Comparing probability forecasts derived from theoretical distributions

Using the wrong forecast error distribution (typically the normal distribution) has been suggested as one reason why prediction intervals are too narrow. Extreme values are especially likely to be drawn from non-normal distributions. A simple way of selecting the appropriate theoretical distribution is to estimate parameters using historical data, transformed if necessary, to make them stationary. The method is demonstrated using daily electricity peak loads, a set of extreme values. Parameters for four specific distributions, the normal, gamma, Cauchy and Weibull, were estimated and used to make probabilistic forecasts. Although none of the distributions produced well-calibrated post-sample forecasts, the Weibull showed the most promise. Probability forecasts calculated from Chebychev’s inequality were the worst-calibrated.