探討雙變量卜瓦松分配與其對應延伸模型

A Study of Bivariate Poisson Distribution and Corresponding Models.

余勇徹(Yu Yung-Che)

指導教授 ： 汪群超

國立臺北大學/商學院/統計學系/碩士(2015年)

摘要

計數資料分析方式目前大部分使用單變數卜瓦松模型，但有鑑於資料收集日益方便，所收集到的計數資料可能超過一個變數，因此雙變數或更多變數聯合的模型才可以有效評估出影響因素。

Edwards 和 Gurland (1961) 提出的複合型雙變量卜瓦松分配，其中雙變數與其相關皆有利用隱藏變數來描述， Kocherlakota 等人 (1973) 提出估計雙變量卜瓦松分配參數的方式。然雙變數之隱藏變數之參數估計並未被 Kocherlakota 等人 (1973) 討論，本論文提出運用 EM 演算法來估計所有參數，再以此雙變數分配假設，建構雙變數廣義線性模型，並利用統計模擬來評估上述模型及方法估計雙變數分配參數的可行性，最後使用此模型來評估一份交通載具使用情況的問卷資料。

關鍵字

廣義線性模型 ； 蒙地卡羅模擬法 ； EM 演算法 ； 拔靴法 ； 雙變量卜瓦松分配

並列摘要

The univariate Poisson model is the most current used analytical methods to analyze count data. Nowadays, data become rather easy to collect. The data structure or information can be complex. Thus, jointly modeling more than one outcome might become necessary to identify possible influential factors. In this paper, we propose using the bivariate compound Poisson distribution proposed by Edward and Gurland (1961) to model multivariate count data. The correlation between
two outcomes is accounted by the compounding part, whereas the variables and correlation between events are characterized by a latent variable. Kocherlakota and Kathleen (1973) provided parameter estimations for the bivariate compound Poisson distribution assuming the parameter in the latent distribution is known. In this paper, assuming the latent variable having the gamma distribution, we suggest using EM algorithm to estimate all the parameters in the bivariate compound Poisson distribution. Furthermore, we construct a bivariate joint log-linear models based on this distribution. Monte Carlo simulations are used to evaluate the performance of the parameter estimations. Finally, a data about the way of commute in Taiwan is used to validate the feasibility of the proposed model.

並列關鍵字

bivariate poisson distribution ； Monte Carlo simulation ； bootstrap ； EM algorithm ； generalized linear model

參考文獻

[1] Edwards, C. B. and Gurland, J. (1961). A class of distributions applicable to accidents, J. Amer. Statist. Assoc., 56, 503-517.

[2] Feller, W. (1957). An introduction to probability theory and its applications, Vol. I, Wiley, New York.

[3] Gurmu, S., Elder, J., (2000). Generalized bivariate count data regression models. Economics Letters, 68, 31 36.

[6] Kawamura, K. (1985). A note on the recurrence relations for the bivariate Poisson distribution, Kodai Math. J., 8 (1), 70-78.

[7] Kocherlakota, S. and Kocherlakota, K. (1992). Bivariate discrete distribution, Marcel Dekker,New York.