探討多變量伯努利分配

Some Discussions on the Multivariate Bernoulli Distribution

梁丞智(CHENG-CHIH LIANG)

指導教授 ： 黃怡婷

國立臺北大學/商學院/統計學系/碩士(2016年)

摘要

二元變量之聯合機率分配有許多設定方式，如 Bahadur (1961) 使用相關係數、 Bishop 等人 (1975) 所提的對數線性模型， Diggle 等人 (2002) 使用勝算比。當所要描述的資料為不同時間點之二元狀態的關聯性時，利用馬可夫假設， Bishop 等人 (1975) 使用移轉機率來設定二元變量之聯合機率分配。在一階馬可夫假設下，本論文探討這些多變量伯努利分配的關聯性，及其最大概似估計量的推導方式與估計量對應的漸進性質。

關鍵字

多變量伯努利分配 ； 馬可夫假設 ； Bahadur 模型 ； 對數線性模型

並列摘要

Many joint distributions for many binary variables have been proposed. For example, Bahadur (1961) used correlations to express the joint distribution, Bishop et al. (1975) used the log linear model to construct the joint distribution, and Diggle et al. (2002) used odds ratios to represent the joint distribution. When the association of binary data in different time periods is of interest, the Markov model is a common model to use (Bishop et al., 1975). This paper discusses relationship among different representations of the joint distribution of many binary variables under the first-order Markov assumption. Furthermore, the maximum likelihood estimator and its corresponding asymptotic distribution are derived.

並列關鍵字

Multivariate Bernoulli distribution ； Markov assumption ； Bahadur model ； log-linear model

參考文獻

Diggle, P. J., Heagerty, P., Liang, K. Y., & Zeger, S. L. (2002). Analysis of longitudinal data. Oxford University Press.

Bishop, Y. M., Fienberg, S. E., & Holland, P. W. (2007). Discrete multivariate analysis: theory and practice. Springer Science & Business Media.

Cheng, J., Levina, E., Wang, P., & Zhu, J. (2014). A sparse ising model with covariates. Biometrika, 70(4), 943-953.

Dai, B., Ding, S., & Wahba, G. (2013). Multivariate Bernoulli distribution. Bernoulli, 19}(4), 1465-1483.

Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80(1), 27-38.