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Given a Gaussian random vector of length N:

$$ X \sim N(\mu,\Sigma) $$

where $\Sigma$ is non-diagonal, what is the probability that a chosen component of $X$ is larger than any other? i.e.

$$P(X_n > max(X_i)) \; \text{ for } \; i = [1,2, ..., n-1,n+1,...,N]$$ For a diagonal covariance matrix, i.e. vector with independent components, we can solve the problem as shown here. I'm not sure how to approach this for the general case, though.

Furthermore -- how can the answer to be extended to include a group of components? e.g.:

$$P(max(X_{n}, X_{n+1}, X_{n+2}) > max(X_i)) \; \text{ for } \; i = [1,2, ..., n-1,n+3,...,N]$$

Jake
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    What does$\Sigma $ look like (particularly diagonal elements)? – herb steinberg Oct 14 '21 at 00:10
  • In the use cases of interest, $\Sigma$ will have most of its energy along the diagonal, but some off-diagonal terms. Enough that ignoring the off-diagonal terms makes the estimate too innaccurate.

    One idea is to form a new random Gaussian vector where we subtract the component of interest from all of the others, then determine the probability that all the elements of this new vector are positive (similar to the approach in the link I shared). I'm not too sure how to go about doing that, though. You could evaluate the CDF at 0, but for larger dimensions this becomes intractable.

     – Jake Oct 14 '21 at 05:46
  • Every difference $X_i-X_j$ is normally distributed. – Max Oct 14 '21 at 05:56

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Expanding on @Max's comment: $Y=X_n-X$ is itself a Gaussian random vector whose covariance matrix can easily be worked out from that of $X$. Then we are looking for the probability that all components of $Y$ are positive, which is an "orthant probability"; closed formulas (for small $N$ only), recursion formulas, and approximation schemes can be found in the literature. See for example: R. H. Bacon, Approximations to multivariate normal orthant probabilities, The Annals of Mathematical Statistics 34 (1963), no. 1, 191–198.

Greg Martin
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  • Thank you for your reply. In my use case of interest, $\mu$ is (mostly, apart from a few components) constant, and $\Sigma$ is (almost) Toeplitz. Do you know of any specific orthant probability approximation methods for this case? It feels as though one could calculate it for a smaller subset and extend that to the higher dimensional case, but I'm not too sure how. – Jake Oct 14 '21 at 18:29