I'm learning Monte-Carlo approach in sampling. There I faced with ways of how to draw samples from given distribution. But can you give me an example of a distribution which can not be trivially simulated as normal or binomial distribution?
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4$\begingroup$ It's still not quite clear what your last sentence is asking. Do you mean "cannot be as trivially simulated as normal or binomial"? What's your trivial means of simulating a binomial? What makes something trivial or non-trivial more generally? $\endgroup$– Glen_bCommented Jun 2, 2015 at 10:46
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$\begingroup$ von Mises-Fisher is also such an example, cf. this previous post on Cross Validated. $\endgroup$– micCommented Jun 18, 2015 at 12:39
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2 Answers
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Drawing gamma random numbers usually requires rejection sampling, it is less trivial.
I assume that you refer to trivial if the CDF is invertible, or conversion from the uniform to the target distribution can be resolved with thresholds.
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In this earlier Cross Validated question, a density defined as $$h_β(r)∝(1−w_{\mu,τ}(r))f_{β_0}(r)+w_{\mu,τ}(r)g_{ϵ,σ}(r)$$ is proposed, with a non-trivial simulation solution.
In my class, I usually give the benchmark density target $$h(x)\propto \{1+\sin^2(2x)+\sin^4(4x)\}\exp\{-x(1+\cos^2(4x)+\cos^4(2x))\}$$ to simulate. You can make similar examples by piling up complex but upper bounded terms.
