# How find confidence regions for multivariate ( or at least bivariate) distributions other than Normal

I am looking for a method which is able to find confidence regions in multivariate distributions with $(1-\alpha)\%$ probability of occurrence. What I mean by probability of occurrence is that, the probability that an event lies in that region should be $(1-\alpha)\%$. I know how to calculate that region for normal distribution. That region for the case of normal distribution would be

$(X-\mu)' \Sigma^{-1} (X-\mu) <= {\chi}^2_{p,(1-\alpha)}$

where X is a vector of p variables , $\mu$ is a vector of p known parameters, $\Sigma$ is a p*p known matrix. (I have one more doubt and that is whether the above region is a confidence region for the mean of the data or any event)

But what about the case when the distribution is not normal or is non-parametric? Any idea is appreciated. Thanks.

• Why must the contours be elliptical? In the case of the multivariate normal the level sets (curves of constant density) are elliptical but this is not the case for most other distributions. – Glen_b -Reinstate Monica Feb 13 '16 at 14:15
• Yes, this is totally true. Only if the distribution is normal those regions can be called ellipsoids – Nile Feb 13 '16 at 18:45
• it's not only in the case of the normal, but the family of ellipitical distributions is small compared to multivariate distributions in general. Your question still asks for ellipsoids -- do you want ellipsoids? – Glen_b -Reinstate Monica Feb 14 '16 at 1:13
• No, Not necessarily elliptical regions, any geometry or region which represents $(1-\alpha)%$ of the probability is what I need! I removed the word " ellipsoid" from the title. Thanks – Nile Feb 14 '16 at 3:09