Hello :) I am currently working on a project where I am stuck calculating the following term:
$u_{i(e)|D(e)} = C_{i(e)|D(e)}(u_{i(e)}|D(e))$
$u_{j(e)|D(e)} = C_{i(e)|D(e)}(j_{i(e)}|D(e))$
The notation comes from the nodes and edges of a regular vine Copula. For example we could be in Tree $T_2$ and want to connect nodes $(1,2)$ and $(2,3)$, the connecting edge would have the notation $(1,3|2)$ with $i(e) = 1$, $j(e) = 3$ and the set $D(e) = {2}$. For example in $T_3$ if we wanted to connect node $(1,3|2)$ with node $(2,4|3)$, the connecting edge would be $(1,4|2,3)$ with $i(e) = 1$, $j(e) = 4$ and the set $D(e) = {2,3}$. $i(e)$ and $j(e)$ are always singletons (dimension 1), whereas $D(e)$ has dimension $k-1$ for Tree $T_k$. Now to my issue:
I know about the h-function and how it is used to calculate conditional copula CDFs:
$h(u_1|u_2; \theta) := P(U_1 \leq u_1 \mid U_2 = u_2) = \frac{\partial C(u_1, u_2; \theta)}{\partial u_2}$
With the h-function I am able to calculate my expression for $k=2$ which looks as follows:
$u_{1|2} = C_{1|2}(u_1|u_2) = h(u_1|u_2)$
$u_{3|2} = C_{3|2}(u_3|u_2) = h(u_3|u_2)$
(maybe here is already my mistake but this is the best I found in the literature and when trying it in my R-Code and compare my result to the results RVineEstimation functions I get the correct results so I guess the calculation goes as planned). Now comes the point where I am stuck: When calculating for $k \geq 3$, e.g. $k=3$ I get the following issue when calculating:
$u_{1|23} = C_{1|23}(u_1|u_2,u_3)$
$u_{4|23} = C_{1|23}(u_4|u_2,u_3)$
At this point I am unable to use the h-function the way I did before, as I am now conditioning under more than one parameter and simply can't get on. In this example I have a D-Vine with 4 dimensional data to it and was able to calculate the correct values for $k=2$ but then had issues with the higher dimension terms and conditions. Maybe some of you can help me and I hope my problem got kind of clear. Otherwise I am happy to explain the issue in further details.
Thanks to everyone trying to help me!! I am really stuck here and it is super frustrating as it is kind of the last step to building a optimized vine model.
Literature that got me really close was the following paper: https://pdf.sciencedirectassets.com/271685/1-s2.0-S0167668709X00028/1-s2.0-S0167668707000194/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjELj%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJGMEQCIHPO5FU%2B93uRW35Adopx2Q85DqIg7VJHunxB%2BMXywQlgAiBsMSqseF%2FBQVb8r8wEIjo8%2B4jsabwkkvdApUDLqx4iLSqzBQhxEAUaDDA1OTAwMzU0Njg2NSIMwHhWRmjRyi79UHPIKpAFLk9h9qLfyH4JYUbNHiVdsnDuwvXnkrkk3KzgPC3fN7aLc74JX0QpylKQMOf6V%2FxU%2FvWprcwNy4snvMFk%2F6YdbigOzH%2BPEiGEgc9ab0aDG0OlnBT1KrPOOZ7IgTyUu66P5EXMK%2BxBhHlqWfKTLLDAefB25EhTA0PqNu%2BtOva%2BxQR3nTMRfGwKbVHp8vP9wNuycus9mP%2BgruexH6DcrFgxD7xCj5icw1rHU6XgXnOxze1pu8shyobTFUlADBvac6NWpoNDJmMucVobhmfTEek5aOU5scPH7Q4q9aaYAvzr%2Fpmow3lpUlmF8FBEGEZQRiTUXsv%2BkOKkPlenKsHsI%2BpPQ1qrIYNc3H6qwavzpjlhaNrPW7BxqZMydjjOt6jeNQeo3F%2FxFcIwvksCi8H0KV6sHFJ2W63LFywpQ9ehqCApHL8sAKiR9zoGo1jNYyLU2UpdQr9zkq2YszJUPxJVCbcrBZ%2FW7gZ4gePFWP%2BoyfGUfTFUMqWP8jRstJdXMO7g48MsKbVERcCRPpJcytn3C5p6tIwAdl8WiS3VW4KFa6zIP7e4XuQhLdiZatAsHmPB52zYfyF5NXh%2FHeX4aWUv9CdgjrCjOfmt384H2uqNC5XY%2FhldAOEZnMpeAyoFVtrgoersc7Sby1Kp1o%2Bvw7nyqVSb5qjNcScrY3yjCHlk3PUaVgRsbENysjNzHovMIn%2BF8Zw%2BILmEup8LtRvwlBV779vjxgCmraDHXI90pHkAfVRLHCuCKu3BCTwnhs8wo7OXSBD4gm0BSjYODDiUfJDVnN3O8DDZ%2FdHZMZwbGamjIMRAVyWAW%2F6VFVgLl6ueb2DFr2RVvcPTLSKJsPC%2B1yG%2BqTUkiqMgkd3N5vxP2ZRgZkgrkKgw74LDrQY6sgF4H%2FLFK9xwm1zjPzPi1gCkVQxju%2FqD%2BYi%2FVU0SVPKcALQQY0zm76lTgvZr7Es8r%2FwiDsyEh3B%2Fvuog5hVoSYkXpUeKoizCkravv3dOMfAHxTz6KIGhzp6mwYkyI4oHLsSdYUYkjolFxkjlt8kIS1MaJNDtZ0ucGVfP9cYR9N763c0A%2B05bfh96zxWi4fXGzPYAhPiNWoa%2BFyRZ0857lIulRHOhE9N1ctZS4HvvNgiIkgor&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20240124T094356Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYU4AKST6O%2F20240124%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=ab06047a2f356d47de8a89126f6caef6d5d071120336289f7ecedfc374148c62&hash=8483dd0f26b546b7dac1f93c3aa4301dd0de8ff32956d4e9fa89fc636cee6092&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0167668707000194&tid=spdf-c54ca59b-bac8-4cc6-967e-54fa5ab900bd&sid=b22b911881ee58409b9a2bb01eca2cbbca27gxrqb&type=client&tsoh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&ua=1e0b5f560454535a0c52&rr=84a755c19dd89b34&cc=de