Fixed Sum Losses in Operational Forecasting

Abstract

In operational applications, forecasts .are normally adjusted by applying safety factors to allow for asymmetry in the underlying loss function. This paper considers an extension of the often used linear loss function to situations in which an error can also result in a fixed sum loss.The need for forecasts as inputs to planning and decision models is widely recognized. Forecasts range from subjective, qualitative predictions of the long range future to short range operational projections which are quantitative and highly structured. In the latter case historical statistics normally are available and form the basis for the projection as in sales forecasting for production and inventory control, cash flow forecasts, and forecasts of labor turnover. Analytic procedures include classical time series analysis, regression models, exponential smoothing, and other adaptive procedures, as well as subjective methods such as the use of executive or sales force opinions.It is generally accepted that forecasts can never be completely accurate. To the extent that the loss functions associated with forecast errors are explicitly considered, the usual assumption is that the loss function is symmetric and often that it is quadratic.' This leads to an emphasis on unbiased predictors. Alternatively, maximum likelihood procedures may be used, implying that a forecast should have a high probability of being (approximately) correct. In practice, the users of forecasts are seldom interested in either an unbiased or a most likely forecast. Instead, a correction usually is added to or subtracted from the forecast to implicitly allow for an intuitively felt asymmetry in the costs of positive and negative forecast errors. As planning and decision models have become more structured, it is frequently possible to objectively include these adjustments or safety factors in the model by formally introducing some measure of forecast uncertainty. Therefore, more and more forecasts are being given as probability distributions rather than as point predictions [4], [ 5 ] , [ 6 ] .In some cases, planning and decision models can directly incorporate these probability distributions by using payoff tables or decision trees. In other cases, the forecast variable is quantitative, and the asymmetric loss function can be made an integral part of the statistical analysis [l]. Since a substantial forecasting methodology already exists based on symmetric loss functions, it is far more practical to consider ways of adjusting these forecasts through correction For example see [7, p. 161. 425

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