Convolution of a multivariate Gaussian and multivariate exponential distribution

Dear all,

I have derived the convolution for a multivariate Gaussian distribution with a multivariate exponential distribution. By having such a distribution in place, one can have the best of both worlds: retain skewness given by the exponential distribution whilst working alongside a Gaussian (i.e. using the mean and covariance). So now, mean of the convolution would include means from both the Gaussian and the Exponential. The same goes for the covariance. This will be a very helpful distribution provided we can estimate distributional parameters ’ 'nicely’.

The joint posterior of this convolution is non conjugate making its integral intractable. Therefore, for inference, I plan to use the ‘noise contrastive estimation’ for parameter estimation used for unnormalised distributions.

My query would be is this the right approach or since this is a convolution, one could just include the extra terms coming from the Exponential distribution into the Bayesian inference thereby not needing to work out the distribution for the sum.

Any advice on this matter would be highly appreciated.