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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?

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    $\begingroup$ Why not take 100 samples from the first random variable and 100 samples from the scale mixture of normals, and then fit to the 10,000 samples that result? The standard way would be take the 0.5th, 1.5th, ... 99.5th percentiles of each distribution, and this approach would still be feasible even if you have to take 100 samples from each of three distributions, and then fit to the million samples that result. $\endgroup$ – Matt F. Jun 16 at 13:08

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