To give a simple hypothetical example, suppose one was interested in using linear regression to predict how horses would perform in a race. Let's suppose the race includes 10 horses and that I have a variety of potential explanatory effects for all of them (e.g. weight, age, record, jockey, and so on) as well as a data set with a year's worth of past results and those same explanatory effects. Let's also suppose that I am only interested in predicting a top three finish; outcomes outside of the top three are of no importance to me.
My first instinct is that ordinary least squares would be inappropriate for this situation because OLS is used to predict the conditional mean, whereas I am interested in predicting a conditional quantile (the 70th percentile, essentially). Am I even correct in thinking that quantile regression is the proper technique for this purpose?
Assuming my choice of quantile regression is appropriate, my primary question is this: which quantile ought I to aim for if attempting to identify a horse that finishes in at least the top three positions?
I might attempt to predict at the 70th percentile, operating under the logic that this is the threshold I am interested in the horse crossing, however that would seem to leave little margin for error, as I'd be predicting right on the boundary of success and failure. Would it be more prudent to predict the 75th, or perhaps even the 90th percentile, operating under the logic that even if the horse predicted to achieve at that level underperforms, there is margin for error because I'm setting the bar higher than necessary?
My first inclination is to go the former route, because I feel like there would have to be some trade-off with predicting a higher quantile than necessary, but I can't identify what that trade-off is. Surely there is a sound reason for not simply attempting to predict the 1st place finisher when a 3rd place finish is equally satisfying to me, no?
Is there a more proper alternative entirely, like composite quantile regression, perhaps? I'm interested to hear how one would approach this situation, as I normally read about quantile regression in the context of comparing the influence of effects across quantiles rather than using quantile regression for prediction per se.
Thanks in advance.