On moments of folded and truncated multivariate Student-t distributions based on recurrence relations

Abstract

The use of the first two moments of the truncated multivariate Student-t distribution has attracted increasing attention from a wide range of applications. This paper develops recurrence relations for integrals that involve the density of multivariate Student-t distributions. The proposed techniques allow for fast computation of arbitrary-order product moments of folded and truncated multivariate Student-t distributions and offer explicit expressions of low-order moments of folded and truncated multivariate Student-t distributions. A real data example containing positive censored responses is applied to illustrate the effectiveness and importance of the proposed methods. An R MomTrunc package is developed and publicly available on the CRAN repository.

Appendix

1.1 Appendix A: Details for the expectations in EM algorithm

To compute the required expected values of all latent data, we find that most of them can be written in terms of $\mathbb {E}(U_i \mid \mathbf{Y }_i)$, and thereby we write ${\widehat{u}}_i= \mathbb {E}\{ \mathbb {E}(U_i\mid \mathbf{Y }_i)\mid \mathbf{V }_i,\mathbf{C }_i,{\widehat{{\varvec{\theta }}}}^{(k)}\}$, where $\mathbb {E}(U_i \mid \mathbf{Y }_i)=(\nu +p)/(\nu +\delta )$ with $\delta =(\mathbf{Y }_i-{\varvec{\mu }})^{\top }{\varvec{\Sigma }}^{-1}(\mathbf{Y }_i-{\varvec{\mu }})$. Subsequently, we discuss the closed-form expressions of conditional expectations as follows:

• 1. If the ith subject has only non-censored components, then

  $$\begin{aligned} \widehat{u\mathbf{y }_i^{2}}^{(k)}= & {} \left\{ \frac{\nu +p}{\nu +{\widehat{\delta }}^{(k)}(\mathbf{y }_i)} \right\} \mathbf{y }_i\mathbf{y }^{\top }_i,\quad \widehat{u\mathbf{y }}^{(k)}_i=\left\{ \frac{\nu +p}{\nu +{\widehat{\delta }}^{(k)}(\mathbf{y }_i)}\right\} \mathbf{y }_i,\quad {\widehat{u}}^{(k)}_i=\left\{ \frac{\nu +p}{\nu +{\widehat{\delta }}^{(k)}(\mathbf{y }_i)}\right\} , \end{aligned}$$

  where ${\widehat{\delta }}^{(k)}(\mathbf{y }_i)=(\mathbf{y }_i-{\widehat{{\varvec{\mu }}}}^{(k)})^{\top } ({\widehat{{\varvec{\Sigma }}}}^{(k)})^{-1}(\mathbf{y }_i-{\widehat{{\varvec{\mu }}}}^{(k)}).$

• 2. If the ith subject has only censored components, from Proposition 3 with $r=1$, we have

  $$\begin{aligned} \widehat{u\mathbf{y }_i^{2}}^{(k)}= & {} \mathbb {E}[\displaystyle U_i\mathbf{Y }_i\mathbf{Y }_i^{\top } \mid \mathbf{V }_i,\mathbf{C }_i,{\widehat{{\varvec{\theta }}}}^{(k)}]={\widehat{\varphi }}^{(k)}(\mathbf{V }_i) {\widehat{\mathbf{w }}}_i^{2 ^{c(k)}},\,\\ \widehat{u\mathbf{y }}^{(k)}_i= & {} \mathbb {E}[\displaystyle U_i\mathbf{Y }_i\mid \mathbf{V }_i,\mathbf{C }_i, {\widehat{{\varvec{\theta }}}}^{(k)}]={\widehat{\varphi }}^{(k)}(\mathbf{V }_i) {\widehat{\mathbf{w }}}_i^{c(k)},\,\, \\ {\widehat{u}}^{(k)}_i= & {} \mathbb {E}[\displaystyle U_i\mid \mathbf{V }_i,\mathbf{C }_i,{\widehat{{\varvec{\theta }}}}^{(k)}]={\widehat{\varphi }}^{(k)}(\mathbf{V }_i), \end{aligned}$$

  where

  $$\begin{aligned} {\widehat{\varphi }}^{(k)}(\mathbf{V }_i)=\frac{L_{p}({\mathbf{V }}_{1i}, {\mathbf{V }}_{2i};{\widehat{{\varvec{\mu }}}}^{(k)}, \quad {\widehat{{\varvec{\Sigma }}}}^{*(k)},\nu +2)}{L_{p}({\mathbf{V }}_{1i}, {\mathbf{V }}_{2i};{\widehat{{\varvec{\mu }}}}^{(k)}, \quad {\widehat{{\varvec{\Sigma }}}}^{(k)},\nu )},\nonumber \\ {\widehat{\mathbf{w }}}_i^{c(k)}=\mathbb {E}[\mathbf{W }_i\mid {\widehat{{\varvec{\theta }}}}^{(k)}], \quad {\widehat{\mathbf{w }}}_i^{2 ^{c(k)}}=\mathbb {E}[\mathbf{W }_i \mathbf{W }_i^\top \mid {\widehat{{\varvec{\theta }}}}^{(k)}] {,} \end{aligned}$$

  (16)

  with $\mathbf{W }_i\sim Tt_{p}({\widehat{{\varvec{\mu }}}}^{(k)},{\widehat{{\varvec{\Sigma }}}}^{*(k)},\nu +2; (\mathbf{V }_{1i},\mathbf{V }_{2i}))$ and ${\widehat{{\varvec{\Sigma }}}}^{*(k)}=\displaystyle \frac{\nu }{\nu +2}{\widehat{{\varvec{\Sigma }}}}^{(k)}$. To compute $\mathbb {E}[\mathbf{W }_i]$ and $\mathbb {E}[\mathbf{W }_i\mathbf{W }_i^{\top }]$ we use the results given in Sect. 3.1.

• 3. If the ith subject has both censored and uncensored components, then $(\mathbf{Y }_i\mid \mathbf{V }_i,\mathbf{C }_i)$, $(\mathbf{Y }_i\mid \mathbf{V }_i,\mathbf{C }_i,\mathbf{y }^o_i)$, and $(\mathbf{Y }^c_i\mid \mathbf{V }_i,\mathbf{C }_i,\mathbf{y }^o_i)$ are equivalent processes. We obtain

  $$\begin{aligned} \widehat{u\mathbf{y }_i^{2}}^{(k)}&=\mathbb {E}(\displaystyle U_i\mathbf{Y }_i\mathbf{Y }_i^{\top }\mid \mathbf{y }^o_i,\mathbf{V }_i,\mathbf{C }_i,{\widehat{{\varvec{\theta }}}}^{(k)}) =\left( \begin{array}{cc} \mathbf{y }^o_i\mathbf{y }^{o\top }_i {\widehat{u}}^{(k)}_i &{} {\widehat{u}}^{(k)}_i\mathbf{y }^o_i \widehat{\mathbf{w }}^{c(k)\top }_i \\ {\widehat{u}}^{(k)}_i\widehat{\mathbf{w }}^{c(k)}_i\mathbf{y }^{o\top }_i &{}{\widehat{u}}^{(k)}_i {\widehat{\mathbf{w }}}_i^{2 ^{c(k)}}\\ \end{array}\right) ,\\ \widehat{u\mathbf{y }}^{(k)}_i&=\mathbb {E}(\displaystyle U_i\mathbf{Y }_i\mid \mathbf{y }^o_i,\mathbf{V }_i,\mathbf{C }_i, {\widehat{{\varvec{\theta }}}}^{(k)})=\mathrm {vec}(\mathbf{y }^o_i{\widehat{u}}^{(k)}_i, {\widehat{u}}^{(k)}_i\widehat{\mathbf{w }}^{c(k)}_i),\,\, \\ {\widehat{u}}^{(k)}_i&=\mathbb {E}(\displaystyle U_i\mid \mathbf{y }^o_i,\mathbf{V }_i,\mathbf{C }_i,{\widehat{{\varvec{\theta }}}}^{(k)})=\left\{ \frac{p^o_i+\nu }{\nu +{\widehat{\delta }}^{(k)}(\mathbf{y }^o_i)}\right\} \\&\displaystyle \frac{L_{p_i^c}(\mathbf{V }_{1i}^c,\mathbf{V }_{2i}^c;{\widehat{{\varvec{\mu }}}}^{co(k)}_i, \widetilde{\mathbf{S }}_i^{co(k)},\nu +p^o_i+2)}{L_{p_i^c}(\mathbf{V }_{1i}^c, \mathbf{V }_{2i}^c;{\widehat{{\varvec{\mu }}}}^{co(k)}_i, {\widetilde{\mathbf{S }}}_i^{co(k)},\nu +p^o_i)}, \end{aligned}$$

  where

  $$\begin{aligned} \widetilde{\mathbf{S }}_i^{co(k)}=\left\{ \displaystyle \frac{\nu +{\widehat{\delta }}^{(k)}(\mathbf{y }^o_i)}{\nu +2+p^o_i}\right\} {\widehat{{\varvec{\Sigma }}}}^{cc.o(k)}_{i}, \quad {\widehat{\delta }}^{(k)}(\mathbf{y }^o_i)=(\mathbf{y }^o_i-{\widehat{{\varvec{\mu }}}}^{o(k)}_i)^{\top } (\widehat{{{\varvec{\Sigma }}}}^{oo(k)}_i)^{-1}(\mathbf{y }^o_i-{\widehat{{\varvec{\mu }}}}^{o(k)}_i), \end{aligned}$$

  ${\widehat{{\varvec{\Sigma }}}}^{cc.o(k)}_{i}$ is defined as in equation (4.22) in the main document, $\widehat{\mathbf{w }}^{c(k)}_i$ and ${\widehat{\mathbf{w }}}_i^{2 ^{c(k)}}$ are defined in (16) with $\mathbf{W }_i\sim Tt_{p^c_i}({\widehat{{\varvec{\mu }}}}^{co(k)}_i,\widetilde{\mathbf{S }}_i^{co(k)}, \nu +p^o_i+2; (\mathbf{V }^c_{1i},\mathbf{V }^c_{2i}))$. Similarly, to compute $\mathbb {E}[\mathbf{W }_i]$ and $\mathbb {E}[\mathbf{W }_i\mathbf{W }_i^{\top }]$, we use the results given in Sect. 3.1.

1.2 Appendix B: Some illustrations using the R MomTrunc package