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I use MCMC to study a distribution. Thanks to the modified Central Limit Theorem found in Robert and Casella's Monte Carlo Statistical methods, one can approach the expectation of the distribution, its variance, actually any integral of the shape $\int g(x) f(x) dx $, $f(\cdot)$ being the density function and $g(\cdot)$ being the function I want to integrate.

However, I do not think that the quantiles of the distribution are an integral like this one. So does it take another result to estimate quantiles from MCMC ? Is there an integral form of quantile estimator that I did not spot ? Or is it impossible ?

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    $\begingroup$ Suppose $f(x)$ is known. Then so is $F(x)=\int_{-\infty}^x f(s) ds$. Thus, you can solve for $q=F^{-1}(p)$, where $p$ is your quantile of interest. $\endgroup$
    – Tim Mak
    Apr 14, 2020 at 9:40
  • $\begingroup$ Thank you very much ! $\endgroup$
    – SebCoube
    Apr 14, 2020 at 12:10
  • $\begingroup$ A difference with an integral estimate is that the quantile estimates are biased. $\endgroup$
    – Xi'an
    Apr 29, 2020 at 13:38
  • $\begingroup$ Thank you so much ! $\endgroup$
    – SebCoube
    Apr 29, 2020 at 16:26

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