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Let's say I have a multivariate distribution $\mathbf{X} \sim \text{MVN}(\mathbf{\mu}, \mathbf{\Sigma})$ and a set of $K$ scalar functions of $\mathbf{X}$, $f_1(\mathbf{X}), \dots, f_K(\mathbf{X})$. Let's say we can make the additional assumption that $f_k$ is smooth and differentiable for all $k$. Is there a way to derive $P(f_1(\mathbf{X}) = \text{max}(f_1(\mathbf{X}), \dots, f_K(\mathbf{X}))$ analytically?

I know one solution would be to sample from $\mathbf{X}$ and in each sample realization $\tilde{\mathbf{X}}$ compute $f_1(\tilde{\mathbf{X}}), \dots, f_K(\tilde{\mathbf{X}})$ and then compute the proportion of samples in which$f_1(\tilde{\mathbf{X}})$ is the largest. I would prefer an analytic solution if possible, even an approximation, e.g., using the delta method, if such exists.

Ultimately the goal is to perform a type of model selection and averaging across parameteric models estimated jointly, where $\mathbf{X}$ is the vector of coefficients from all models (estimated jointly to get their joint distribution). $f_k(\mathbf{X})$ is a quantity computed on each model which is used to select the best model. Rather than choosing one single model as the best, I am interested in the probability that each model is best given the randomness in the data. These probabilities will be used in a type of model averaging.

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    $\begingroup$ In such generality the answer is obviously not, because you are asking for a solution for a family that includes, but is not limited to, absolutely continuous multivariate distributions with a Gaussian copula. If you could specify the $f_i$ within narrow limits there might be some hope of an analytical solution--but it would depend on the $f_i.$ $\endgroup$
    – whuber
    Commented Jul 10 at 17:59

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