# How to calculate $P(f_1(X) = \text{max}(f_1(X), \dots, f_K(X))$ when $X$ is multivariate Normal?

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.

• 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 $f_i.$
– whuber
Commented Jul 10 at 17:59