Our simulation experiment involved a certain statistical measure M of a certain distribution F. We decided to estimate M using a quantile (Q_{p}, 0
Clearly your p is a proportion because it is within the limits of 0 and 1. Being so it is the mean of a Bernoulli trial. Therefore, asymptotically, i.e. if the sample size increases indefinitely, the Central Limit Theorem guarantees an approximate normal distribution for p.
Thanks Ette Etuk. But here, {p_{i}} is a bounded population of proportions as you might like to call it. While the Bernoulli trial assume a distribution of a population x with a fixed parameter p (proportion), thus the asymptotic distribution of x (not p) may be approximated as N(np,np(1-p)) precisely when p>>>0.5. So, here we are interested in the distribution of {p_{i}} with no x involved. Regards
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p_(i) is indeed a F(x) (assuming F is the probability density)
couldn't you consider going from the asymptotic distribution of a quantile x (a normal law) to the asymptotic distribution of p as the image through F of this asymptotic distribution ?
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I like what you said, Fabrice. The scenario here is; we avoid using F in defining Q_{p} because we are trying to have ideas about the values that M can assume in a sample even if the parameters of F are not known/estimated. However, the simulated result shows that the values of p \in (0.25,0.6).
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