討論零膨脹計數資料之對數線性模型參數估計

Evaluation of Parameter Estimations in Log-Linear Model under Zero-Inflated Count Data

宋志雄(Chin-Hsiung Sung)

指導教授 ： 黃怡婷

國立臺北大學/商學院/統計學系/碩士(2015年)

摘要

在電子錢包逐漸取代傳統貨幣的現今社會，平均每個人擁有一張以上的信用卡，但許多信用卡辦了卻很少使用它，當分析銀行信用卡消費次數資料時，鮮少使用的卡就會產生很多零值，這些零值過多的計數資料稱之為零膨脹計數資料。卜瓦松分配是分析計數資料最常使用的母數假設，為處理零膨脹問題，Lambert (1992) 提出的零膨脹卜瓦松分配，該分配是目前分析零膨脹計數資料最常使用的母數假設。大部分探討消費者行為都是由Ehrenberg (1959) 提出的純香草模型修正得來，根據該模型的設定，結合貝塔分配，吳佩琪 (2008)提出貝塔二項分配來分析零膨脹消費者行為資料。
根據吳佩琪(2008)，本論文結合邏輯式函數，延伸出負二項邏輯式模型用來處理零膨脹問題，本論文的目的是三種模型在不同的零膨脹計數資料下，考慮四種形態的資料設定，利用統計模擬方式探討資料特性對母數分配假設的敏感性，比較各參數估計的絕對偏誤與各模型的誤差來衡量分配假設與零膨脹計數資料的敏感性。最後將模型應用到實務信用卡消費資料。

關鍵字

對數線性模型 ； 負二項分配 ； 零膨脹卜瓦松分配 ； 卜瓦松分配 ； 邏輯式模型

並列摘要

More and more customers use credit cards or electronic purse to pay their bills instead of real money. Quite frequently, people have more than one credit card on average. Nevertheless, only a few credit cards are used. To analyze the consumer behavior in using credit cards, there exists many zeros. Such a data with many zeros are called zero-inflated count data. To deal with the excess zeros, Lambert (1992) proposed a zero-inflated Poisson distribution. The most popular model for consumer consumption behavior was proposed by Ehrenberg (1959) which is called the plain vanilla model. To take into account of excess zeros, Wu (2008) combined Beta distribution and the plain vanilla model and proposed a beta-binomial model.
Based on the derivation in Wu (2008), this thesis proposes combining Beta distribution with logistic model to deal with excess zeros. To understand the sensitivity of the distributional assumption, Monte Carlo simulation is conducted. Under various settings, the absolute bias and the prediction error are used to evaluate the performance of the estimators. A real data is used to illustrate the feasibility of the proposed model.

並列關鍵字

Negative Binomial Distribution ； Zero-inflated Poisson Distribution ； Poisson Distribution ； Log-Linear Model ； Logistic Model.

參考文獻

Danie, B. H. and Kenneth, S. B. (2002). Score tests for heterogeneity and overdispersion in zero-inflated Poisson and binomial regression models. The Canadian Journal of Statistics, 30 (3), 415-430.

Donald, G. M. and Schmittlein, D. C. (1988). Generalizing the NBD model for customer purchases: what arethe implications and is it worth the effort? Journal of Business and Economic Statistics, 6, 145-159.

Ehrenerg, A. S. C. (1959). The pattern of consumer purchases. Applied Statistics, 8, 26-41.

Greenwood, M. and Yule, G. U. (1920). An onquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents. Journal of Royal Statistical Society, 83, 255-79.

Greenwood, M. and Yule, G. U. (1920). An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or repeated accidents. Journal of Royal Statistical Society B, 83, 255-279.