Cannot use the MultinormalDistribution code

Question

I am trying to use the MultinormalDistribution code with the Quantile function to get the 0.01 quartile of the multinormal distribution function with a certain mean and covariance as the following:

 Quantile[MultinormalDistribution[0, Covariance[{rc1, rc2, rc3, rc4}]], 0.01]

Here, rc1, rc2, rc3, rc4 are any arbitrary list of numbers.

However, this code does not seem to give any results but just a repetition of its own self. (i.e. the result is Quantile[MultinormalDistribution[0, Covariance[{rc1, rc2, rc3, rc4}]], 0.01])

I would appreciate if I could get some help on this if possible. Thank you.

Answer 1

It seems like Quantile doesn't work with a symbolic MultinormalDistribution.
You can get an approximate numerical result for a specific {rc1, rc2, rc3, rc4}

{rc1, rc2, rc3, rc4} = RandomReal[1, {4, 4}]

using

Quantile[
 RandomVariate[
  MultinormalDistribution[{0, 0, 0, 0}, Covariance[{rc1, rc2, rc3, rc4}]], 
  10^5], 
 0.01]

Answer 2

It makes sense that Quantile shouldn't work with any multivariate distributions. Effectively Quantile uses the Inverse CDF which cannot exist for multivariate distributions, since you can produce an infinite number of rectangles or other shapes around the median (or mean) point which contain $q$% of the probability mass. Inverse CDFs only can exist in 1D when they are bijective monotonic functions $F:X\rightarrow\{0,1\}$. Multivariate distributions are never invertible because they have a mass function $F:X_1\times X_2 \times \dots X_n \rightarrow\{0,1\}$. For any $p = F(x_1,x_2,\dots,x_n)$ there could be many points $F(x_1,x_2,\dots,x_n)$ where this is true.

Consider the standard multinormal distribution. A "quantile" around the peak would not be described as an interval (as in the 1D case) but instead by circles, forming level sets. But you don't necessarially have to use circles, you could use any squiggly region where the interior made up for a $q$% quantile of probability mass. This is why Mathematica won't return anything meaningful.