C-NORTA: A Rejection Procedure for Sampling from the Tail of Bivariate Distributions

We propose C-NORTA, an exact algorithm to generate random variates from the tail of a bivariate NORTA random vector. (A NORTA random vector is specified by a pair of marginals and a rank or product-moment correlation, and is sampled using the popular NORmal-To-Anything procedure.) At the core of this method lies the question of sampling from a piecewise-linear connected region in the tail of a bivariate joint-normal distribution. The proposed sampling technique employs a combination of joint-distribution factorization and acceptance/rejection steps. We demonstrate that, in a certain precise asymptotic sense, the sampling efficiency of C-NORTA is exponentially larger than what is achievable through a näıve application of NORTA to such constrained problems. Furthermore, for at least a certain class of problems, the acceptance probability within C-NORTA is shown to decay only linearly with respect to a defined rarity parameter. The corresponding decay rate achievable through a näıve adaptation of NORTA is shown to be exponential.