Derivative of the expected gpl loss — dexp_gpl

For the linear increment function this is $$\kappa\, g'(x)\, (F(x) - \alpha),$$ the quantity whose root gives the unconstrained optimal allocation.

Usage

dexp_gpl_loss(g = "x", dg = NULL, F, kappa = 1, alpha = NA, O = NA, U = NA)

g: a non-decreasing increment function, supplied either as a function or as a string in the variable x such as "log(x)".
dg: derivative of g; if NULL or NA it is derived from g.
F: predictive cdf of the outcome.
kappa: scale factor.
alpha: normalized loss when the outcome y exceeds the allocation x. Exactly one of alpha and U must be supplied.
O: cost incurred when the allocation x exceeds the outcome y; equals kappa * (1 - alpha).
U: cost incurred when the outcome y exceeds the allocation x; equals kappa * alpha.

A function of an allocation x giving the derivative of the expected loss with respect to the distribution F.

# zero at the median of the predictive distribution
dexp_gpl_loss(F = pnorm, alpha = 0.5)(0)
#> [1] 0