連鎖銷售商店動輒上百家分店，商店銷售額的預測是重要的目的，一般是以個別分店的銷售資料，找出統計模型，對個別分店的銷售額做預測是一種簡單的方法，然而，因為這些分店之間可能有某些相似性，若能找到一個可以同時運用多個店家資料的統計模型，可能有機會改進模型的預測能力與模型係數的適切性，有助商家因應節日及進行促銷時的行銷策略。本論文使用回歸分析對銷售資料進行預測，對不同店家的銷售額所做的回歸分析的參數，用貝氏方法來做進一步的處理，透過將多家店家的回歸係數縮減(shrinkage)，以達到較合理的參數，此方法的主要目的是尋找較合理的參數，其次則是探討迴歸係數縮減下模型預測能力的表現。
本研究發現在多家分店的原始迴歸係數相當接近時，使用貝氏方法的改進空間有限，其中階層貝氏方法能夠將若干家商店資訊納入，能對迴歸係數產生較大的縮減，因此有機會改進預測能力，而James-Stein 估計並沒有參考多家商店資訊，因此對於迴歸係數產生較小的縮減，故其預測能力並無太大改進。

The prediction of sales is important. It is common to do regression analysis to predict sales for a store using its own data. However, for a chain with hundreds of stores, it may be possible to improve prediction accuracy and obtain more reasonable regression coefficients by combining data from different stores. We propose to achieve these goals by using two shrinkage methods: hierarchical Bayesian method and James-Stein estimator.
We found that the shrinkage methods yield limited improvement when the regression coefficients in separate models are rather close. Moreover, the hierarchical method incorporated data from different stores and improve predictions, while James-Stein estimator did not improve much.

1. Jun Shao, Mathematical Statistics, 1999
2. Robert C. Blattberg and Edward I. George,1991, Shrinkage Estimation of Price and Promotional Elasticities: Seemingly Unrelated Equations
3. Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B.Rubin 2013, Bayesian Data Analysis, Third Edition
4. Donald B. Rubin 1980, Using Empirical Bayes Techniques in the Law School Validity Studies
5. Andrew McCallum, Ronald Rosenfeld, Tom Mitchell, Andrew Y. Ng, Improving Text Classification by Shrinkage in a Hierarchy of Classes
6. John Barnard, Robert McCulloch and Xiao-Li Meng 1999, Modeling Covariance Matrices in Terms of standard deviations and correlations, with application to shrinkage