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Journal of Statistical Distributions and Applications

Table 3 Conditional PMF of (Y1, Y3) given Y2

From: Multivariate distributions of correlated binary variables generated by pair-copulas

 

Probability

(Y1,Y3)|Y2=0

 

(0, 0)

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

(0, 1)

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

(1, 0)

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

(1, 1)

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

(Y1,Y3)|Y2=1

 

(0, 0)

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

(0, 1)

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

(1, 0)

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

(1, 1)

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

  1. Note: P(Y1,Y3|Y2=0) has 6 parameters; marginal means p1,p2,p3 and copula parameters \(\phantom {\dot {i}\!}\theta _{12}, \theta _{23}, \theta _{13|Y_{2}=0}\). P(Y1,Y3|Y2=1) needs the same first five parameters and \(\phantom {\dot {i}\!}\theta _{13|Y_{2}=1}\)
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