Quantile Forecasts
Asymmetric costs have an asymmetric answer. If being short costs three times as much as being long, the target is not the mean.
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Quantile Level
A quantile level selects a position in the outcome distribution. Exact exceedance probabilities require continuity; discrete outcomes can have ties.
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Pinball Loss (a.k.a. Quantile Loss)
At p = 0.5 it is just absolute error. Away from the median it weights misses unevenly, which is what makes a quantile the thing being estimated.