Comparison of Hierarchical Bayesian Interpurchase Time Models

指數分配（exponential distribution）、Erlang分配、伽馬分配（gamma distribution）與半常態分配（Half-Normal distribution）是一般化伽馬分配（general gamma distribution）之特例，用以描述購買期間（interpurchase time）的分配時，各自具有不同形態的危險率函數與購買行為上之意涵：指數分配具無記憶性（memoryless），隱含購買發生是隨機性的；Erlang分配適用無法作品牌移轉（brand switching）之情況；伽馬分配型狀較具彈性，可包含指數分配與Erlang分配；半常態分配符合一般具有品牌移轉之情況。為了研究危險率函數型態與實際購買行為型態的配適，是否具有較好的模型預測能力，本研究針對不同分配危險率函數之購買行為意涵進行研究，並比較其預測能力。 本研究使用層級貝氏購買期間模型（Hierarchical Bayesian interpurchase time model）作為主要研究方法，並使用馬可夫鏈蒙地卡羅法（MCMC, Markov Chain Monte Carlo）模擬參數之後驗分配（posterior distribution）。另外，為解決不具有特定分配型態（close form）後驗分配之問題，使用反累積機率函數法（inverse cumulative density function method）搭配資料摸擬產生後驗分配之亂數值。 實證結果顯示，雖然伽馬分配具有兩個參數，在危險率函數型態上較具有彈性，但同時也需要較多的樣本作參數估，在個人樣本資料不足的情況下，伽馬分配的預測的表現並不是最好的。在其他單一參數的分配中，Erlang分配之危險率函數型態與實際資料之購買行為型態最符合，預測的效率最好。Exponential, Erlang, gamma, and Half-Normal distributions of general gamma distribution with different shapes of hazard rate functions and marketing intent. Exponential distribution with memoryless hazard rate function implies a random purchase behavior; Erlang distribution is suitable for the monopoly case, where no brand switching is allowed; Gamma distribution is quite flexible, which covers the shapes of exponential and Erlang distribution; Half-Normal distribution is a common case, where brand switching exists. To examine whether the consistency between purchase behavior intent of model and actual data behavior would lead to a better prediction, we analyze the behavioral meaning of hazard rate functions and compare its performance in empirical research. Hierarchical Bayesian interpurchase time models are built and MCMC (Markov Chain Monte Carlo) method is used to simulate the posterior distributions of parameters. To deal with the distribution with no close form, we use inverse cumulative density function method with data simulation method to generate random numbers. The comparison results showed that in small sample cases, gamma distribution with two parameters, though more unconstrained in shape of hazard rate function, is not efficient in prediction for lack of enough sample information to estimate. Moreover, among distributions with one parameter containing exponential, Erlang, and Half-Normal distributions, the Erlang outstanding in prediction is because of the consistency between the marketing intent of hazard rate function and the actual purchase behavior of data, which is a case without brand switching behavior.ACKNOWLEDGEMENTS 1 中文摘要 2 ABSTRACT 3 TABLE OF CONTENT 4 LIST OF TABLES 7 LIST OF FIGURES 8 CHAPTER 1: PREFACE 9 1.1 RESEARCH MOTIVES 9 1.2 RESEARCH SCOPE 11 I. Purchase behavior assumptions of hazard rate function and its prediction performance 11 II. Prediction performance in small sample 11 III.Prediction performance in out sample 11 1.3 RESEARCH OBJECTIVES 12 1.4 FRAMEWORK 12 CHAPTER 2: LITERATURE REVIEW 15 2.1 ALTERNATIVE APPROACHES OF PURCHASE TIME MODEL15 I. Linear Regression 15 II. Modeling the Buy/Not Buy Decision 15 III.Negative Binomial Distribution model 16 IV. Stochastic Interpurchase Time 19 2.2 HIERARCHICAL BAYESIAN MODEL 24 I. Bayesian Statistics 24 II. Gibbs sampling 25 CHAPTER 3: RESEARCH METHODOLOGY 27 3.1 POOLED GENERAL GAMMA MODEL 29 I. Model 29 II. Estimation Algorithms-Maximum Likelihood 30 3.2 INDIVIDUAL GENERAL GAMMA MODEL 33 I. Model 33 II. Estimation Algorithms-Maximum Likelihood 34 3.3 HIERARCHICAL BAYESIAN GENERAL GAMMA DISTRIBUTION MODEL 37 I. Model 37 II. Estimation Algorithms 40 3.4 HIERARCHICAL BAYESIAN GENERAL GAMMA REGRESSION MODEL 45 I. Model 45 II. Estimation Algorithms 47 III.Independent variables X (covariates) 53 3.5 GENERAL GAMMA DISTRIBUTION HAZARD RATE FUNCTIONS 55 I. General Gamma Distribution 56 II. Exponential Distribution 59 III.Erlang Distribution 64 IV. Gamma Distribution 69 V. Half-Normal Distribution 74 CHAPTER 4: EMPIRICAL RESEARCH 79 4.1 SAMPLE 79 I. Source 79 II. Descriptive Statistics 79 III.Difference across Groups 83 4.2 ESTIMATION RESULT 86 4.3 PREDICTION 88 I. Prediction Estimator – Expectation 88 II. Prediction Efficiency Index 89 III. Prediction Result 90 4.4 HYPOTHESIS TESTING 94 CHAPTER 5: CONCLUSIONS AND SUGGESTIONS 99 5.1 CONCLUSIONS AND FINDINGS 99 5.2 MANAGEMENT INSIGHT 100 5.3 RESEARCH CONSTRAINT AND FURTHER RESEARCH 100 REFERENCES 101 APPENDIX 103 APPENDIX 1: GAUSS PROGRAM 103 Main Program 103 Data Process Procedures 107 Pooled Model Procedures 110 Individual Model Procedures 117 Prediction Procedures 124 APPENDIX 2: MATLAB PROGRAM 125 Main Program 125 Data Load Functions 130 MCMC Functions 132 Model 3 Functions 136 Model 4 Functions 141 APPENDIX 3: MCMC RESULT 14