Copula Model

Definition

Let $T_1,T_2,\dots ,T_n$ be i.i.d. survival time with CDF $F(t_1,t_2,\dots ,t_k)$, PDF $f(t_1,t_2,\dots ,t_k)$, and Survival Function $S(t_1,t_2,\dots ,t_k)$, $S_j(t_j)=P(T_j>t_j)$ be marginal survival function, $F_j(t_j)=P(T_j\le t_j)$ be marginal CDF, and $C_1,C_2,\dots ,C_n$ be i.i.d. censoring time.

Copula model define the joint survival function as the copula function whose arguments are marginal survival functions.

When $k=2$, $S(t_1,t_2)=K(S_1(t_1),S_2(t_2))$ where $K$ is the copula function with uniform marginals.

Copula Functions

Clayton’s copula function $S(t_1,t_2)=(S_1(t_1{)}^{-\varphi }+S_2(t_2)-1{)}^{-1/\varphi }$ where $\varphi \in {\mathbb{R}}^{+}$

Crowder’s copula function $S(t_1,t_2)=(1+{\lambda }_1{t}_1^{{\gamma }_1}+{\lambda }_2{t}_2^{{\gamma }_2}{)}^{-\nu }$

Hougaard’s copula function $S(t_1,t_2)=(1+\varphi {\Lambda }_1(t_1)+\varphi {\Lambda }_2(t_2){)}^{-1/\varphi }$