From a certain point of view, a standard linear regression finds a model which minimizes the Pearson correlation between the predicted value of the response variable and the true value of the response variable. Are there any (standard) modifications to the procedure which instead minimize a rank-order correlation (e.g. the Spearman rho or Kendall tau)? Or more generally, are there approaches to generalized linear models where the functional form of the link function is unknown/unspecified?
I have a situation where I think a generalized linear model is likely to be appropriate ... except that I don't know what the functional form for the link function would be. I can, however, specify the constraint that the link function will be monotonic. Hence the hope that I can fit a (generalized) linear model to rank ordering (which will just enforce monotonicity) rather than a standard fit (which will also attempt to match absolute magnitudes of the response variable).
An approach which can be regularized (as in lasso or ridge) would be much preferred.
Note: My situation is distinct from that for ordinal regression. My response variable is a continuously-valued real function, rather than a small set of discrete integer levels as is the case for ordinal regression.