I need to bound the empirical quantiles for a *noncentral* chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality parameter.

In case it helps; I can make the following assumptions:

- $5000>\nu>50$
- $\lambda \in [\frac{\sqrt{\nu-1}}{2}-2,\frac{\sqrt{\nu-1}}{2}+2]$
- number of samples $10^2<n<10^4$
- Interested in q-quantile and (1-q)-quantile for $q\in[0.01,0.05]$

My current approach involves getting bounds on the theoretical quantiles and applying Chernoff to get bounds on empirical quantiles given enough samples.

The closest thing I've found is Approximations to the Non-Central Chi-Squared Distribution. Note it's for Non-central Chi-Squared, not Non-central Chi. Modifying the math in the paper a bit (setting b=0 in its initial formula) gives me a Gaussian approximation for non-central chi with approximate cumulants; but I'm not sure if this is a promissing approach since this is far from my normal research area.

Is there a better approach to bound the theoretical quantiles of the non-central chi-squared than by trying to approximate it with a Gaussian?