In Section 4.1 of Analyzing Dependent Data with Vine Copulas (Czado), the author decomposes a three-dimensional joint density function into bivariate copula densities and marginal density functions. I’m having trouble with this derivation and would appreciate any help.
The first step taken is to factorise the joint density as $$f\left(x_1, x_2, x_3\right)=f_{3 \mid 12}\left(x_3 \mid x_1, x_2\right) f_{2 \mid 1}\left(x_2 \mid x_1\right) f_1\left(x_1\right).$$
The author then considers each resulting density function separately, starting with $f_{3 \mid 12}\left(x_3 \mid x_1, x_2\right)$. In the second step, the author considers $f_{13 \mid 2}\left(x_1, x_3 \mid x_2\right)$ and uses Sklar’s theorem to obtain $$f_{13 \mid 2}\left(x_1, x_3 \mid x_2\right)=c_{13 ; 2}\left(F_{1 \mid 2}\left(x_1 \mid x_2\right), F_{3 \mid 2}\left(x_3 \mid x_2\right) ; x_2\right) f_{1 \mid 2}\left(x_1 \mid x_2\right) f_{3 \mid 2}\left(x_3 \mid x_2\right) .$$
In the third step, the author uses Lemma 1.15 of the textbook which is $$f_{1 \mid 2}\left(x_1 \mid x_2\right)=c_{12}\left(F_1\left(x_1\right), F_2\left(x_2\right)\right) f_2\left(x_2\right)$$ and applies it to the expression obtained in the second step to arrive at $$f_{3 \mid 12}\left(x_3 \mid x_1, x_2\right)=c_{13 ; 2}\left(F_{1 \mid 2}\left(x_1 \mid x_2\right), F_{3 \mid 2}\left(x_3 \mid x_2\right) ; x_2\right) f_{3 \mid 2}\left(x_3 \mid x_2\right).$$
The derivation then continues but I do not understand this third step. I understand Lemma 1.15 and its proof. I do not understand how its application to the expression obtained for $f_{13 \mid 2}\left(x_1,x_3 \mid x_2\right)$ yields the final expression for $f_{3 \mid 12}\left(x_3 \mid x_1, x_2\right)$.
Thank you for taking the time to consider my question!
