Table 3.

The posterior mean E(ω_k|X_k, δ_k, Z_k, X_k0, δ_k0, Z_k0) under gamma, inverse Gaussian, positive stable and discrete.

Gamma

{1+ θHk.(t)}–1/θ

inverse Gaussian

$\{\begin{array}{cc}\frac{1}{\theta }{(\frac{1}{{\theta }^2}+\frac{H_{k.}\left(t\right)}{\theta })}^{-\frac{1}{2}}\hfill & \text{if}{d}_k=0\hfill \\ \frac{1}{\theta }{(\frac{1}{{\theta }^2}+\frac{H_{k.}\left(t\right)}{\theta })}^{-\frac{1}{2}}[1+\frac{1}{2}{(\frac{1}{{\theta }^2}+\frac{H_{k.}\left(t\right)}{\theta })}^{-\frac{1}{2}}]\hfill & \text{if}{d}_k=1\hfill \\ \frac{1}{\theta }{(\frac{1}{{\theta }^2}+\frac{H_{k.}\left(t\right)}{\theta })}^{-\frac{1}{2}}[\{1+{(\frac{1}{{\theta }^2}+\frac{H_{k.}\left(t\right)}{\theta })}^{-\frac{1}{2}}\}+\frac{1}{4}{(\frac{1}{{\theta }^2}+\frac{H_{k.}\left(t\right)}{\theta })}^{-1}{\{1+\frac{1}{2}{(\frac{1}{{\theta }^2}+\frac{H_{k.}\left(t\right)}{\theta })}^{-\frac{1}{2}}\}}^{-1}]\hfill & \text{if}{d}_k=2\hfill \end{array}\phantom{\}}$

Positive Stable

$\{\begin{array}{cc}\theta H_{k.}{\left(t\right)}^{\theta -1}\hfill & \text{if}{d}_k=0\hfill \\ \theta H_{k.}{\left(t\right)}^{\theta -1}-(\theta -1)H_{k.}{\left(t\right)}^{-1}\hfill & \text{if}{d}_k=1\hfill \\ \frac{{\theta }^2{H}_{k.}{\left(t\right)}^{2\theta }{\left(t\right)}^{2\theta }-3\theta (\theta -1)H_{k.}{\left(t\right)}^{\theta }+(\theta -1)(\theta -2)}{H_{k.}\left(t\right)(\theta H_{k.}{\left(t\right)}^{\theta }-\theta +1)}\hfill & \text{if}{d}_k=2\hfill \end{array}\phantom{\}}$

Discrete

$\frac{{(1+\theta )}^{d_k+1}\text{exp}\{-(1+\theta )H_{k.}\left(t\right)\}+{(1-\theta )}^{d_k+1}\text{exp}\{-(1-\theta )H_{k.}\left(t\right)\}}{{(1+\theta )}^{d_k}\text{exp}\{-(1+\theta )H_{k.}\left(t\right)\}+{(1-\theta )}^{d_k}\text{exp}\{-(1-\theta )H_{k.}\left(t\right)\}}$