Normal variate prediction under linear-quadratic loss

The prediction of the value of a normal random variable is considered in the presence of combined linear and quadratic loss. It is shown that the optimal prediction is an additive adjustment to the predictive mean, the adjustment being the product of the predictive standard deviation and an appropriate adjustment factor. An extensive table of adjustment factor values is presented, and it is noted that for given loss parameters the adjustment is positive unless the predictive variance is sufficiently large.