Table 7 Conditional PMF of bivariate binary variables generated by Frank copula

(Y₁,Y₃|Y₂=0)

(0, 0)

\(\phantom {\dot {i}\!} C_{13|0}(q_{1|0}, q_{3|0};\theta _{13|Y_{2}=0})\)

\(=-\frac {1}{0.95}\log \left (1+\frac {(e^{-0.95*0.5057}-1)(e^{-0.95*0.699}-1)}{e^{-0.95}-1}\right)=0.3782\)

(0, 1)

\(\phantom {\dot {i}\!}q_{1|0}-C_{13|0}(q_{1|0}, q_{3|0};\theta _{13|Y_{2}=0})=0.5057-0.3782=0.1275\)

(1, 0)

\(\phantom {\dot {i}\!}q_{3|0}-C_{13|0}(q_{1|0}, q_{3|0};\theta _{13|Y_{2}=0})=0.699-0.3782=0.3208\)

(1, 1)

\(\phantom {\dot {i}\!}1-q_{1|0}-q_{3|0}+C_{13|0}(q_{1|0}, q_{3|0};\theta _{13|Y_{2}=0})=0.1735\)

(Y₁,Y₃|Y₂=1)

(0, 0)

\(\phantom {\dot {i}\!} C_{13|1}(q_{1|1}, q_{3|1};\theta _{13|Y_{2}=1})\)

\(=-\frac {1}{0.85}\log \left (1+\frac {(e^{-0.85*0.069}-1)(e^{-0.85*0.2719}-1)}{e^{-0.85}-1}\right)=0.0244\)

(0, 1)

\(\phantom {\dot {i}\!}q_{1|1}-C_{13|1}(q_{1|1}, q_{3|1};\theta _{13|Y_{2}=1})=0.069-0.0244=0.0446\)

(1, 0)

\(\phantom {\dot {i}\!}q_{3|1}-C_{13|1}(q_{1|1}, q_{3|1};\theta _{13|Y_{2}=1})=0.2719-0.0244=0.2475\)

(1, 1)

\(\phantom {\dot {i}\!}1-q_{1|1}-q_{3|1}+C_{13|1}(q_{1|1}, q_{3|1};\theta _{13|Y_{2}=1})=0.6835\)