# Bounding quantiles of the noncentral chi distribution

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?

• How big is your sample size $n$? If $(1-q)$-quantiles are the ones of interest, what is the range of the values of $q$? Do you want asymptotic or non-asymptotic bounds? – Iosif Pinelis Apr 9 '18 at 17:59
• @IosifPinelis I've added two extra assumptions refering to q and n; thanks for pointing this out. – etal Apr 9 '18 at 18:23
• I think you can use the fact that, if $X$ is a nonnegative random variable and $x_q$ is its $(1-q)$-quantile, then $x_q^2$ is a $(1-q)$-quantile of $X^2$, and vice versa. Thus, you reduce your noncentral chi distribution to the known case of a noncentral chi-squared distribution. – Iosif Pinelis Apr 9 '18 at 19:10