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Are there any good approximations or tail bounds for the first-order statistic of the folded normal, or the closely related chi-square distribution with $k$ degrees of freedom? It seems that the $n$-th order statistic is well studied (Gumbel), but not the minimum.

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One can obtain an exact closed-form solution:

If $X \sim N(\mu, \sigma^2)$, then $|X|$ has a folded Normal distribution with pdf $f(x)$:

enter image description here

Then, the pdf of the $1^{st}$ order statistic, in a sample of size $n$, is:

enter image description here

with domain of support on the positive real line,

where:

  1. I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty-gritties,

  2. Erf[z] denotes the error function $\frac{2}{\sqrt{\pi }}\int _0^z e^{-t^2} d t$

To illustrate, here is a plot of the pdf of the sample minimum, when $\mu=2$, $\sigma=1$ and for various sample sizes $n$:

enter image description here

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