# Multivariate Normal Distribution – Ruling Out Specific Summation Terms in Probability Analysis

distributionsmultivariate normal distributionprobability

I am curious about orthant probabilities for the multivariate normal distribution for any finite dimension $$n$$. While Wikipedia currently doesn't seem mention these quantities the Wolfram Mathworld entry for the bivariate normal gives

$$\frac{1}{4} + \frac{1}{2 \pi} \left( \sin^{-1} \rho \right)$$

as the quadrant probability and similarly the trivariate normal octant probability is given by

$$\frac{1}{8} + \frac{1}{4 \pi} \left( \sin^{-1} \rho_{1,2} + \sin^{-1} \rho_{1,3} + \sin^{-1} \rho_{2,3} \right).$$

The more general case appears to be difficult. It was asked about here in Multivariate Normal Orthant Probability, but has equivalently been asked about in math.se in Probability that multi-dimensional random variable is positive? and Multivariate gaussian integral over positive reals.

Sometimes it is easier to evaluate a candidate guess to a problem than to rigorously derive something from first principles and definitions. Looking at the bivariate and trivariate cases led me to guess an n-dimensional generalization of these equations for the orthant probability.

$$\frac{1}{2^n} + \frac{1}{2 \pi (n-1)} \left( \sum_{\substack{i,j \in \{ 1, \cdots, n \} \\ i < j}} \sin^{-1} (\rho_{i,j}) \right)$$

I have started with some sanity checks.

• This equation agrees with the bivariate and trivariate cases.
• When there is no correlation we have $$\frac{1}{2^n}$$ probability equally for all orthants, which makes sense for a symmetric distribution.

The main irregularity that jumps out at me is the equation is undefined for $$n=1$$, but this is not necessarily a problem if we simply limit the generalization to $$n \geq 2$$. Of course in the $$n=1$$ case we should have $$\frac{1}{2}$$ probability either left or right of the mean, which agrees with the $$\frac{1}{2^n}$$ term.

Can we can rule this guess out?

Another check is what happens when you set one of the correlations to 1, effectively reducing the dimension from $$n$$ to $$n-1$$ (because two variables will be the same if the correlation is 1).

### Examples of checks that work

• Reducing from $$n = 3$$ to $$n=2$$

With $$n=3$$, if $$\rho_{1,2}=1$$ then $$\rho_{2,3}=\rho_{1,3}=\rho$$, so you get $$\frac{1}{8}+\frac{1}{2\pi(3-1)}\left(\sin^{-1}\rho+\sin^{-1}\rho+\frac{\pi}{2}\right)=\frac{1}{8}+\frac{1}{2\pi}\sin^{-1}\rho+ \frac{1}{8}$$ which is equal to the formula of the $$n=2$$ case.

• Reducing from $$n=2$$ to $$n=1$$

And with $$n=2$$, setting the correlation to 1 reduces the formula to $$1/2$$, which is the $$n=1$$ case.

### Counter example that does not work

• Reducing from $$n=4$$ to $$n=3$$

With $$n=4$$, if you set $$\rho_{1,2}=1$$, you get three distinct $$\rho$$s: $$\rho_{2,3}=\rho_{1,3}$$, $$\rho_{2,4}=\rho_{1,4}$$ and $$\rho_{3,4}$$. Two of these are pairs; one isn't. So, the formula will reduce to something of the form $$A+B\left(2\sin^{-1}\rho_{2,3}+2\sin^{-1}\rho_{2,4}+\sin^{-1}\rho_{3,4}\right)$$ which can't match the $$n=3$$ formula.

So this general formula must be wrong.