third
The third order loss function, (1/6)Emax(X-x,0)^3.
The partial-expectation expansion of secondOrderLossFunction carried one order up: expanding (u-x)^3 inside the integral leaves H3 - 3xH2 + 3x^2H1 - x^3G0, where H_k(x) = EX^k * Phi(ksigma - z) is the kth partial expectation above x.
Note that EX^3 = exp(3mu + 4.5sigma^2) on the log scale, which is the first place this family's moments become large: a heavy-tailed fit can carry it past the range of a Double while the lower orders stay finite, so the result is reported as infinite rather than as a silently wrong finite number.