I would like to compute a quantile Z of a conditional cumulative bivariate normal distribution. I have two questions: 1) Do I formulate the problem correctly? 2) How would I compute this quantile Z with R?

1) Problem Formulation: The joint distribution of X,Y is given by: \begin{align} \left(X,Y \right) \sim N\left(0,\Sigma \right) \end{align} The conditioning set for Y is that Y is below its q% quantile $\{Y < \phi^{-1}(q)\sigma_{22}\}$. The conditonal distribution should equal p. Im interested in the p$\%$ quantile Z. Using the general law of conditional probabilities I get:

\begin{align*} p= \Pr \left(X < Z |Y < \phi^{-1}(q)\sigma_{22}\right) = \frac{\int_{-\infty}^{Z} \int_{-\infty}^{\phi^{-1}\sigma_{22}} f_{X,Y}(u,v)\, du \,dx}{q}. \end{align*}

Here $f_{X,Y}$ is the joint normal density which is known. This is how far I get. Can I summarize this expression any further?

2) Given $\Sigma,q,p$, how can I compute Z? I know that I could just simulate the distribution and form the empirical conditional distribution function. Are there any other, maybe more elegant ways that use the nice properties of the multivariate normal distribution?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.