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This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research.

Let $(X_i:1\le i\le N)$ be a multivariate normal random vector where i) each coordinate $X_i$ is standard normal and ii) $\mathbb{E}[X_iX_j]=\rho$ for every $1\le i<j\le N$.

My question. Is there a symbolic expression (as a function of $N$ and $\rho$ only) for the following probability: $$ \mathbb{P}\left(X_1\ge 0,X_2\ge 0,\dots,X_N\ge 0\right). $$ Some further notes:

  • For $N=2$, using the fact $X_1$ and $\frac{X_2-\rho X_1}{\sqrt{1-\rho^2}}$ are i.i.d. standard normal, one can indeed reach to $\frac14+\frac{\sin^{-1}\rho}{2\pi}$.
  • For $N=3$, such formulas are available in standard textbooks, and is given by $\frac18+\frac{3\sin^{-1}\rho}{4\pi}$.

Situation gets more involved beyond $N\ge 4$.

Edit. Looking at the structure of the solution for $N=2$ and $N=3$, I postulate that the answer is $$ \frac{1}{2^N}+\binom{N}{2}\frac{\sin^{-1}\rho}{\eta} $$ where $$ \eta = \binom{N}{2}\pi \frac{2^{N-1}}{2^{N-1}-1}. $$ My rationale is a) for $\rho=0$ it must be $2^{-N}$ as everybody is independent, b) the term, $\sin^{-1}\rho$, should appear exactly once for each pair $1\le i<j\le N$, and c) for $\rho=1$, it should be $\frac12$.

This obviously simplifies to $$ 2^{-N}+\frac{\sin^{-1}\rho\left(2^{N-1}-1\right)}{2^{N-1}\pi} $$

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E.g., Ruben, formulas (102) and (102') on p. 220 has a recurrence for your probability $$ p_n:=P(X_1\ge0,\dots,X_n\ge 0). $$ It is stated there, on p. 213: "For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas can no longer be expressed in terms of elementary functions." See also e.g. Plackett and references there for more general results.


Anyhow, your conjectured expression for $p_n$ cannot be true in general. Indeed, the eigenvalues of your covariance matrix of $(X_1,\dots,X_n)$ are $1+(n-1)\rho$ (with the eigenspace spanned by the vector $\mathbf1:=(1,\dots,1)$) and $1-\rho$. Since the covariance matrix must be positive semidefinite, necessarily $1+(n-1)\rho\ge0$, that is, $\rho\ge-1/(n-1)$.

In particular, if $\rho=-1/(n-1)$, then the random point $(X_1,\dots,X_n)$ is with probability $1$ on the hyperplane passing through the origin and orthogonal to the vector $\mathbf1$. Therefore, then $p_n=0$, which does not agree with your conjectured expression for $p_n$.

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Others who have looked at this problem found no closed-form expression. E.g., Shanti Gupta surveyed results in this area in 1963; he said on p. 800: "the integrals can be evaluated in closed form for n=1, 2, and 3."

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When $\rho > 0$ such r.v.s can be represented as $Z + X_i$ where $Z$ independent normal and the $X_i$ are i.i.d. normal and you can immediately write down a one dimensional integral for the expression you are looking for, but I personally can't integrate it, except maybe in case $Z$ has the same distribution as well, which is $\rho = .5$ and for which I get 1/(N+1). You might browse things related to a model of Vasicek, which is described here: http://dse.univr.it/safe/documents/SSEFCANAZEI2012/07_correlation_-_modeling.pdf

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  • $\begingroup$ What is the one-dimensional integral for this? $\endgroup$ – Matt F. Nov 24 '20 at 19:33

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