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題名 基於Sarmanov分佈的二元縱向計數資料模型
A Sarmanov Distribution Based Model for Bivariate Panel Count Data作者 陳盈諳
Chen, Ying-An貢獻者 黃佳慧
Huang, Chia-Hui
陳盈諳
Chen, Ying-An關鍵詞 非齊次Poisson過程
縱向計數資料
比例均值迴歸模型
隨機效應
Sarmanov分佈
Nonhomogeneous Poisson process
Panel count data
Proportional mean regression model
Random effect
Sarmanov distribution日期 2022 上傳時間 1-Aug-2022 17:16:07 (UTC+8) 摘要 在本文中，我們為二元縱向計數資料建立聯合模型，此類資料只能在特定時間點上被蒐集。在模型的架構中，我們假設每一種事件類型的計數服從一個非齊次Poisson過程，並使用比例均值迴歸模型建構事件發生率。為了將二元資料內的關聯性納入模型，我們考慮在每一個事件類型的均值函數內存在隨機效應，而提出的模型允許此二元縱向計數資料可以透過包含於均值函數內的隨機效應使其建立相依性，這些隨機效應服從一個邊際分佈為Gamma分佈的Sarmanov分佈。在此隨機模型假設下，我們推導出二元縱向計數資料的聯合機率分佈，並利用最大概似估計法取得參數估計。我們使用模擬比較兩種估計方法下所得之估計量的表現，從模擬的結果中可以觀察到兩者的表現相似。最後，本文所提出之模型套用在內政部警政署的交通資料，估計協變量與季節對於車禍發生率的影響。
In this work, we consider a joint model for panel count data with bivariate event types, which are only collected at particular time points. We assume that the counts follow a nonhomogeneous Poisson process for each event type, and a proportional mean regression model is specified. To account for the association, we further impose a positive random effect on each of the mean functions. The proposed model allows for the dependence of event types through random effects that follow the bivariate Sarmanov distribution with gamma marginals. The estimations of the parameters are based on the maximum likelihood method. We use two estimation methods and compare the performance of the estimators based on several simulation studies, which result in similar performance. An application to traffic accident data is presented.參考文獻 Abdallah, A., Boucher, J.-P., and Cossette, H. (2016). Sarmanov family of multivariate distributions for bivariate dynamic claim counts model. Insurance: Mathematics and Economics, 68:120–133.Bahraoui, Z., Bolancé, C., Pelican, E., and Vernic, R. (2015a). On the bivariate sarmanov distribution and copula. an application on insurance data using truncated marginal dis- tributi. SORT, 39(2):209–230.Bahraoui, Z., Bolancé, C., Pelican, E., and Vernic, R. (2015b). On the bivariate sarmanov distribution and copula. an application on insurance data using truncated marginal dis- tributions. Statistics and Operations Research Transactions, SORT, 39(2):209–230.Bairamov, I., Altinsoy, B., and Kerns, G. J. (2011). On generalized sarmanov bivariate distributions. TWMS Journal of Applied and Engineering Mathematics, 1(1):86–97.Bairamov, I., Kotz, S., and Gebizlioglu, O. L. (2001). The sarmanov family and its gen- eralization: theory and methods. South African Statistical Journal, 35(2):205–224.Bolancé, C. and Vernic, R. (2019). Multivariate count data generalized linear models: Three approaches based on the sarmanov distribution. Insurance: Mathematics and Economics, 85:89–103.Dias, A., Embrechts, P., et al. (2004). Dynamic copula models for multivariate high- frequency data in finance. Manuscript, ETH Zurich, 81.Hashorva, E. and Ratovomirija, G. (2015). On sarmanov mixed erlang risks in insurance applications. ASTIN Bulletin: The Journal of the IAA, 45(1):175–205.Huang, C.-Y., Wang, M.-C., and Zhang, Y. (2006). Analysing panel count data with in- formative observation times. Biometrika, 93(4):763–775.Joe, H. (1997). Multivariate models and multivariate dependence concepts. CRC press. Kotz, S., Balakrishnan, N., and Johnson, N. L. (2004). Continuous multivariate distribu-tions, Volume 1: Models and applications, volume 1. John Wiley & Sons.Nelsen, R. B. (2006). An introduction to copulas. Springer Science & Business Media.Ross, S. M., Kelly, J. J., Sullivan, R. J., Perry, W. J., Mercer, D., Davis, R. M., Washburn, T. D., Sager, E. V., Boyce, J. B., and Bristow, V. L. (1996). Stochastic processes, volume 2. Wiley New York.Sarmanov, O. V. (1966). Generalized normal correlation and two-dimensional fréchet classes. In Doklady Akademii Nauk, volume 168, pages 32–35. Russian Academy of Sciences.Shi, P. and Zhao, Z. (2020). Regression for copula-linked compound distributions with applications in modeling aggregate insurance claims. The Annals of Applied Statistics, 14(1):357–380.Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. Publ. inst. statist. univ. Paris, 8:229–231.Sun, J., Zhao, X., et al. (2013). Statistical analysis of panel count data. Springer.Ting Lee, M.-L. (1996). Properties and applications of the sarmanov family of bivariate distributions. Communications in Statistics-Theory and Methods, 25(6):1207–1222.Wellner, J. A. and Zhang, Y. (2000). Two estimators of the mean of a counting process with panel count data. The Annals of statistics, 28(3):779–814.Wellner, J. A. and Zhang, Y. (2007). Two likelihood-based semiparametric estimation methods for panel count data with covariates. The Annals of Statistics, 35(5):2106– 2142. 描述 碩士
國立政治大學
統計學系
109354017資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109354017 資料類型 thesis dc.contributor.advisor 黃佳慧 zh_TW dc.contributor.advisor Huang, Chia-Hui en_US dc.contributor.author (Authors) 陳盈諳 zh_TW dc.contributor.author (Authors) Chen, Ying-An en_US dc.creator (作者) 陳盈諳 zh_TW dc.creator (作者) Chen, Ying-An en_US dc.date (日期) 2022 en_US dc.date.accessioned 1-Aug-2022 17:16:07 (UTC+8) - dc.date.available 1-Aug-2022 17:16:07 (UTC+8) - dc.date.issued (上傳時間) 1-Aug-2022 17:16:07 (UTC+8) - dc.identifier (Other Identifiers) G0109354017 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/141009 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計學系 zh_TW dc.description (描述) 109354017 zh_TW dc.description.abstract (摘要) 在本文中，我們為二元縱向計數資料建立聯合模型，此類資料只能在特定時間點上被蒐集。在模型的架構中，我們假設每一種事件類型的計數服從一個非齊次Poisson過程，並使用比例均值迴歸模型建構事件發生率。為了將二元資料內的關聯性納入模型，我們考慮在每一個事件類型的均值函數內存在隨機效應，而提出的模型允許此二元縱向計數資料可以透過包含於均值函數內的隨機效應使其建立相依性，這些隨機效應服從一個邊際分佈為Gamma分佈的Sarmanov分佈。在此隨機模型假設下，我們推導出二元縱向計數資料的聯合機率分佈，並利用最大概似估計法取得參數估計。我們使用模擬比較兩種估計方法下所得之估計量的表現，從模擬的結果中可以觀察到兩者的表現相似。最後，本文所提出之模型套用在內政部警政署的交通資料，估計協變量與季節對於車禍發生率的影響。 zh_TW dc.description.abstract (摘要) In this work, we consider a joint model for panel count data with bivariate event types, which are only collected at particular time points. We assume that the counts follow a nonhomogeneous Poisson process for each event type, and a proportional mean regression model is specified. To account for the association, we further impose a positive random effect on each of the mean functions. The proposed model allows for the dependence of event types through random effects that follow the bivariate Sarmanov distribution with gamma marginals. The estimations of the parameters are based on the maximum likelihood method. We use two estimation methods and compare the performance of the estimators based on several simulation studies, which result in similar performance. An application to traffic accident data is presented. en_US dc.description.tableofcontents 摘要 . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . .. . . . iiContents . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . .. . . vList of Tables . . . . . . . . . . . . . . .. . vi1 Introduction . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . .32.1 Poisson Process . . . . . . . . . . . . . . .32.2 Sarmanov Distribution . . . . . . . . . . . .63 Statistical Model and Estimation . . . . . . .113.1 Notation and Model . . . . . . . . . . . . .113.2 Estimation and Asymptotic Properties . . . .153.2.1 Maximum Likelihood Estimation . . . . . . 153.2.2 Method of Inference Functions for Margins 174 Simulations and Data Analysis . . . . . . . . 204.1 Simulation Studies . . . . . . . . . . . . .204.1.1 Data Generation . . . . . . . . . . . . . 204.1.2 Maximum Likelihood Estimation . . . . . . 224.1.3 IFM Estimation . . . . . . . . . . . . . .254.2 Real Data Analysis . . . . . . . . . . . . .285 Conclusion . . . . . . . . . . . . . . . . . .33References . . . . . . . . . . . . . . . . . . .35 zh_TW dc.format.extent 559514 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109354017 en_US dc.subject (關鍵詞) 非齊次Poisson過程 zh_TW dc.subject (關鍵詞) 縱向計數資料 zh_TW dc.subject (關鍵詞) 比例均值迴歸模型 zh_TW dc.subject (關鍵詞) 隨機效應 zh_TW dc.subject (關鍵詞) Sarmanov分佈 zh_TW dc.subject (關鍵詞) Nonhomogeneous Poisson process en_US dc.subject (關鍵詞) Panel count data en_US dc.subject (關鍵詞) Proportional mean regression model en_US dc.subject (關鍵詞) Random effect en_US dc.subject (關鍵詞) Sarmanov distribution en_US dc.title (題名) 基於Sarmanov分佈的二元縱向計數資料模型 zh_TW dc.title (題名) A Sarmanov Distribution Based Model for Bivariate Panel Count Data en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) Abdallah, A., Boucher, J.-P., and Cossette, H. (2016). Sarmanov family of multivariate distributions for bivariate dynamic claim counts model. Insurance: Mathematics and Economics, 68:120–133.Bahraoui, Z., Bolancé, C., Pelican, E., and Vernic, R. (2015a). On the bivariate sarmanov distribution and copula. an application on insurance data using truncated marginal dis- tributi. SORT, 39(2):209–230.Bahraoui, Z., Bolancé, C., Pelican, E., and Vernic, R. (2015b). On the bivariate sarmanov distribution and copula. an application on insurance data using truncated marginal dis- tributions. Statistics and Operations Research Transactions, SORT, 39(2):209–230.Bairamov, I., Altinsoy, B., and Kerns, G. J. (2011). On generalized sarmanov bivariate distributions. TWMS Journal of Applied and Engineering Mathematics, 1(1):86–97.Bairamov, I., Kotz, S., and Gebizlioglu, O. L. (2001). The sarmanov family and its gen- eralization: theory and methods. South African Statistical Journal, 35(2):205–224.Bolancé, C. and Vernic, R. (2019). Multivariate count data generalized linear models: Three approaches based on the sarmanov distribution. Insurance: Mathematics and Economics, 85:89–103.Dias, A., Embrechts, P., et al. (2004). Dynamic copula models for multivariate high- frequency data in finance. Manuscript, ETH Zurich, 81.Hashorva, E. and Ratovomirija, G. (2015). On sarmanov mixed erlang risks in insurance applications. ASTIN Bulletin: The Journal of the IAA, 45(1):175–205.Huang, C.-Y., Wang, M.-C., and Zhang, Y. (2006). Analysing panel count data with in- formative observation times. Biometrika, 93(4):763–775.Joe, H. (1997). Multivariate models and multivariate dependence concepts. CRC press. Kotz, S., Balakrishnan, N., and Johnson, N. L. (2004). Continuous multivariate distribu-tions, Volume 1: Models and applications, volume 1. John Wiley & Sons.Nelsen, R. B. (2006). An introduction to copulas. Springer Science & Business Media.Ross, S. M., Kelly, J. J., Sullivan, R. J., Perry, W. J., Mercer, D., Davis, R. M., Washburn, T. D., Sager, E. V., Boyce, J. B., and Bristow, V. L. (1996). Stochastic processes, volume 2. Wiley New York.Sarmanov, O. V. (1966). Generalized normal correlation and two-dimensional fréchet classes. In Doklady Akademii Nauk, volume 168, pages 32–35. Russian Academy of Sciences.Shi, P. and Zhao, Z. (2020). Regression for copula-linked compound distributions with applications in modeling aggregate insurance claims. The Annals of Applied Statistics, 14(1):357–380.Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. Publ. inst. statist. univ. Paris, 8:229–231.Sun, J., Zhao, X., et al. (2013). Statistical analysis of panel count data. Springer.Ting Lee, M.-L. (1996). Properties and applications of the sarmanov family of bivariate distributions. Communications in Statistics-Theory and Methods, 25(6):1207–1222.Wellner, J. A. and Zhang, Y. (2000). Two estimators of the mean of a counting process with panel count data. The Annals of statistics, 28(3):779–814.Wellner, J. A. and Zhang, Y. (2007). Two likelihood-based semiparametric estimation methods for panel count data with covariates. The Annals of Statistics, 35(5):2106– 2142. zh_TW dc.identifier.doi (DOI) 10.6814/NCCU202200735 en_US
