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Let's say I have a k-dimensional multivariate normal distribution $MVN(0,\Sigma)$. Denote random vector $X \sim MVN(0,\Sigma)$ as $X = (x_1, x_2, \dots, x_k)$. It is trivial to find $P(x_1 \leq c_1, x_2 \leq c_2, \dots, x_k \leq c_k$) (i.e. CDF) for some chosen $C = (c_1, c_2,\dots,c_k)$ using Monte Carlo methods (assuming $\Sigma$ is known and you can you can sample from the multivariate normal distribution). Is it possible to find the quantile function using sampling? In other words, given some value for $P(x_1 \leq c_1, x_2 \leq c_2, \dots, x_k \leq c_k$), say 0.05, can I find $C$. Perhaps C is not unique. Then could I find/sampling from the space of $C$?

David Wang
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    Your question asks how to solve a single equation in the unknown vector $(c_1,c_2,\ldots, c_k)$ for all components. As a general proposition of arithmetic and algebra, that's not possible when $k\gt 1.$ This impossibility, along with apparent corruption in your final statement of the question, make it difficult to understand what you are trying to ask. Could you clarify? – whuber Sep 07 '23 at 18:17
  • After doing a bit more digging with improved key words, I believe this is what I was looking for. From this, it appears that any $C$ in the space that satisfies $C^T\Sigma^{-1} C \leq \chi^2_k(0.05)$ would work. I believe I asked for the space of possible C, not a single C. Maybe I'm misunderstanding what was unclear. Sorry. – David Wang Sep 07 '23 at 19:05
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    The final sentence is ungrammatical and confusing -- I still cannot tell what it is trying to ask. – whuber Sep 07 '23 at 19:15

1 Answers1

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Here is a simple example.

Consider the space where $P[X<b,Y<c]=1/8$, for independent $X$ and $Y$. This includes points such as

  • (values just above the first octile of $X$, high values of $Y$)
  • (first quartile of $X$, median of $Y$)
  • (median of $X$, first quartile of $Y$)
  • (high values of $X$, values just above the first octile of $Y$)

To sample $(b,c)$ in this space, we first sample $u$ from a uniform distribution on $[0,1]$, and then find $b,c$ where $$P[X<b]=(1/8)^{1-u}$$ $$P[Y<c]=(1/8)^u$$

This is the limiting distribution of the regions where $$P[X<b,Y<c]\in[\frac18-\epsilon,\frac18+\epsilon]$$

For more independent variables, we can similarly sample from the simplex where $0\le u_i\le 1$, $u_1+\dots+u_n=1$, and solve $$P[X_i<c_i]=p^{u_i}$$

Perhaps someone else will see how to generalize this nicely to correlated normal variables.

Matt F.
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