I am trying to sample from a known distribution (somewhat complicated in that a transformed random variable has random noise from a scale mixture of normals added to it and is then back-transformed - so I don't think I can analytically approach my problem by minimizing e.g. KL-divergence or something like that) in order to then fit a simple two-parameter distribution to the sample.
What I am wondering is whether there's some general rule that would tell me how to construct samples that allow this fitting to be more efficiently than if I randomly sample. I'm thinking of some of the examples of MCMC samplers having effective sample size > number of MCMC samples or whether using the distribution quantiles as my samples (if I can calculate them somehow) or something like that.
Or is this incredibly specific to the specific case?