The random variable I simulate is the mean value of 5 draws from an i.i.d. range statistic

*Y*. I.e., I have \(Y(\sigma) \sim \sigma F()\), and I simulated the value of \(X(\sigma=1) \sim \sum_{1}^{5} y(1)_i / 5\).

Is it valid for me to use this quantile function for statistical inference on real-world samples from

*X*? In particular, given a sample \(x_i\):

1. Is it correct to say that \(x_i\) is an estimator of the mean of \(\bar X\), and therefore our best guess is that \(\hat\sigma = x_i/\overline{\hat X(1)}\)?

2. Is it correct to say that our confidence intervals for \(x_i\) are given by the simulated quantile function? E.g., the 90% confidence range on \(x_i\) is \([\hat X_{.05}(\hat \sigma), \hat X_{.95}(\hat \sigma)]\)?