消費者購物次數之階層式貝氏模型模擬分析

A simulation study of Hierarchical Bayesian model for modeling the number of purchases

張聖賢(Sheng-Hsien Chang)

指導教授 ： 黃怡婷

國立臺北大學/商學院/統計學系/碩士(2014年)

摘要

消費者購物模式研究可以提供接近實際的消費預測結果，提供給工商界管理階層掌握公司的成本、提升經營效能、為公司創造更多的利潤，造福員工。因此許多學者研究向來是以預測模型為主題，本文嘗試透過模擬方法探討統計分析模型的表現，期盼精進模型預測的能力。

本文以 Jen, Chou 與 Allenby (2003) 的「建構購買頻率之貝氏方法」為基礎，該模型假設消費者的購物次數符合波松分配，並假設做對數轉換後的平均購物次數服從伽瑪分配，並且假設型態參數服從均勻分配，而對數轉換後的尺度與類別形式的自變項有關，再者，假設對數轉換後的類別自變數的係數服從反伽瑪分配 (inverse gamma distribution)。由於 HB模型的建構非常複雜，本論文將詳細的參數估計流程，提出三種修正模型，主要修正先驗分配的假設與先驗分配的參數估計方式，並提出利用拔靴法的參數估計的標準誤。最後利用統計模擬討論其中一種修正模型參數估計的表現，其中評估指標包含偏誤、標準誤及信賴區間 (confidence interval) 和可靠區間 (credible interval) 並計算其覆蓋率。

關鍵字

Gibbs 取樣程序 ； 邊際最大概似估計法 ； 階層式貝氏模型 ； 可靠區間 ； 拔靴法

並列摘要

Understanding the pattern of the consumer’s purchase provides the most important information for the managers to control the cost and enhance the efficacy of management such that the business can have more profit and provide more benefit to employees. Hence, many researchers have focused on discussing the performance of the prediction model.

Based on the Jen, Chou and Allenby (2003) proposed a hierarchical Bayesian (HB) approach to predict purchase frequency. The HB approach assumes first that the number of purchases for a particular store or items follows a Poisson distribution, where the parameter of the average number of purchase follows a gamma distribution with shape and scale parameter. Furthermore, the log transformation of the scale parameter is related to a set of covariates, where the log transformation of the corresponding coefficient follows an inverse gamma distribution. Also, the shape parameter follows an uniform distribution with ranged 0 and M, where M is pre-specified. HB approach intends to take into account of heterogeneity of purchase frequency of customers. However, the estimation procedure and the model features are not discussed thoroughly in Jen et al. (2003). Based on the Monte Carlo simulation, this paper will provide the detailed estimation procedure of HB approach. Three modified procedures are provided and the standard error of parameter estimators is obtained by bootstrap approaches. Finally, the performance of estimators is evaluated by bias, standard errors and the coverage of the confidence intervals and credible intervals.

並列關鍵字

Gibbs Sampler ； Marginal Maximum Likelihood Estimate ； Hierarchical Bayesian Model ； Credible Interval ； Bootstrap

參考文獻

Brokett, Patrick L., Golden, Linda L., Panjer, Harry H. (1996). ”Flexible Purchase Frequency Modeling”,

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