathrm{tr}\left( {\varvec{\varOmega }}_i^{-1}{\varvec{\varLambda }}_{ij}\right) \right\} \nonumber \\= & {} \frac{1}{2}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}\Big [2{\varvec{\varOmega }}_i-\mathrm{Diag}\{{\varvec{\varOmega }}_i\}-\big (2{\varvec{\varLambda }}_{ij}-\mathrm{Diag}\{{\varvec{\varLambda }}_{ij}\}\big )\Big ]. \end{aligned}$$

  (C.4)

  Equating (C.4) to the zero vector gives

  $$\begin{aligned} {\hat{{\varvec{\varOmega }}}_i}^{(k+1)}=\frac{\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\hat{{\varvec{\varLambda }}}_{ij}}^{(k+1)}}{\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}}. \end{aligned}$$

• (e) Taking the partial derivative of $Q({\varvec{\varTheta }}\mid \hat{{\varvec{\varTheta }}}^{(k)})$ with respect to ${\varvec{D}}_i$ yields

  $$\begin{aligned} \frac{\partial Q}{\partial {\varvec{D}}_i^{-1}}= & {} \frac{1}{2}\frac{\partial }{\partial {\varvec{D}}_i^{-1}}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}\left[ \log |{\varvec{D}}_i^{-1}|-\mathrm{tr}\left( {\varvec{D}}_i^{-1}{\varvec{\varUpsilon }}_{ij}\right) \right] \nonumber \\= & {} \frac{1}{2}\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}({\varvec{D}}_i-{\varvec{\varUpsilon }}_{ij}). \end{aligned}$$

  (C.5)

  We have the following estimator

  $$\begin{aligned} {\hat{{\varvec{D}}}_i}^{(k+1)}=\frac{\mathrm{Diag}\left\{ \sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}{\hat{{\varvec{\varUpsilon }}}_{ij}}^{(k+1)}\right\} }{\sum _{j=1}^n{{\hat{z}}_{ij}}^{(k)}} \end{aligned}$$

  obtained by equating (C.5) to the zero matrix.

Appendix D: Parameter estimation for the MCghstFA model using the ECM algorithm

Table 5 Comparison of some characterizations between the MCrstFA and MCghstFA models

According to Table 5, the MCghstFA model admits a three-level hierarchy:

$$\begin{aligned} \left[ \begin{array}{c}{\varvec{Y}}_{j} \\ {\varvec{U}}_{ij} \end{array}\right] \Big | (W_{j}, Z_{ij}=1)\sim & {} N_{p+q} \left( \left[ \begin{array}{c}{\varvec{A}} {\varvec{\xi }}_i + W_{j}{\varvec{A}}{\varvec{\lambda }}_i \\ {\varvec{\xi }}_i + W_{j}{\varvec{\lambda }}_i \end{array}\right] , W_{j}\left[ \begin{array}{cc}{\varvec{A}}{\varvec{{\varOmega }}}_i{\varvec{A}}+{\varvec{D}} &{} {\varvec{A}}{\varvec{{\varOmega }}}_i \\ {\varvec{{\varOmega }}}_i{\varvec{A}}^\top &{} {\varvec{{\varOmega }}}_i \end{array}\right] \right) , \nonumber \\ W_{j}\mid Z_{ij}=1\sim & {} {\varGamma }^{-1}\left( \frac{\nu _i}{2}, \frac{\nu _i}{2}\right) ,\nonumber \\ {\varvec{Z}}_j\sim & {} {\mathscr {M}}(1;\pi _1,\ldots ,\pi _g). \end{aligned}$$

(D.1)

From (D.1), it can be verified that

$$\begin{aligned} {\varvec{Y}}_j \mid (W_{j}, Z_{ij}=1) \sim N_p \big ({\varvec{A}} {\varvec{\xi }}_i + W_{j}{\varvec{A}}{\varvec{\lambda }}_i, W_{j}({\varvec{A}}{\varvec{{\varOmega }}}_i{\varvec{A}}+{\varvec{D}})\big ), \end{aligned}$$

and

$$\begin{aligned} {\varvec{U}}_{ij}\mid ({\varvec{y}}_j,W_j,Z_{ij}=1) \sim N_q({\varvec{\mu }}_{2\cdot 1},{\varvec{{\varSigma }}}_{22\cdot 1}), \end{aligned}$$

(D.2)

where ${\varvec{\mu }}_{2\cdot 1}={\varvec{\xi }}_i+W_j{\varvec{\lambda }}_i+ {\varvec{{\varOmega }}}_i{\varvec{A}}^{\top }({\varvec{A}}{\varvec{{\varOmega }}}_i{\varvec{A}}^{\top }+{\varvec{D}})^{-1}({\varvec{y}}_j-{\varvec{A}}{\varvec{\xi }}_i-W_j{\varvec{A}}{\varvec{\lambda }}_i)$ and ${\varvec{{\varSigma }}}_{22\cdot 1}=W_j({\varvec{{\varOmega }}}_i-{\varvec{{\varOmega }}}_i{\varvec{A}}^{\top }({\varvec{A}}{\varvec{{\varOmega }}}_i{\varvec{A}}^{\top }+{\varvec{D}})^{-1}{\varvec{A}}{\varvec{{\varOmega }}}_i)= W_j({\varvec{{\varOmega }}}^{-1}_i-{\varvec{A}}^{\top }{\varvec{D}}^{-1}{\varvec{A}})^{-1}$.

A positive random variable X is said to follow the Generalized Inverse Gaussian (GIG) distribution (Good 1953), denoted by $W\sim \mathrm{GIG}(\psi ,\chi ,r)$, if it has the pdf

$$\begin{aligned} f_{GIG}(w;\psi ,\chi ,r)=\frac{\chi ^{-r}(\sqrt{\chi \psi })^{r} }{2K_{r}(\sqrt{\chi \psi }) }w^{q-1}\exp \left\{ -\frac{1}{2}(\chi w^{-1}+\psi w) \right\} , \end{aligned}$$

(D.3)

where $\psi ,\chi \in {\mathbb {R}}^+$, $r\in {\mathbb {R}}$, and $K_q$ is the modified Bessel function of the third kind with index r. Some particular moments of the GIG distribution have tractable forms, for instance,

$$\begin{aligned} E(W^a)= (\chi /\psi )^{a/2} \frac{K_{r+a}(\sqrt{\psi \chi })}{K_r(\sqrt{\psi \chi })},~a\in {\mathbb {R}}, \end{aligned}$$

(D.4)

and

$$\begin{aligned} E(\log W)= \log (\chi /\psi )^{1/2} +\frac{K^{\prime }_{r}(\sqrt{\psi \chi })}{K_r(\sqrt{\psi \chi })}, \end{aligned}$$

(D.5)

where

$$\begin{aligned} K'_{r}(x)=\frac{dK_{r}(x)}{dr}=\frac{1}{2}\int _{0}^{\infty }\log (y)y^{r-1}\exp \left\{ -\frac{x}{2}\left( y+\frac{1}{y} \right) \right\} d y, \quad x>0.\nonumber \\ \end{aligned}$$

(D.6)

By Bayes’ Theorem, the conditional pdf of $W_j$ given ${\varvec{y}}_j$ can be written as

$$\begin{aligned} f(w_j\mid {\varvec{y}}_j,Z_{ij}=1)\propto & {} w_j^{-(\nu _i+p)/2-1} \exp \Big \{-\frac{1}{2}\Big [(\nu _i+{\varDelta }_{ij})w_j^{-1}\\&+\,({\varvec{\lambda }}^{\top }_i{\varvec{A}}^{\top }\big ({\varvec{A}}{\varvec{\varOmega }}_i{\varvec{A}}^{\top }+{\varvec{D}})^{-1}{\varvec{A}}{\varvec{\lambda }}_i\big )w_j\Big ]\Big \}, \end{aligned}$$

where ${\varDelta }_{ij}=({\varvec{y}}_j-{\varvec{A}}{\varvec{\xi }}_i)^{\top }({\varvec{A}}{\varvec{\varOmega }}_i{\varvec{A}}^{\top }+{\varvec{D}})^{-1}({\varvec{y}}_j-{\varvec{A}}{\varvec{\xi }}_i)$. It follows from (D.3) that

$$\begin{aligned} W_j\mid ({\varvec{y}}_j,Z_{ij}=1)\sim \mathrm{GIG}\left( {\varvec{\lambda }}^{\top }_i{\varvec{A}}^{\top }({\varvec{A}}{\varvec{\varOmega }}_i{\varvec{A}}^{\top }+{\varvec{D}})^{-1}{\varvec{A}}{\varvec{\lambda }}_i,\nu _i+{\varDelta }_{ij},-\frac{\nu _i+p}{2}\right) . \end{aligned}$$

Alternatively, the MCghstFA model can be represented by a four-level hierarchy:

$$\begin{aligned} {\varvec{Y}}_{j} \mid ({\varvec{U}}_{ij}, W_{j}, Z_{ij}=1)\sim & {} N_p({\varvec{A}}{\varvec{U}}_{ij}, W_{j}{\varvec{D}}),\nonumber \\ {\varvec{U}}_{ij} \mid (W_{j}, Z_{ij}=1)\sim & {} N_q({\varvec{\xi }}_i + W_{j} {\varvec{\lambda }}_i, W_{j}{\varvec{{\varOmega }}}_i),\nonumber \\ W_{j}\mid Z_{ij}=1\sim & {} {\varGamma }^{-1}\left( \frac{\nu _i}{2}, \frac{\nu _i}{2}\right) ,\nonumber \\ {\varvec{Z}}_j\sim & {} {\mathscr {M}}(1;\pi _1,\ldots ,\pi _g). \end{aligned}$$

(D.7)

From (D.7), the complete-data log-likelihood function for ${\varvec{\varTheta }}$ on the basis of ${\varvec{Y}}_{c}=\{{\varvec{y}}_{j},{\varvec{U}}_{ij},W_{j},{\varvec{Z}}_j\}^{n}_{j=1}$, for $i=1,\ldots ,g$, is given by

$$\begin{aligned}&\ell _c({\varvec{{\varTheta }}}\mid {\varvec{Y}}_{c}) = \sum ^{g}_{i=1} \sum ^{n}_{j=1} Z_{ij} \log \bigg \{ \pi _i \log \phi _p({\varvec{y}}_{j}\mid {\varvec{A}}{\varvec{U}}_{ij}, W_{j}{\varvec{D}}) \phi _p({\varvec{U}}_{ij}\mid {\varvec{\xi }}_i \nonumber \\&\qquad + \,W_{j} {\varvec{\lambda }}_i, W_{j}{\varvec{{\varOmega }}}_i)\times \, f\left( w_{j}\mid \frac{\nu _i}{2},\frac{\nu _i}{2}\right) \bigg \}\nonumber \\&\quad =\sum ^{g}_{i=1} \sum ^{n}_{j=1} Z_{ij}\bigg \{\log \pi _i-\frac{1}{2}\log |W_j{\varvec{D}}|-\frac{1}{2}\log |W_j{\varvec{{\varOmega }}}_i|\nonumber \\&\qquad -\,\frac{W^{-1}_j}{2}\Big [({\varvec{y}}_j-{\varvec{A}}{\varvec{U}}_{ij})^{\top }{\varvec{D}}^{-1}({\varvec{y}}_j-{\varvec{A}}{\varvec{U}}_{ij})\nonumber \\&\qquad +\,({\varvec{U}}_{ij}-{\varvec{\xi }}_{i}-W_j {\varvec{\lambda }}_{i})^{\top }{\varvec{{\varOmega }}}^{-1}_i({\varvec{U}}_{ij}-{\varvec{\xi }}_{i}-W_j {\varvec{\lambda }}_{i})\Big ]\nonumber \\&\qquad + \,\frac{\nu _i}{2}\log \left( \frac{\nu _i}{2}\right) - \log {\varGamma }\left( \frac{\nu _i}{2}\right) -\left( \frac{\nu _i}{2}+1\right) \log W_{j} -\frac{\nu _i}{2W_{j}}\bigg \}. \end{aligned}$$

(D.8)

To evaluate the expected value of (D.8), called the Q function, we first calculate

$$\begin{aligned} {\hat{z}}_{ij}=E\left( Z_{ij}\mid {\varvec{y}}_{j},\hat{{\varvec{\varTheta }} }\right) =\frac{\hat{\pi }_{i}\zeta \left( {\varvec{y}}_{j};\hat{{\varvec{A}}}\hat{{\varvec{\xi }}}_i,\hat{{\varvec{A}}}\hat{{\varvec{\varOmega }}}_i\hat{{\varvec{A}}}^{\top }+\hat{{\varvec{D}}}, \hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i,\hat{\nu }_i\right) }{f({\varvec{y}}_j; \hat{{\varvec{\varTheta }}})}, \end{aligned}$$

(D.9)

which is the posterior probability of ${\varvec{y}}_j$ belonging to the ith component of the mixture. In addition, we utilize the results (D.4) and (D.5) to calculate of the following conditional expectations:

$$\begin{aligned} {\hat{s}}_{1ij}= & {} E(W_{j} \mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}})\nonumber \\= & {} \left( \frac{\hat{\nu }_i+\hat{{\varDelta }}_{ij}}{\hat{{\varvec{\lambda }}}^{\top }_i\hat{{\varvec{A}}}^{\top }(\hat{{\varvec{A}}}\hat{{\varvec{\varOmega }}}_i\hat{{\varvec{A}}}^{\top }+\hat{{\varvec{D}})}^{-1}\hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i}\right) ^{1/2} \frac{K_{-\frac{(\hat{\nu }+p)}{2}+1}(\hat{\omega }_{ij})}{K_{-\frac{(\hat{\nu }+p)}{2}}(\hat{\omega }_{ij})},\nonumber \\ {\hat{s}}_{2ij}= & {} E(W_{j}^{-1} \mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}})\nonumber \\= & {} \left( \frac{\hat{\nu }_i+\hat{{\varDelta }}_{ij}}{\hat{{\varvec{\lambda }}}^{\top }_i\hat{{\varvec{A}}}^{\top }(\hat{{\varvec{A}}}\hat{{\varvec{\varOmega }}}_i\hat{{\varvec{A}}}^{\top }+\hat{{\varvec{D}})}^{-1}\hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i}\right) ^{-1/2} \frac{K_{-\frac{(\hat{\nu }+p)}{2}-1}(\hat{\omega }_{ij})}{K_{-\frac{(\hat{\nu }+p)}{2}}(\hat{\omega }_{ij})},\nonumber \\ {\hat{s}}_{3ij}= & {} E(\log W_{j} \mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}})\nonumber \\= & {} \log \left( \frac{\hat{\nu }_i+\hat{{\varDelta }}_{ij}}{\hat{{\varvec{\lambda }}}^{\top }_i\hat{{\varvec{A}}}^{\top } (\hat{{\varvec{A}}}\hat{{\varvec{\varOmega }}}_i\hat{{\varvec{A}}}^{\top }+\hat{{\varvec{D}})}^{-1}\hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i}\right) ^{1/2} +\frac{K^{\prime }_{-\frac{(\hat{\nu }+p)}{2}}(\hat{\omega }_{ij})}{K_{-\frac{(\hat{\nu }+p)}{2}}(\hat{\omega }_{ij})}, \end{aligned}$$

(D.10)

where $\hat{\omega }_{ij}=\sqrt{(\hat{\nu }_i+\hat{{\varDelta }}_{ij})\hat{{\varvec{\lambda }}}^{\top }_i\hat{{\varvec{A}}}^{\top }(\hat{{\varvec{A}}}\hat{{\varvec{\varOmega }}}_i\hat{{\varvec{A}}}^{\top }+\hat{{\varvec{D}}})^{-1}\hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i}$ and $K^{\prime }_{-\frac{(\hat{\nu }+p)}{2}}(\hat{\omega }_{ij})$ is evaluated via (D.6). By (D.2), we obtain

$$\begin{aligned} \hat{{\varvec{u}}}_{ij}= & {} E\left( {\varvec{U}}_{ij} \mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}}\right) =E\left[ E({\varvec{U}}_{ij} \mid {\varvec{y}}_{j}, W_j,Z_{ij}=1)\mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}} \right] \nonumber \\= & {} \hat{{\varvec{\xi }}}_i+{\hat{s}}_{1ij}\hat{{\varvec{\lambda }}}_i+\hat{{\varvec{\gamma }}}^{\top }_i({\varvec{y}}_j-\hat{{\varvec{A}}}\hat{{\varvec{\xi }}}_i-{\hat{s}}_{1ij} \hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i), \end{aligned}$$

(D.11)

$$\begin{aligned} \hat{{\varvec{\eta }}}_{ij}= & {} E\left( W^{-1}_j{\varvec{U}}_{ij} \mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}}\right) \nonumber \\= & {} E\left[ W^{-1}_jE({\varvec{U}}_{ij} \mid {\varvec{y}}_{j}, W_j,Z_{ij}=1)\mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}} \right] \nonumber \\= & {} \hat{{\varvec{\lambda }}}_i-\hat{{\varvec{\gamma }}}^{\top }_i\hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i+{\hat{s}}_{2ij}\Big (\hat{{\varvec{\xi }}}_i+\hat{{\varvec{\gamma }}}^{\top }_i({\varvec{y}}_j-\hat{{\varvec{A}}}\hat{{\varvec{\xi }}}_i)\Big ), \end{aligned}$$

(D.12)

and

$$\begin{aligned} \hat{{\varvec{{\varPsi }}}}_{ij}= & {} E\left( W^{-1}_j{\varvec{U}}_{ij} {\varvec{U}}^{\top }_{ij}\mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}}\right) \nonumber \\= & {} E\left[ W^{-1}_jE({\varvec{U}}_{ij}{\varvec{U}}^{\top }_{ij} \mid {\varvec{y}}_{j}, W_j,Z_{ij}=1)\mid {\varvec{y}}_{j}, Z_{ij}=1, \hat{{\varvec{{\varTheta }}}} \right] \nonumber \\= & {} E\left[ W^{-1}_j\left[ E({\varvec{U}}_{ij}\mid \right. {\varvec{y}}_{j},\right. W_j,Z_{ij}=1)E({\varvec{U}}^{\top }_{ij}\mid {\varvec{y}}_{j}, W_j,Z_{ij}=1)\nonumber \\&+\,\mathrm{cov}\left. ({\varvec{U}}_{ij}\mid {\varvec{y}}_{j}, W_j,Z_{ij}=1)\right] \mid {\varvec{y}}_{j}, \left. Z_{ij}=1, \hat{{\varvec{{\varTheta }}}} \right] \nonumber \\= & {} \hat{{\varvec{\eta }}}_{ij}\Big (\hat{{\varvec{\xi }}}_i+\hat{{\varvec{\gamma }}}^{\top }_i({\varvec{y}}_j-\hat{{\varvec{A}}}\hat{{\varvec{\xi }}}_i)\Big )^{\top } +\hat{{\varvec{u}}}_{ij}\left( \hat{{\varvec{\lambda }}}_i-\hat{{\varvec{\gamma }}}^{\top }_i\hat{{\varvec{A}}}\hat{{\varvec{\lambda }}}_i\right) ^{\top }\nonumber \\&+\left( \hat{{\varvec{{\varOmega }}}}^{-1}_i+\,\hat{{\varvec{A}}}^{\top }\hat{{\varvec{D}}}^{-1}\hat{{\varvec{A}}}\right) ^{-1}, \end{aligned}$$

(D.13)

where $\hat{{\varvec{\gamma }}}_i = (\hat{{\varvec{A}}} \hat{{\varvec{{\varOmega }}}}_i \hat{{\varvec{A}}}^{\top } + \hat{{\varvec{D}}})^{-1}\hat{{\varvec{A}}}\hat{{\varvec{{\varOmega }}}}_i$.

After some algebraic manipulations, the resulting Q function that gets rid of the constants is given by

$$\begin{aligned} Q({\varvec{{\varTheta }}}\mid \hat{{\varvec{{\varTheta }}}})= & {} \sum ^{g}_{i=1}\sum ^{n}_{j=1}{\hat{z}}_{ij}\Bigg \{\log \pi _i -\frac{1}{2}\log |{\varvec{D}}| -\frac{1}{2}\log |{\varvec{{\varOmega }}}_i| -\frac{1}{2} \mathrm{tr}\Big ({\varvec{{\varvec{D}}}}^{-1}\Big [{\hat{s}}_{2ij}{\varvec{y}}_j{\varvec{y}}^{\top }_j\nonumber \\&-\,{\varvec{y}}_j\hat{{\varvec{\eta }}}_{ij}^{\top }{\varvec{A}}^{\top } -{\varvec{A}}\hat{{\varvec{\eta }}}_{ij}{\varvec{y}}^{\top }_j+{\varvec{A}}\hat{{\varvec{\varPsi }}}_{ij}{\varvec{A}}^{\top }\Big ]\Big )\nonumber \\&-\,\frac{1}{2}\mathrm{tr}\bigg ({\varvec{{\varvec{{\varOmega }}}}}^{-1}_i\Big [\hat{{\varvec{{\varPsi }}}}_{ij}-\hat{{\varvec{\eta }}}_{ij}{\varvec{\xi }}_{i}^{\top }-{\varvec{\xi }}_{i}\hat{{\varvec{\eta }}}_{ij}^{\top }\nonumber \\&+\,{\hat{s}}_{2ij}{\varvec{\xi }}_{i}{\varvec{\xi }}_{i}^{\top }+{\hat{s}}_{1ij}{\varvec{\lambda }}_i{\varvec{\lambda }}^{\top }_i-(\hat{{\varvec{u}}}_{ij}-{\varvec{\xi }}_i){\varvec{\lambda }}^{\top }_i-{\varvec{\lambda }}_i(\hat{{\varvec{u}}}_{ij}-{\varvec{\xi }}_i)^{\top }\Big ]\bigg )\nonumber \\&+ \,\left( \frac{\nu _i}{2}\right) \log \left( \frac{\nu _i}{2}\right) - \frac{\nu _i}{2}{\hat{s}}_{2ij}-\frac{\nu _i}{2}{\hat{s}}_{3ij}- \log {\varGamma }\left( \frac{\nu _i}{2}\right) \Bigg \}. \end{aligned}$$

(D.14)

Taking partial derivatives of (D.14) with respect to ${\varvec{\xi }}_i$ and ${\varvec{\lambda }}_i$ and equating them to zero vectors yield

$$\begin{aligned}&\sum ^{n}_{j=1}{\hat{z}}_{ij}\left( \hat{{\varvec{\eta }}}_{ij}-{\hat{s}}_{2ij}{\varvec{\xi }}_{i}-{\varvec{\lambda }}_{i}\right) ={\varvec{0}},\end{aligned}$$

(D.15)

$$\begin{aligned}&\sum ^{n}_{j=1}{\hat{z}}_{ij}\left( \hat{{\varvec{u}}}_{ij}-{\varvec{\xi }}_{i}-{\hat{s}}_{1ij}{\varvec{\lambda }}_{i}\right) ={\varvec{0}}. \end{aligned}$$

(D.16)

In summary, the ECM algorithm for estimating the parameters of MCghstFA proceeds as follows:

• E-step: Given the current value ${\varvec{\varTheta }}=\hat{{\varvec{\varTheta }}}$, compute ${\hat{z}}_{ij}$, ${\hat{s}}_{1ij}$, ${\hat{s}}_{2ij}$, ${\hat{s}}_{3ij}$, $\hat{{\varvec{u}}}_{ij}$, $\hat{{\varvec{\eta }}}_{ij}$ and $\hat{{\varvec{{\varPsi }}}}_{ij}$ as defined in (D.9)–(D.13) for $i=1,\ldots ,g$ and $j=1,\ldots ,n$.

• CM step 1: Maximizing (D.14) with respect to $\pi _i$ and using the Lagrange multiplier method, this gives $\hat{\pi }_i={\hat{n}}_i/n$, where ${\hat{n}}_i=\sum _{j=1}^n {\hat{z}}_{ij}$.

• CM step 2: Update parameters ${\varvec{\xi }}_i$ and ${\varvec{\lambda }}_i$ by solving simultaneous Eqs. (D.15) and (D.16). Simple matrix algebra yields

  $$\begin{aligned} \hat{{\varvec{\xi }}}_i= & {} \frac{\big (\sum _{j=1}^n{\hat{z}}_{ij}{\hat{s}}_{1ij}\big )\big (\sum _{j=1}^n{\hat{z}}_{ij}\hat{{\varvec{\eta }}}_{ij}\big )- {\hat{n}}_i(\sum _{j=1}^n{\hat{z}}_{ij}\hat{{\varvec{u}}}_{ij}\big )}{\big (\sum _{j=1}^n{\hat{z}}_{ij}{\hat{s}}_{1ij}\big )\big (\sum _{j=1}^n{\hat{z}}_{ij}{\hat{s}}_{2ij}\big )-{\hat{n}}^2_i} \end{aligned}$$

  and

  $$\begin{aligned} \hat{{\varvec{\lambda }}}_i= & {} \frac{\big (\sum _{j=1}^n{\hat{z}}_{ij}{\hat{s}}_{2ij}\big )\big (\sum _{j=1}^n{\hat{z}}_{ij}\hat{{\varvec{u}}}_{ij}\big )- {\hat{n}}_i(\sum _{j=1}^n{\hat{z}}_{ij}\hat{{\varvec{\eta }}}_{ij}\big )}{\big (\sum _{j=1}^n{\hat{z}}_{ij}{\hat{s}}_{1ij}\big )\big (\sum _{j=1}^n{\hat{z}}_{ij}{\hat{s}}_{2ij}\big )-{\hat{n}}^2_i}. \end{aligned}$$

• CM-step3: The updates for ${\varvec{A}}$, ${\varvec{{\varOmega }}}_i$ and ${\varvec{D}}$ are given by

  $$\begin{aligned} \hat{{\varvec{A}}}= & {} \left( \sum _{i=1}^g\sum _{j=1}^n{\hat{z}}_{ij}\hat{{\varvec{y}}}_j\hat{{\varvec{\eta }}}_{ij}^{\top }\right) \left( \sum _{i=1}^g\sum _{j=1}^n{\hat{z}}_{ij}\hat{{\varvec{{\varPsi }}}}_{ij}\right) ^{-1},\\ \hat{{\varvec{{\varOmega }}}}_i= & {} \frac{1}{{\hat{n}}_i}\sum _{j=1}^n{\hat{z}}_{ij}\Big [\hat{{\varvec{{\varPsi }}}}_{ij}-\hat{{\varvec{\eta }}}_{ij}\hat{{\varvec{\xi }}}_{i}^{\top } -\hat{{\varvec{\xi }}}_{i}\hat{{\varvec{\eta }}}_{ij}^{\top }+{\hat{s}}_{2ij}\hat{{\varvec{\xi }}}_{i}\hat{{\varvec{\xi }}}_{i}^{\top }+{\hat{s}}_{1ij}\hat{{\varvec{\lambda }}}_i\hat{{\varvec{\lambda }}}^{\top }_i\\&-\,(\hat{{\varvec{u}}}_{ij}-\hat{{\varvec{\xi }}}_i)\hat{{\varvec{\lambda }}}^{\top }_i-\hat{{\varvec{\lambda }}}_i(\hat{{\varvec{u}}}_{ij}-\hat{{\varvec{\xi }}}_i)^{\top }\Big ],\\ \hat{{\varvec{D}}}= & {} \frac{1}{n}\mathrm{Diag}\Bigg \{\sum _{i=1}^g\sum _{j=1}^n{\hat{z}}_{ij}\big ({\hat{s}}_{2ij}{\varvec{y}}_j{\varvec{y}}^{\top }_j-{\varvec{y}}_j\hat{{\varvec{\eta }}}_{ij}^{\top }\hat{{\varvec{A}}}^{\top }\big )\Bigg \}. \end{aligned}$$

• CM step 4: Calculate $\hat{\nu }_i$ by solving the root of the following equation:

  $$\begin{aligned} \log \left( \frac{\nu _i}{2}\right) - \mathrm{DG}\left( \frac{\nu _i}{2}\right) + 1 -\frac{1}{{\hat{n}}_i} \sum _{j=1}^n {\hat{z}}_{ij}({\hat{s}}_{2ij} + {\hat{s}}_{3ij})=0. \end{aligned}$$