# Loss function/error measurement for allocation problem

I'm working on a prediction rule for an allocation problem. So, it's data like this:

| group | allocation (actual)  | allocation (predicted) |
|-------|----------------------|------------------------|
| 1     | 0.2                  | 0.15                   |
| 1     | 0.5                  | 0.6                    |
| 1     | 0.3                  | 0.25                   |
| 2     | 0.1                  | 0.0                    |
| 2     | 0.2                  | 0.4                    |
| 2     | 0.4                  | 0.45                   |
| 2     | 0.3                  | 0.15                   |
| ...   | ...                  | ...                    |


The distribution of actual allocations for each group happens to be highly skewed:

I'm unsure which loss function to use to tune my prediction rule. I definitely care about discrimination of very small values from large ones, but I don't really care whether values above 0.25 are accurately predicted as long as they end up in a > 0.25 bucket.

Among the various forms of squared error (MSE, RMSE, weighted MSE, etc.) and absolute error (MAE, MAD), is there one that would make the most sense for my purposes?

$$L_\delta(y, F) = \begin{cases} \frac{1}{2}(y - F)^2 & \textrm{for } |y - F| \le \delta, \\ \delta\, |y - F| - \frac{1}{2}\delta^2 & \textrm{otherwise.} \end{cases}$$