InverseSamplingBivariate
A bivariate sampler to generate samples from a bivariate distribution
Some theory
For a univariate distribution with CDF , we can generate a sample from said distribution via the inverse sampling theorem; if the random variable U is drawn from a uniform distribution between 0 and 1, then a realization of the random variable X is:
,
or, the inverse of the CDF of X calculated in a random outcome of the uniform distribution between 0 and 1.
For a bivariate distribution with CDF , we can approximate the generation of two samples by following those two steps:
- Generate a sample from X with the Inverse sampling theorem, using the marginal CDF
- Then, generate a sample of Y by using the conditional CDF of Y given that X=x, i.e.
.
In case the marginal is not readily available, one can calculate it as:
.
In practice this means calculating the Joint CDF at a very high value of y (in my code I used 1e5).
Files description
- generate_sample generates a sample from a univariate distribution given its CDF
- generate_sample_bivariate generates two samples given the joint CDF, and, optionally, the marginal CDF
- test_generate_sample tests the generation against some univariate distributions
- test_bivariate tests the bivariate distribution against a multivariate exponential type II
Validation
Here two plots for the bivariate testing:
