Complexity of simple nonlogarithmic loss functions

The loss complexity for nonlogarithmic loss functions is defined analogously to the stochastic complexity for logarithmic loss functions such that its mean provides an achievable lower bound for estimation, the mean taken with respect to the worst case data generating distribution. The loss complexity also provides a lower bound for the worst case mean prediction error for all predictors. For the important α-loss functions |y-ˆy|^α, where y-yˆ denotes the prediction or fitting error and α is in the interval, an accurate asymptotic formula for the loss complexity is given.