What is the covariance matrix of the normal order statistics? I would like to test if a sample comes from a standard normal distribution. I want to do that by sorting the sample values, and measuring the Mahalanobis distance to the expected order statistics from this distribution.
To compute the Mahalanobis distance, we need to subtract the ordered sample vector by the expected values, and multiply by the inverse of the covariance matrix. Investigating this problem numerically, I noticed this matrix seems to be symmetric tridiagonal. The second diagonal also seems to be merely the diagonal times -1/2.
My question is basically how can I compute this vector of expected values and inverse of the covariance from the order statistics of a sample from the standard normal distribution with size N? (I'm interested in N = 16)
I hope there is a closed formula for this. If this isn't available, though, can we at least prove that the matrix is tridiagonal, and the coefficients of the secondary diagonals are proportional to the principal diagonal?

The exact covariance $\operatorname{Cov}(U_{(k)},U_{(j)})$ between the order statistics of uniform random variables is given wikipedia.  From that you should at least be able to obtain a reasonable approximation of
$$\operatorname{Cov}(X_{(k)},X_{(j)})=\operatorname{Cov}(\Phi^{-1}(U_{(k)}),\Phi^{-1}(U_{(j)}))
$$
via linearizations of $\Phi^{-1}$ around the expected values of $U_{(j)}$ and $U_{(k)}$. This indeed seems to be approach taken in https://stats.stackexchange.com/a/397749/77222 and in references therein.
