Appendix 1
Derivation of Conditional posterior distribution of
\({\varvec{\uprho}}\)
given
\(\left({\varvec{\upbeta}},{\varvec{\upvartheta}},{\varvec{\updelta}}\right)\)
:
Let us write
$${\mathfrak{z}}_{t}\left(\beta \right)={y}_{t}-{x}_{t}^{^{\prime}}\beta .$$
Further, we define
$${\widehat{\rho }}^{*}\equiv {\widehat{\rho }}^{*}\left(\beta ,\vartheta \right)=\frac{{B}_{1}}{{A}_{1}},$$
$${A}_{1}\equiv {A}_{1}(\beta )=\sum_{t=1}^{n}{\mathfrak{z}}_{t-1}^{2}\left(\beta \right)$$
$${B}_{1}\equiv {B}_{1}(\beta ,\vartheta )=\sum_{t=1}^{{n}_{1}}{\mathfrak{z}}_{t}\left(\beta \right){\mathfrak{z}}_{t-1}\left(\beta \right)+\sum_{t={n}_{1}+1}^{n}\left({\mathfrak{z}}_{t}\left(\beta \right)-\vartheta {\mathfrak{z}}_{t-1}\left(\beta \right)\right){\mathfrak{z}}_{t-1}\left(\beta \right)$$
$${\phi }_{1}\left(\beta ,\vartheta \right)=\sum_{t=1}^{{n}_{1}}{{\mathfrak{z}}_{t}}^{2}\left(\beta \right)+\sum_{t={n}_{1}+1}^{n}{\left({\mathfrak{z}}_{t}\left(\beta \right)-\vartheta {\mathfrak{z}}_{t-1}\left(\beta \right)\right)}^{2}-\frac{{B}_{1}^{2}}{{A}_{1}}$$
$$\widehat{\rho }\equiv \widehat{\rho }\left(\beta ,\delta \right)=\frac{{B}_{2}}{{A}_{2}},$$
$${A}_{2}\equiv {A}_{2}\left(\beta ,\delta \right)=\sum_{t=1}^{{n}_{1}}{\mathfrak{z}}_{t-1}^{2} \left(\beta \right)+\delta \sum_{t={n}_{1}+1}^{n}{\mathfrak{z}}_{t-1}^{2} \left(\beta \right),$$
$${B}_{2}\equiv {B}_{2}\left(\beta ,\delta \right)=\sum_{t=1}^{{n}_{1}}{\mathfrak{z}}_{t}\left(\beta \right){\mathfrak{z}}_{t-1}\left(\beta \right)+\delta \sum_{t={n}_{1}+1}^{n}{\mathfrak{z}}_{t}\left(\beta \right){\mathfrak{z}}_{t-1}\left(\beta \right)$$
$${\phi }_{2}\left(\beta ,\delta \right)=\sum_{t=1}^{{n}_{1}}{{\mathfrak{z}}_{t}}^{2}\left(\beta \right)+\delta \sum_{t={n}_{1}+1}^{n}{{\mathfrak{z}}_{t}}^{2}\left(\beta \right)-\frac{{B}_{2}^{2}}{{A}_{2}},$$
The likelihood function (9) can be written as
$$p\left(y|X,\beta ,\tau ,\delta ,\rho ,\vartheta \right)$$
$$=\left(1-\epsilon \right){\left(\frac{\tau }{2\pi }\right)}^\frac{n}{2}exp\left[-\frac{\tau }{2}\left\{\sum_{t=1}^{{n}_{1}}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\rho {\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}+\sum_{t={n}_{1}+1}^{n}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\left(\rho +\vartheta \right){\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}\right\}\right]$$
$$+\epsilon {\delta }^{\frac{{n}_{2}}{2}}{\left(\frac{\tau }{2\pi }\right)}^\frac{n}{2}exp\left[-\frac{\tau }{2}\left\{\sum_{t=1}^{{n}_{1}}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\rho {\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}+\delta \sum_{t={n}_{1}+1}^{n}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\rho {\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}\right\}\right]$$
(33)
Notice that the first part of the likelihood function with \(\left(1-\epsilon \right)\) gives the likelihood under model \({M}_{1}\), whereas the second part with \(\epsilon\) gives the likelihood function under model \({M}_{2}\). For obtaining the conditional \(\uprho\) given \(\left(\upbeta ,\updelta ,\mathrm{\vartheta }\right)\), combining the likelihood under the model \({M}_{1}\) with the prior distributions \(p\left(\rho |\vartheta \right), p\left(\tau \right)\), and integrating with respect to \(\tau\) we obtain
$${\pi }_{1}\left(\rho |\beta ,\delta ,\vartheta \right)$$
$$\propto \frac{1}{1-\vartheta }{\int }_{0}^{\infty }{\tau }^{\frac{n}{2}-1}exp\left[-\frac{\tau }{2}\left\{\sum_{t=1}^{{n}_{1}}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\rho {\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}+\sum_{t={n}_{1}+1}^{n}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\left(\rho +\vartheta \right){\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}\right\}\right]d\tau$$
We observe that
$$\sum_{t=1}^{{n}_{1}}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\rho {\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}+\sum_{t={n}_{1}+1}^{n}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\left(\rho +\vartheta \right){\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}$$
$$={\rho }^{2}\sum_{t=1}^{n}{\mathfrak{z}}_{t-1}{\left(\beta \right)}^{2}-2\rho \left\{\sum_{t=1}^{{n}_{1}}{\mathfrak{z}}_{t}\left(\beta \right){\mathfrak{z}}_{t-1}\left(\beta \right)+\sum_{t={n}_{1}+1}^{n}\left({\mathfrak{z}}_{t}\left(\beta \right)-\vartheta {\mathfrak{z}}_{t-1}\left(\beta \right)\right){\mathfrak{z}}_{t-1}\left(\beta \right)\right\}+\sum_{t=1}^{{n}_{1}}{\mathfrak{z}}_{t}{\left(\beta \right)}^{2}+\sum_{t={n}_{1}+1}^{n}{\left({\mathfrak{z}}_{t}\left(\beta \right)-\vartheta {\mathfrak{z}}_{t-1}\left(\beta \right)\right)}^{2}$$
$$={\left(\rho -{\widehat{\rho }}^{*}\right)}^{2}{A}_{1}+{\phi }_{1}(\beta ,\vartheta )$$
Hence, we obtain
$${\pi }_{1}\left(\rho |\beta ,\vartheta \right)$$
$$\propto \frac{1}{1-\vartheta }{\int }_{0}^{\infty }{\tau }^{\frac{n}{2}-1}exp\left[-\frac{\tau }{2}\left\{{\left(\rho -{\widehat{\rho }}^{*}\right)}^{2}{A}_{1}+{\phi }_{1}\left(\beta ,\vartheta \right)\right\}\right]d\tau$$
$$\propto \frac{1}{1-\vartheta }\frac{1}{{\left\{{\left(\rho -{\widehat{\rho }}^{*}\right)}^{2}{A}_{1}+{\phi }_{1}\left(\beta ,\vartheta \right)\right\}}^\frac{n}{2}}.$$
Therefore
$${\pi }_{1}\left(\rho |\beta ,\delta ,\vartheta \right)={C}_{1\rho }^{-1}\frac{1}{1-\vartheta }\frac{1}{{\left\{{\left(\rho -{\widehat{\rho }}^{*}\right)}^{2}{A}_{1}+{\phi }_{1}\left(\beta ,\vartheta \right)\right\}}^\frac{n}{2}},$$
(34)
where
$${C}_{1\rho }=\frac{1}{1-\vartheta }{\int }_{0}^{1-\vartheta }\frac{1}{{\left\{{\left(\rho -{\widehat{\rho }}^{*}\right)}^{2}{A}_{1}+{\phi }_{1}\left(\beta ,\vartheta \right)\right\}}^\frac{n}{2}}d\rho$$
$$=\frac{\mathrm{\rm B}\left(\frac{1}{2},\frac{n-1}{2}\right)}{\left(1-\vartheta \right){\phi }_{1}{\left(\beta ,\vartheta \right)}^{\frac{n-1}{2}}{A}_{1}^\frac{1}{2}}{\int }_{-\widehat{\rho }\sqrt{\frac{\left(n-1\right){A}_{1}}{{\phi }_{1}\left(\beta ,\vartheta \right)}}}^{\left(1-\vartheta -\widehat{\rho }\right)\sqrt{\frac{\left(n-1\right){A}_{1}}{{\phi }_{1}\left(\beta ,\vartheta \right)}}}{f}_{n-1}\left(t\right)dt$$
$$=\frac{\mathrm{\rm B}\left(\frac{1}{2},\frac{n-1}{2}\right)}{\left(1-\vartheta \right){\phi }_{1}{\left(\beta ,\vartheta \right)}^{\frac{n-1}{2}}{A}_{1}^\frac{1}{2}}\left[{F}_{n-1}\left(\left(1-\vartheta -\widehat{\rho }\right)\sqrt{\frac{\left(n-1\right){A}_{1}}{{\phi }_{1}\left(\beta ,\vartheta \right)}}\right)+{F}_{n-1}\left(\widehat{\rho }\sqrt{\frac{\left(n-1\right){A}_{1}}{{\phi }_{1}\left(\beta ,\vartheta \right)}}\right)-1\right],$$
(35)
where \({f}_{n-1}\left(t\right)\) and \({F}_{n-1}(t)\) denote, respectively, the pdf and cdf of t-distribution with (n − 1) degrees of freedom.
Further, we have
$$\sum_{t=1}^{{n}_{1}}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\rho {\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}+\delta \sum_{t={n}_{1}+1}^{n}{\left\{{\mathfrak{z}}_{t}\left(\beta \right)-\rho {\mathfrak{z}}_{t-1}\left(\beta \right)\right\}}^{2}={A}_{2}{\left(\rho -\widehat{\rho }\right)}^{2}+{\phi }_{2}\left(\beta ,\delta \right)$$
Hence, under model \({M}_{2}\) the posterior density of \(\rho\) given \((\beta ,\delta )\) is
$${\pi }_{2}\left(\rho |\beta ,\delta \right)\propto {\int }_{0}^{\infty }{\tau }^{\frac{n}{2}-1}exp\left[-\frac{\tau }{2}\left\{{\left(\rho -\widehat{\rho }\right)}^{2}{A}_{2}+{\phi }_{2}\left(\beta ,\delta \right)\right\}\right]d\tau$$
$$\propto \frac{1}{{\left\{{\left(\rho -\widehat{\rho }\right)}^{2}{A}_{2}+{\phi }_{2}\left(\beta ,\delta \right)\right\}}^\frac{n}{2}} ,$$
so that,
$${\pi }_{2}\left(\rho |\beta ,\delta \right)={\mathrm{C}}_{2\uprho }^{-1}\frac{1}{{\left\{{\left(\rho -\widehat{\rho }\right)}^{2}{A}_{2}+{\phi }_{2}\left(\beta ,\delta \right)\right\}}^\frac{n}{2}}$$
(36)
with
$${C}_{2\rho }={\int }_{0}^{1}\frac{1}{{\left\{{\left(\rho -\widehat{\rho }\right)}^{2}{A}_{2}+{\phi }_{2}\left(\beta ,\delta \right)\right\}}^\frac{n}{2}}d\rho$$
$$=\frac{\mathrm{\rm B}\left(\frac{1}{2},\frac{n-1}{2}\right)}{{\phi }_{2}{\left(\beta ,\delta \right)}^{\frac{n-1}{2}}{A}_{2}^\frac{1}{2}}{\int }_{-\widehat{\rho }\sqrt{\frac{\left(n-1\right){A}_{2}}{{\phi }_{2}\left(\beta ,\delta \right)}}}^{\left(1-\widehat{\rho }\right)\sqrt{\frac{\left(n-1\right){A}_{2}}{{\phi }_{2}\left(\beta ,\delta \right)}}}{f}_{n-1}\left(t\right)dt$$
$$=\frac{\mathrm{\rm B}\left(\frac{1}{2},\frac{n-1}{2}\right)}{{\phi }_{2}{\left(\beta ,\delta \right)}^{\frac{n-1}{2}}{A}_{2}^\frac{1}{2}}\left[{F}_{n-1}\left(\left(1-\widehat{\rho }\right)\sqrt{\frac{\left(n-1\right){A}_{2}}{{\phi }_{2}\left(\beta ,\delta \right)}}\right)+{F}_{n-1}\left(\widehat{\rho }\sqrt{\frac{\left(n-1\right){A}_{2}}{{\phi }_{2}\left(\beta ,\delta \right)}}\right)-1\right].$$
(37)
Further
$${\lambda }_{\rho }\left(y\right)=\frac{\left(1-\epsilon \right){m}_{1\rho }\left(y\right)}{\left(1-\epsilon \right){m}_{1\rho }\left(y\right)+\epsilon {m}_{2\rho }\left(y\right)},$$
where
$${m}_{1\rho }\left(y\right)=\frac{1}{1-\vartheta }{\int }_{0}^{1-\vartheta }{\int }_{0}^{\infty }\frac{{\tau }^{\frac{n}{2}-1}}{{\left(2\pi \right)}^\frac{n}{2}}exp\left[-\frac{\tau }{2}\left\{{\left(\rho -{\widehat{\rho }}^{*}\right)}^{2}{A}_{1}+{\phi }_{1}\left(\beta ,\vartheta \right)\right\}\right]d\tau d\rho =\frac{\Gamma \left(\frac{n}{2}\right)}{{\pi }^\frac{n}{2}}{C}_{1\rho }$$
$${m}_{2\rho }\left(y\right)={\int }_{0}^{1}{\int }_{0}^{\infty }\frac{{\tau }^{\frac{n}{2}-1}}{{\left(2\pi \right)}^\frac{n}{2}}\mathrm{exp}[-\frac{\tau }{2}\left\{{\left(\rho -\widehat{\rho }\right)}^{2}{A}_{2}+{\phi }_{2}\left(\beta ,\delta \right)\right\}]d\tau d\rho =\frac{\Gamma \left(\frac{n}{2}\right)}{{\pi }^\frac{n}{2}}{C}_{2\rho }.$$
Thus
$${\lambda }_{\rho }\left(y\right)=\frac{\left(1-\epsilon \right){C}_{1\rho }}{\left(1-\epsilon \right){C}_{1\rho }+\epsilon {C}_{2\rho }}$$
(38)
Then the posterior density of \(\uprho\) given \(\left(\upbeta ,\mathrm{\vartheta },\updelta \right)\) is
$${\pi }^{*}\left(\rho |\beta ,\vartheta ,\delta \right)={\lambda }_{\rho }\left(y\right){\pi }_{1}\left(\rho |\beta ,\vartheta \right)+\left(1-{\lambda }_{\rho }\left(y\right)\right){\pi }_{2}\left(\rho |\beta ,\delta \right)$$
(39)
Derivation of Conditional Posterior Density of
\({\varvec{\beta}}\)
given
\(\left({\varvec{\uprho}},{\varvec{\upvartheta}},{\varvec{\updelta}}\right)\)
:
For deriving the conditional posterior density of \(\beta\) given \(\left(\uprho ,\mathrm{\vartheta },\updelta \right)\), we define
$$\mathcal{y}\left(\rho \right)={y}_{t}-\rho {y}_{t-1};t=1,\dots ,n$$
$${\mathcal{y}}_{t}\left(\rho +\vartheta \right)={y}_{t}-\left(\rho +\vartheta \right){y}_{t-1};t={n}_{1}+1,\dots ,n; \left(\mathrm{under model }{\mathrm{M}}_{1}\right)$$
$${\mathcal{x}}_{t}\left(\rho \right)={x}_{t}-\rho {x}_{t-1};t=1,\dots ,{n}_{1}$$
$${\mathcal{x}}_{t}\left(\rho +\vartheta \right)={x}_{t}-\left(\rho +\vartheta \right){x}_{t-1};t={n}_{1}+1,\dots ,n; \left(\mathrm{under model }{\mathrm{M}}_{1}\right)$$
$${A}_{3}\left(\rho ,\vartheta \right)\equiv {A}_{3}=\left(\sum_{t=1}^{{n}_{1}}{\mathcal{x}}_{t}\left(\rho \right){\mathcal{x}}_{t}{\left(\rho \right)}^{\mathrm{^{\prime}}}+\sum_{t={n}_{1}+1}^{n}{\mathcal{x}}_{t}\left(\rho +\vartheta \right){\mathcal{x}}_{t}{\left(\rho +\vartheta \right)}^{\mathrm{^{\prime}}}\right)$$
$${A}_{4}\left(\rho ,\delta \right)\equiv {A}_{4}=\left(\sum_{t=1}^{{n}_{1}}{\mathcal{x}}_{t}\left(\rho \right){\mathcal{x}}_{t}{\left(\rho \right)}^{\mathrm{^{\prime}}}+\delta \sum_{t={n}_{1}+1}^{n}{\mathcal{x}}_{t}\left(\rho \right){\mathcal{x}}_{t}{\left(\rho \right)}^{\mathrm{^{\prime}}}\right)$$
$${\mathcal{w}}_{3}\left(\rho ,\vartheta \right)=\left(\sum_{t=1}^{{n}_{1}}{\mathcal{x}}_{t}\left(\rho \right){\mathcal{y}}_{t}\left(\rho \right)+\sum_{t={n}_{1}+1}^{n}{\mathcal{x}}_{t}\left(\rho +\vartheta \right){\mathcal{y}}_{t}\left(\rho +\vartheta \right)\right)$$
$${\mathcal{w}}_{4}\left(\rho ,\delta \right)=\left(\sum_{t=1}^{{n}_{1}}{\mathcal{x}}_{t}\left(\rho \right){\mathcal{y}}_{t}\left(\rho \right)+\delta \sum_{t={n}_{1}+1}^{n}{\mathcal{x}}_{t}\left(\rho \right){\mathcal{y}}_{t}\left(\rho \right)\right)$$
$$\widehat{\beta }\left(\rho ,\vartheta \right)={\left({A}_{3}+V\right)}^{-1}\left({\mathcal{w}}_{3}\left(\rho ,\vartheta \right)+V{\beta }_{0}\right)$$
$$\widehat{\beta }\left(\rho ,\delta \right)={\left({A}_{4}+V\right)}^{-1}\left({\mathcal{w}}_{4}\left(\rho ,\delta \right)+V{\beta }_{0}\right)$$
$${\phi }_{3}\left(\rho ,\vartheta \right)=\sum_{t=1}^{{n}_{1}}{{\mathcal{y}}_{t}\left(\rho \right)}^{2}+\sum_{t={n}_{1}+1}^{n}{{\mathcal{y}}_{t}\left(\rho +\vartheta \right)}^{2}+{\beta }_{0}^{^{\prime}}V{\beta }_{0}-\widehat{\beta }{\left(\rho ,\vartheta \right)}^{^{\prime}}\left({A}_{3}+V\right)\widehat{\beta }\left(\rho ,\vartheta \right)$$
$${\phi }_{4}\left(\rho ,\delta \right)=\sum_{t=1}^{{n}_{1}}{{\mathcal{y}}_{t}\left(\rho \right)}^{2}+\delta \sum_{t={n}_{1}+1}^{n}{{\mathcal{y}}_{t}\left(\rho \right)}^{2}+{\beta }_{0}^{^{\prime}}V{\beta }_{0}-\widehat{\beta }{\left(\rho ,\delta \right)}^{^{\prime}}\left({A}_{4}+V\right)\widehat{\beta }\left(\rho ,\delta \right)$$
Then, under the model \({M}_{1}\), combining the likelihood with the prior distributions of \((\beta ,\tau )\), gives the posterior distribution of \(\beta\) given (\(\rho ,\vartheta )\) as
$${\pi }_{1}\left(\beta |\rho ,\vartheta \right)$$
$$={C}_{1\beta }^{-1}\frac{1}{{\left(2\pi \right)}^\frac{k}{2}}{\int }_{0}^{\infty }{\tau }^{\frac{n+k}{2}-1}exp\left[-\frac{\tau }{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)}^{^{\prime}}\left({A}_{3}+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)\right\}\right]d\tau$$
$$={C}_{1\beta }^{-1}\frac{{2}^{\frac{\mathrm{n}}{2}}\Gamma \left(\frac{n+k}{2}\right)}{{{\pi }^\frac{k}{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)}^{^{\prime}}\left({A}_{3}+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)\right\}}^{\frac{n+k}{2}}}$$
(40)
where
$${C}_{1\beta }\equiv {C}_{1\beta } (\rho ,\vartheta )$$
$$={\int }_{0}^{\infty }\frac{1}{{\left(2\pi \right)}^\frac{k}{2}}{\int }_{{R}^{k}}{\tau }^{\frac{n+k}{2}-1}exp\left[-\frac{\tau }{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)}^{^{\prime}}\left({A}_{3}+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)\right\}\right]d\beta d\tau$$
$$=\frac{{2}^{\frac{\mathrm{n}}{2}}\Gamma \left(\frac{n}{2}\right)}{{\left|{A}_{3}+V\right|}^\frac{1}{2}{{\phi }_{3}\left(\rho ,\vartheta \right)}^\frac{n}{2}}$$
(41)
Further, under model \({M}_{2}\), the posterior distribution of \(\beta\) given (\(\rho ,\delta )\) is obtained as
$${\pi }_{2}\left(\beta |\rho ,\delta \right)={C}_{2\beta }^{-1}\frac{{2}^{\frac{\mathrm{n}}{2}}\Gamma \left(\frac{n+k}{2}\right)}{{{\pi }^\frac{k}{2}\left\{{\phi }_{4}\left(\rho ,\delta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)}^{\mathrm{^{\prime}}}\left({A}_{4}+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)\right\}}^{\frac{n+k}{2}}}$$
(42)
where
$${C}_{2\beta }\equiv {C}_{2\beta } (\rho ,\delta )$$
$$={\int }_{0}^{\infty }\frac{1}{{\left(2\pi \right)}^\frac{k}{2}}{\int }_{{R}^{k}}{\tau }^{\frac{n+k}{2}-1}exp\left[-\frac{\tau }{2}\left\{{\phi }_{4}\left(\rho ,\delta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)}^{^{\prime}}{\left({A}_{4}+V\right)}^{-1}\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)\right\}\right]d\beta d\tau$$
$$=\frac{{2}^{\frac{\mathrm{n}}{2}}\Gamma \left(\frac{n}{2}\right)}{{\left|{A}_{4}+V\right|}^\frac{1}{2}{{\phi }_{4}\left(\rho ,\delta \right)}^\frac{n}{2}}$$
(43)
Then the posterior density of \(\upbeta\) given \(\left(\uprho ,\mathrm{\vartheta },\updelta \right)\) is
$${\pi }^{*}\left(\beta |\rho ,\vartheta ,\delta \right)={\lambda }_{\beta }\left(y\right){\pi }_{1}\left(\beta |\rho ,\vartheta \right)+\left(1-{\lambda }_{\beta }\left(y\right)\right){\pi }_{2}\left(\beta |\rho ,\delta \right)$$
(44)
where
$${\lambda }_{\beta }\left(y\right)\equiv {\lambda }_{\beta }\left(y|\rho ,\vartheta ,\delta \right)=\frac{\left(1-\epsilon \right){C}_{1\beta }}{\left(1-\epsilon \right){C}_{1\beta }+\epsilon {C}_{2\beta }}.$$
Derivation of conditional posterior distribution of
\({\varvec{\tau}}\)
given
\(\left({\varvec{\rho}},\boldsymbol{\vartheta },{\varvec{\delta}}\right):\)
Under model \({M}_{1}\), the conditional posterior distribution of \(\tau\) given \(\left(\rho ,\vartheta ,\delta =1\right)\) is
$${\pi }_{1}\left(\tau |\rho ,\vartheta \right)$$
$$\propto \frac{1}{{\left(2\pi \right)}^\frac{k}{2}}{\int }_{{R}^{k}}{\tau }^{\frac{n+k}{2}-1}exp\left[-\frac{\tau }{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)}^{^{\prime}}\left({A}_{3}+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)\right\}\right]d\beta$$
$$\propto {\tau }^{\frac{n}{2}-1}exp\left[-\frac{\tau }{2}{\phi }_{3}\left(\rho ,\vartheta \right)\right]$$
Hence
$${\pi }_{1}\left(\tau |\rho ,\vartheta \right)=\frac{{\phi }_{3}{\left(\rho ,\vartheta \right)}^\frac{n}{2}{\tau }^{\frac{n}{2}-1}}{{2}^\frac{n}{2}\Gamma (\frac{n}{2})}exp\left[-\frac{\tau }{2}{\phi }_{3}\left(\rho ,\vartheta \right)\right]$$
(45)
Similarly, under model \({M}_{2}\), the conditional posterior distribution of \(\tau\) given \(\left(\rho ,\delta ,\vartheta =0\right)\) is obtained as
$${\pi }_{2}\left(\tau |\rho ,\delta \right)=\frac{{\phi }_{4}{\left(\rho ,\delta \right)}^\frac{n}{2}{\tau }^{\frac{n}{2}-1}}{{2}^\frac{n}{2}\Gamma (\frac{n}{2})}exp\left[-\frac{\tau }{2}{\phi }_{4}\left(\rho ,\delta \right)\right]$$
(46)
Further, the conditional posterior distribution of \(\tau\) given \(\left(\rho ,\delta ,\vartheta \right)\) is given by
$${\pi }^{*}\left(\tau |\rho ,\vartheta ,\delta \right)={\lambda }_{\tau }\left(y\right){\pi }_{1}\left(\tau |\rho ,\vartheta \right)+\left(1-{\lambda }_{\tau }\left(y\right)\right){\pi }_{2}\left(\tau |\rho ,\delta \right)$$
(47)
where
$${\lambda }_{\tau }\left(y\right)=\frac{\frac{\left(1-\epsilon \right)}{{\phi }_{3}{\left(\rho ,\vartheta \right)}^\frac{n}{2}}}{\frac{\left(1-\epsilon \right)}{{\phi }_{3}{\left(\rho ,\vartheta \right)}^\frac{n}{2}}+\frac{\epsilon }{{\phi }_{4}{\left(\rho ,\delta \right)}^\frac{n}{2}}}$$
$$= \frac{\left(1-\epsilon \right){\phi }_{4}{\left(\rho ,\delta \right)}^\frac{n}{2}}{\left(1-\epsilon \right){\phi }_{4}{\left(\rho ,\delta \right)}^\frac{n}{2}+\epsilon {\phi }_{3}{\left(\rho ,\vartheta \right)}^\frac{n}{2}}$$
(48)
Derivation of posterior distributions of
\(\boldsymbol{\vartheta }\)
and
\({\varvec{\delta}}\)
:
We have
$$P\left(\vartheta =0|y\right)=\frac{p\left(y|\vartheta =0\right)P\left(\vartheta =0\right)}{p\left(y|\vartheta =0\right)P\left(\vartheta =0\right)+p\left(y|\delta =1\right)P\left(\delta =1\right)}$$
$$=\frac{p\left(y|\vartheta =0\right)\left(1-\epsilon \right)}{p\left(y|\vartheta =0\right)\left(1-\epsilon \right)+p\left(y|\delta =1\right)\epsilon }$$
(49)
and
$$P\left(\delta =1|y\right)=\frac{p\left(y|\delta =1\right)P\left(\delta =1\right)}{p\left(y|\vartheta =0\right)P\left(\vartheta =0\right)+p\left(y|\delta =1\right)P\left(\delta =1\right)}$$
$$=\frac{p\left(y|\delta =1\right)\epsilon }{p\left(y|\vartheta =0\right)\left(1-\epsilon \right)+p\left(y|\delta =1\right)\epsilon }$$
(50)
Now
$$p\left(y|\vartheta =0\right)={\int }_{0}^{1}{\int }_{0}^{1}{\int }_{0}^{\infty }{\int }_{{R}^{k}}\frac{{\tau }^{\frac{n+k}{2}-1}{\delta }^{\frac{{n}_{2}}{2}}{\left|V\right|}^\frac{1}{2}}{{\left(2\pi \right)}^{\frac{n+k}{2}}}\mathrm{exp}\left[-\frac{\tau }{2}\left\{{\phi }_{4}\left(\rho ,\delta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)}^{^{\prime}}\left({A}_{4}\left(\rho ,\delta \right)+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)\right\}\right]d\beta d\tau d\rho d\delta$$
$$=\frac{\Gamma \left(\frac{n}{2}\right){\left|V\right|}^\frac{1}{2}}{{\pi }^\frac{n}{2}}{\int }_{0}^{1}{\int }_{0}^{1}\frac{{\delta }^{\frac{{n}_{2}}{2}}}{{\phi }_{4}{\left(\rho ,\delta \right)}^\frac{n}{2}{\left|{A}_{4}\left(\rho ,\delta \right)+V\right|}^\frac{1}{2}}d\rho d\delta$$
$$=\frac{\Gamma \left(\frac{n}{2}\right){\left|V\right|}^\frac{1}{2}}{{\pi }^\frac{n}{2}}{\Upsilon }_{1}$$
(51)
Further
$$p\left(y|\delta =1\right)={\int }_{0}^{1}{\int }_{0}^{1-\vartheta }{\int }_{0}^{\infty }{\int }_{{R}^{k}}2\frac{{\tau }^{\frac{n+k}{2}-1}{\left|V\right|}^\frac{1}{2}}{{\left(2\pi \right)}^{\frac{n+k}{2}}}\mathrm{exp}\left[-\frac{\tau }{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)}^{^{\prime}}\left({A}_{3}\left(\rho ,\vartheta \right)+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)\right\}\right]d\beta d\tau d\rho d\vartheta$$
$$=\frac{\Gamma \left(\frac{n}{2}\right){\left|V\right|}^\frac{1}{2}}{{\pi }^\frac{n}{2}}{\int }_{0}^{1}{\int }_{0}^{1-\vartheta }\frac{2}{{{\phi }_{3}\left(\rho ,\vartheta \right)}^\frac{n}{2}{\left|{A}_{3}\left(\rho ,\vartheta \right)+V\right|}^\frac{1}{2}}d\rho d\vartheta$$
$$=\frac{\Gamma \left(\frac{n}{2}\right){\left|V\right|}^\frac{1}{2}}{{\pi }^\frac{n}{2}}{\Upsilon }_{2}$$
(52)
Here
$${\Upsilon }_{1}={\int }_{0}^{1}{\int }_{0}^{1}\frac{{\delta }^{\frac{{n}_{2}}{2}}}{{\phi }_{4}{\left(\rho ,\delta \right)}^\frac{n}{2}{\left|{A}_{4}\left(\rho ,\delta \right)+V\right|}^\frac{1}{2}}d\rho d\delta$$
$${\Upsilon }_{2}={\int }_{0}^{1}{\int }_{0}^{1-\vartheta }\frac{2}{{{\phi }_{3}\left(\rho ,\vartheta \right)}^\frac{n}{2}{\left|{A}_{3}\left(\rho ,\vartheta \right)+V\right|}^\frac{1}{2}}d\rho d\vartheta$$
Hence
$$P\left(\vartheta =0|y\right)=\frac{{\Upsilon }_{1}\left(1-\epsilon \right)}{{\Upsilon }_{1}\left(1-\epsilon \right)+{\Upsilon }_{2}\epsilon }$$
$$={\lambda }_{{M}_{1}\left(y\right)} (\mathrm{say})$$
$$P\left(\delta =1|y\right)=\frac{{\Upsilon }_{2}\epsilon }{{\Upsilon }_{1}\left(1-\epsilon \right)+{\Upsilon }_{2}\epsilon }$$
$$=1-{\lambda }_{{M}_{1}\left(y\right)}={\lambda }_{{M}_{2}\left(y\right)} (\mathrm{say})$$
The posterior density of \(\vartheta\), when \(\delta =1\), is given by
$${p}^{*}\left(\vartheta |y\right)$$
$$\propto \frac{1}{1-\vartheta }{\int }_{0}^{1-\vartheta }{\int }_{0}^{\infty }{\int }_{{R}^{k}}{\tau }^{\frac{n+k}{2}-1}\mathrm{exp}\left[-\frac{\tau }{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)}^{^{\prime}}\left({A}_{3}\left(\rho ,\vartheta \right)+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)\right\}\right]d\beta d\tau d\rho$$
$$\propto \frac{1}{1-\vartheta }{\int }_{0}^{1-\vartheta }\frac{1}{{\left|{A}_{3}\left(\rho ,\vartheta \right)+V\right|}^\frac{1}{2}}{\int }_{0}^{\infty }{\tau }^{\frac{n}{2}-1}\mathrm{exp}\left[-\frac{\tau }{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)\right\}\right]d\tau \mathrm{d\rho }$$
$$\propto \frac{1}{1-\vartheta }{\int }_{0}^{1-\vartheta }{{\phi }_{3}\left(\rho ,\vartheta \right)}^{-\frac{n}{2}}{\left|{A}_{3}\left(\rho ,\vartheta \right)+V\right|}^{-\frac{1}{2}}d\rho$$
Hence
$${p}^{*}\left(\vartheta |y\right)=\frac{\frac{1}{1-\vartheta }{\int }_{0}^{1-\vartheta }{{\phi }_{3}\left(\rho ,\vartheta \right)}^{-\frac{n}{2}}{\left|{A}_{3}\left(\rho ,\vartheta \right)+V\right|}^{-\frac{1}{2}}d\rho }{{\int }_{0}^{1}\frac{1}{1-\vartheta }{\int }_{0}^{1-\vartheta }{{\phi }_{3}\left(\rho ,\vartheta \right)}^{-\frac{n}{2}}{\left|{A}_{3}\left(\rho ,\vartheta \right)+V\right|}^{-\frac{1}{2}}d\rho d\vartheta };\left(0<\vartheta <1\right)$$
(53)
Therefore, the posterior distribution of \(\vartheta\) is a mixture of discrete and continuous distribution and given by
$${\pi }^{*}\left(\vartheta |y\right)=\left\{\begin{array}{c}{\lambda }_{{M}_{1}\left(y\right)}; if \vartheta =0\\ \left(1-{\lambda }_{{M}_{1}\left(y\right)}\right){p}^{*}\left(\vartheta |y\right); if 0<\vartheta <1\end{array}\right.$$
(54)
The posterior density of \(\delta\), when \(\vartheta =0\), is obtained as
$${p}^{*}\left(\delta |y\right)\propto {\delta }^{\frac{{n}_{2}}{2}}{\int }_{0}^{1}{\int }_{0}^{\infty }{\int }_{{R}^{k}}{\tau }^{\frac{n+k}{2}-1}\mathrm{exp}\left[-\frac{\tau }{2}\left\{{\phi }_{4}\left(\rho ,\delta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)}^{^{\prime}}{\left({A}_{4}(\delta ,\rho )+V\right)}^{-1}\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)\right\}\right]d\beta d\tau d\rho$$
$$\propto {\int }_{0}^{1}{\delta }^{\frac{{n}_{2}}{2}}{{\phi }_{4}\left(\rho ,\delta \right)}^{-\frac{n}{2}}{\left|{A}_{4}\left(\delta ,\rho \right)+V\right|}^{-\frac{1}{2}}d\rho$$
Hence
$${p}^{*}\left(\delta |y\right)=\frac{{\int }_{0}^{1}{\delta }^{\frac{{n}_{2}}{2}}{{\phi }_{4}\left(\rho ,\delta \right)}^{-\frac{n}{2}}{\left|{A}_{4}\left(\delta ,\rho \right)+V\right|}^{-\frac{1}{2}}d\rho }{{\int }_{0}^{1}{\int }_{0}^{1}{\delta }^{\frac{{n}_{2}}{2}}{{\phi }_{4}\left(\rho ,\delta \right)}^{-\frac{n}{2}}{\left|{A}_{4}\left(\delta ,\rho \right)+V\right|}^{-\frac{1}{2}}d\rho d\delta }$$
(55)
Again, the posterior distribution of \(\delta\) is a mixture of discrete and continuous distribution and given by
$${\pi }^{*}\left(\delta |y\right)=\left\{\begin{array}{c}1-{\lambda }_{{M}_{1}\left(y\right)}; if \delta =1\\ {\lambda }_{{M}_{1}\left(y\right)}{p}^{*}\left(\delta |y\right); if 0<\delta <1\end{array}\right.$$
(56)
Derivation of Posterior Odds Ratio
For obtaining the posterior odds, let us consider the numerator of the Eq. (27)
$$\pi \left({H}_{0}\right)$$
$$={\int }_{0}^{1}{\int }_{0}^{1}{\int }_{0}^{\infty }{\int }_{{R}^{k}}\frac{{\tau }^{\frac{n+k}{2}-1}{\left|V\right|}^\frac{1}{2}{\delta }^{\frac{{n}_{2}}{2}}}{{\left(2\pi \right)}^{\frac{n+k}{2}}}exp\left[-\frac{\tau }{2}\left\{{\phi }_{4}\left(\rho ,\delta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)}^{^{\prime}}\left({A}_{4}\left(\rho ,\delta \right)+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\delta \right)\right)\right\}\right]d\beta d\tau d\delta d\rho$$
$$=\frac{\Gamma \left(\frac{n}{2}\right){\left|V\right|}^\frac{1}{2}}{{\pi }^\frac{n}{2}}{\Upsilon }_{1}$$
(57)
Further, the denominator of (27) is given by
$$\pi \left({H}_{1}\right)$$
$$={\int }_{0}^{1}{\int }_{0}^{1-\vartheta }{\int }_{0}^{\infty }{\int }_{{R}^{k}}2\frac{{\tau }^{\frac{n+k}{2}-1}{\left|V\right|}^\frac{1}{2}}{{\left(2\pi \right)}^{\frac{n+k}{2}}}exp\left[-\frac{\tau }{2}\left\{{\phi }_{3}\left(\rho ,\vartheta \right)+{\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)}^{^{\prime}}\left({A}_{3}\left(\rho ,\vartheta \right)+V\right)\left(\beta -\widehat{\beta }\left(\rho ,\vartheta \right)\right)\right\}\right]d\beta d\tau d\rho d\vartheta$$
$$=\frac{\Gamma \left(\frac{n}{2}\right){\left|V\right|}^\frac{1}{2}}{{\pi }^\frac{n}{2}}{\Upsilon }_{2}$$
(58)
Utilization of (57) and (58) in (27) leads to the required expression (28) for the posterior odds ratio.
