# Inference using simulated quantile function

#### dbooksta

##### New Member
I generated a quantile function $$\hat X$$ using Monte Carlo simulation.

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 $$[\hat X_{.05}(\hat \sigma), \hat X_{.95}(\hat \sigma)]$$?

#### maartenbuis

##### TS Contributor
Sounds similar to a Bootstrap test to me. I would read up on that, e.g. in Chapter 16 of Efron, Bradley, and Robert J. Tibshirani. An introduction to the bootstrap. CRC press, 1994.