I'm working on a mean-variance optimization problem, but instead of financial securities I'm choosing a 'portfolio' of N athletes. It is a 1-period optimization problem over one generic statistic which I'll call
performance here. I'm assuming
athlete_performance is an N length random vector distributed as multivariate-normal:
athlete_performance $\sim MVN(\mu, \Sigma)$
Where $\mu$ is the 1xN vector of means (or expected performance)
and where $\Sigma$ is an NxN matrix with variance on the diagonal ( $\Sigma[i,i]$ = $Var(i)$ ) and covariance on the off-diagonal ( $\Sigma[i,j]$ = $Cov(i,j)$ ).
My question is about the options available for estimating the covariance (off-diagonal) part of the matrix.
My main concern is the predictiveness of my covariance matrix. If I were working with securities that had all been listed together for 10 years, then "sample covariance" might be predictive of future covariance, but in sports it's not that simple.
Imagine a quarterback and a wide receiver in American Football. How well their performance correlates is dependent on the quality of the pass defense they're playing against. Or in F1 racing, if driver A and B are both strong on straighter tracks, but only driver B is strong on tight cornered tracks, their performances will correlate much differently based on whether the track is straight or zigzag-ing.
I'm aware of "sample covariance" which in my case would look at the historical overlap between two athletes. I'm also aware of "shrunk covariance". I was wondering if there are more robust methods for calculating covariance that would be more predictive of future covariance, possibly using some sort of regression or MCMC.
Thank you for reading the question and for your time!