COMBINING FORECASTING PROCEDURES: SOME THEORETICAL RESULTS
研究了组合预测连续随机变量的方法，推导了平方误差损失下的风险界，表明组合预测能自动达到最佳候选方法的性能，但盲目组合可能因权重估计的变异性而降低精度，理论证明了自动组合方法能平衡潜在收益与复杂度惩罚。
We study some methods of combining procedures for forecasting a continuous random variable. Statistical risk bounds under the square error loss are obtained under distributional assumptions on the future given the current outside information and the past observations. The risk bounds show that the combined forecast automatically achieves the best performance among the candidate procedures up to a constant factor and an additive penalty term. In terms of the rate of convergence, the combined forecast performs as well as if the best candidate forecasting procedure were known in advance.Empirical studies suggest that combining procedures can sometimes improve forecasting accuracy over the original procedures. Risk bounds are derived to theoretically quantify the potential gain and price of linearly combining forecasts for improvement. The result supports the empirical finding that it is not automatically a good idea to combine forecasts. Indiscriminate combining can degrade performance dramatically as a result of the large variability in estimating the best combining weights. An automated combining method is shown in theory to achieve a balance between the potential gain and the complexity penalty (the price of combining), to take advantage (if any) of sparse combining, and to maintain the best performance (in rate) among the candidate forecasting procedures if linear or sparse combining does not help.This research was supported by U.S. National Security Agency Grant MDA9049910060 and U.S. National Science Foundation CAREER Grant DMS0094323. The author sincerely thanks three reviewers and Poti Giannakouros for their very valuable comments, which led to a substantial improvement of the paper.