Approximating the mathematical expectation of the argmax of a Gaussian random vector

Question

Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$.

$I$ has probability mass function

$\mathbb{P}\left( {I = i} \right) = \mathbb{P}\left( {{X_i} = \mathop {\max {X_j}}\limits_{j = 1,n} } \right) = \mathbb{P}\left( {{X_i} - \mathop {\max {X_j}}\limits_{j \ne i} > 0} \right)$

and mathematical expectation

$\mathbb{E}I = \sum\limits_{i = 1}^n {i\mathbb{P}\left( {I = i} \right)} $

Generally speaking, for large $n$ and arbitrary covariance matrix ${\mathbf{\Sigma }}$ , computing $\mathbb{E}I$ is very difficult because it requires the numerical evaluation of high-dimensional normal orthant integrals. So, apart from the IID and INID cases with a diagonal covariance matrix ${\mathbf{\Sigma }}$, banded covariance matrices and degenerate cases such as ${\mu _j} \gg {\mu _{i \ne j}}$ , under which conditions on ${\mathbf{\Sigma }}$ (e.g. correlation decay) can we get simple, easy-to-evaluate numerical approximations to $\mathbb{E}I$ (and $\mathbb{V}I$ as well)?

The covariance matrices ${\mathbf{\Sigma }}$ I'm interested in look like this:

Until now, I’ve not been able to find anything about this problem.

Related question: Maximal component of a multivariate Gaussian distribution

Answer 1

The integrals one needs to evaluate are of the form $${\mathbb P}(I=1)=\int_{-\infty}^\infty dx\int_{-\infty}^x dX_2\int_{-\infty}^x dX_3\cdots \int_{-\infty}^x dX_n \,P(x,X_2,X_3,\ldots X_n)$$ and similarly for $I=2,3,\ldots n$. An efficient method to evaluate these integrals numerically for large $n$ has been developed in Estimating orthant probabilities of high dimensional Gaussian vectors with an application to set estimation (2016). The method has been tested for $n$ as large as $10^4$.