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題名 小區域生命表編製與死亡率模型估計
A Study of Life Table Construction and Mortality Model for Small Areas作者 謝靖惟
Hsieh, Ching-Wei貢獻者 余清祥
Yue, Ching-Syang
謝靖惟
Hsieh, Ching-Wei關鍵詞 死亡率模型
小區域估計
生命表
修勻
電腦模擬
Life table
Mortality rate estimation for small area
Population projection
Graduation
Computer simulation日期 2022 上傳時間 1-八月-2022 17:31:42 (UTC+8) 摘要 死亡率模型可用於推估人們的未來壽命，有助於政府擬定社福政策及產業發展計畫，以及個人安排退休生活的規劃，然而臺灣各地人口特性差異不小，用於全國的方法未必可直接套用至縣市、鄉鎮市區等小區域。以常見的Lee-Carter死亡率模型（Lee and Carter, 1992）為例，這個模型用於全國層級時相當準確，但當人數較少時參數估計值有明顯偏誤，甚至產生不收斂的現象，必須適度調整以取得較為穩定的估計值。有鑑於此，本文以臺灣縣市、鄉鎮市區層級為研究目標，希冀可修正Lee-Carter模型在人數少時的估計偏誤，並將研究結果應用至估計臺灣人數較少之縣市或鄉鎮市區等行政區域層級，解決死亡人數為零的死亡率震盪問題，以利小區域生命表的編製。Lee-Carter模型的參數估計偏誤多半導因於死亡觀察人數為零，通常發生在死亡率較低的年齡層（如5～19歲），傳統修勻方法未必適用，本文引進修勻方法穩定小區域死亡率，藉由人口數較多的參考地區，降低死亡模型的估計偏誤。本文以臺灣鄉鎮市區的人口資料為依據，透過電腦模擬評估修勻方法的優劣，考慮方法包含部分標準死亡比（Partial Standard Mortality Ratio）、Whittaker比值法等方法，藉此調整死亡人數為零（或偏低）的現象。研究發現修勻方法確實可以降低死亡模型的估計偏差，人口數越少時扮演角色越重，以20～49歲年齡層的參數改善最為顯著。此外，參考地區與小區域是否類似、參考地區的人口數都與降低偏誤有關，且兩者彼此會相互影響死亡率估計的準確度。
Mortality models can be used to predict future life expectancy and help governments to design welfare policies and national development plans, as well as providing information for planning individuals’ retirement lives. However, the population characteristics of local areas, such as counties and townships, are very different and the methods used in the national level may not be applicable to small areas. Taking Lee-Carter model as example, it may not be feasible to small populations since its parameters’ estimates are likely to be under-biased. Thus, we aim to explore the possible modifications for reducing the bias of estimates, if the Lee-Carter model is applied to small populations.The bias of parameters’ estimates is likely caused zero number of deaths, which usually occurs in age groups with low mortality rates (e.g., ages 5-19), and the traditional smoothing method may not be applicable. In this paper, we consider a larger population as the reference population, together with graduation methods, such as Partial Standard Mortality Ratio (PSMR) and Whittaker method, to reduce the estimation bias of the mortality model. Based on Taiwan’s data at county and township level to evaluate the proposed approach. We found the proposed approach can reduce the bias of estimates, especially for the cases of smaller populations and age groups 20-49. In addition, both the similarity between the reference population and small population and the size of reference population have a impact on the accuracy of mortality estimation.參考文獻 中文文獻：王信忠、余清祥、王子瑜（2017）。「臺灣原住民死亡率暨生命表編撰研究」。《人口學刊》，55，99-131。王信忠、金碩、余清祥（2012）。「小區域死亡率推估之研究」。《人口學刊》，45，121-154。余清祥（1997）。《修勻：統計在保險的應用》。臺北：雙葉書廊。余清祥、王信忠、陳譽騰（2021）。「年輪變動比用於小區域人口推估的探討」。《人口學刊》，63，99-133。余清祥、梁穎誼、林沛柔（印刷中）。「健康、醫療利用與人口移動的關聯」。《地理學報》。林正祥、張怡陵（2020）。「影響平均餘命增長之生命表特性及其相關死亡率模式分析」。《臺灣公共衛生雜誌》，39-1，74-89。林正祥、劉士嘉、劉于琪（2014）。「臺灣事故傷害對潛在生命年數、工作年數及社會經濟損失影響探討」。《人口學刊》，48，141-171。林志軒（2014）。小區域死亡率模型的探討。國立政治大學商學院統計學系碩士論文。凃明蕙（2020）。臺灣居民健康與壽命之空間分析。國立政治大學商學院統計學系碩士論文。陳芝嘉、余清祥、蔡偉德（2015）。「921震災對中老年人死亡風險的影響」。《人口學刊》，50，61-99。陳政勳、余清祥（2010）。「小區域人口推估研究：臺北市、雲嘉兩縣、澎湖縣的實證分析」。《人口學刊》，41，153-183。曾奕翔（2002）。「台灣地區死亡率推估的實證方法之研究與相關年金問題之探討」。國立政治大學商學院風險管理與保險學系碩士班碩士論文。劉士嘉、林正祥（2017）。「人類壽命上限值探討——以臺灣為例」。《人口學刊》，55，133-163。英文文獻：Alexander, M., Zagheni, E., and Barbieri, M. (2017). A Flexible Bayesian Model for Estimating Subnational Mortality. Demography, Volume 54, 2025-2041.Arias, E., Escobedo, L. A., Kennedy, J., Fu, C., and Cisewki, J. (2018). U.S. Small-area Life Expectancy Estimates Project: Methodology and Results Summary. Vital and Health Statistics. Volume 181, 1-40.Bacaër, N. (2011). A Short History of Mathematical Population Dynamics. London, England: Springer.Bravo, J. M. and Malta, J. (2010). Estimating Life Expectancy in Small Population Areas. Joint Eurostat / UNECE Work Session on Demographic Projections. https://unece.org/fileadmin/DAM/stats/documents/ece/ces/ge.11/2010/wp.10.e.pdfBrouhns, N., M. Denuit, and J. K. Vermunt (2002). A Poisson Log-Bilinear Regression Approach to the Construction of Projected Lifetables. Insurance: Mathematics and Economics, Volume 31, Issue 3, 373-393.Coulomb, J.-B., Salhi, Y., and Thérond, P.-E. (2020). Credibility Adjustment of the Lee-Carter Longevity Model for Multiple Populations. HAL Open Science. https://hal.archives-ouvertes.fr/hal-02557208/file/CredibilityLeeCarter.pdfCopas, J. B. and Haberman, S. (1983). Non-parametric Graduation Using Kernel Methods. Journal of the Institute of Actuaries, Volume 110, Issue 1, 135-156.Forfar, D., McCutcheon, J., and Wilkie, D. (1988). On Graduation by Mathematical Formula. Journal of the Institute of Actuaries. Volume 115, Issue 1, 1-149.Gompertz, B. (1825). On the Nature of the Nature of the Function Expressive of the Law on a New Model of Determining Life Contingencies. Phil. Trans. R. Soc. Volume 115, 513-585.Greville, T.N.E. (1941). Short Methods of Constructing Abridged Life Tables. The American Institute of Actuaries, Part 1. Volume 32, 29, 34-40.Haberman, S. and Renshaw, A. (1996). Generalized Linear Models and Actuarial Science. The Statistician, Volume 45, 407-436.Hamilton, C. H. and J. Perry (1962). A Short Method for Projecting Population by Age from One Decennial Census to Another. Social Forces. Volume 41, Issue 2, 163-170.Henderson, R. (1924). A New Method of Graduation. Transactions of the Actuarial Society of America, Volume 25, 29-40.Henderson, R. (1925). Further Remarks on Graduation. Transactions of the Actuarial Society of America, Volume 26, 52-57.Henderson, R. (1938). Mathematical Theory of Graduation. New York, NY: Actuary Society of America.Kimeldorf, G. S. and Jones, D. A. (1967). Bayesian Graduation. Transactions of the Society of Actuaries, Volume 19, 66-112.Lawson, C. L., and Hanson, R. J. (1974). Solving Least Squares Problems. New York, NY: Prentice-Hall.Lee, R. D. and Carter, L. R. (1992). Modeling and Forecasting US Mortality, Journal of the American Statistical Association, Volume 87, Issue 419, 659-671.Lee, W. C. (2003). A Partial SMR Approach to Smoothing Age-specific Rates, Annals of Epidemiology, Volume 13, Issue 2, 89-99.Lewis, C. D. (1982). Industrial and Business Forecasting Methods: A Practical Guide to Exponential Smoothing and Curve Fitting. London, England: Butterworth Scientific.Li, N. and Lee, R. (2005). Coherent Mortality Forecasts for a Group of Populations – An Extension of the Lee-Carter Method. Demography. Volume 42, Issue 3, 575-594.Macaulay, F. R. (1931). The Smoothing of Time Series. New York, NY: National Bureau of Economic Research.Mantel, N. and Myers, M. (1971). Maximum Likelihood Iterative Procedures in Multiparameter Situations. Volume 66, Issue 335, 484-491.Massimiliano, M., Maria, F. M., and Manuela, S. (2019). Mortality Projections for Small Populations: An Application to the Maltese Elderly. Risks, Volume 7, Issue 2, 35-59.Makeham, W. M. (1860). On the Law of Mortality and the Construction of Annuity Tables. The Assurance Magazine, and Journal of the Institute of Actuaries, Volume 8, Issue 6, 301-310.Ramlau-Hansen, H. (1983). Smoothing Counting Process Intensities by Means of Kernel Functions. The Annals of Statistics, Volume 11, Issue 2, 453-466.Renshaw, A. and Haberman, S. (2006). A Cohort-Based Extension to the Lee-Carter Model for Mortality Reduction Factors. Insurance: Mathematics and Economics, Volume 3, Issue 3, 556-570.Stephens, A. S., Purdie, S., Yang, B., and Moore, H. (2013). Life Expectancy Estimation in Small Administrative Areas with Non-uniform Population Sizes: Application to Australian New South Wales Local Government Areas. British Medical Journal Open, Volume 3.Su, K. C. and Yue, J. C. (2019). A Synthesis Mortality Model for the Elderly. North American Actuarial Journal, 1-25.Tsai, S., and Wen, C. (1989). Mortality Trends in a Rapidly Developing Economy in Taiwan: Part I: Comparison with the USA and Japan 1976-1983. Asia Pacific Journal of Public Health, Volume 3, Issue 1, 41-50.Wang, H.-C., Yue, C. J. (2015). Mortality, Health and Marriage: A Study Based on Taiwan’s Population Data. North American Actuarial Journal, 1-13.Wang, H.-C., Yue, C. J., and Chong, C.-T. (2018). Mortality Models and Longevity Risk for Small Populations. Insurance: Mathematics and Economics, Volume 78, 351-359.Wang, H.-C., Yue, C. J., and Wang, T.-Y. (2019). Do Domestic Immigrants Live Longer? An Approach for Estimating the Life Expectancy of Small Populations. Migration Letters, Volume 16, Issue 3, 399-416.Whittaker, E. T. (1922). On a New Method of Graduation. Proceedings of the Edinburgh Mathematical Society, Volume 41, 63-75.Wu, J.-C and Chiang, T.-L. (2007).Comparing Child Mortality in Taiwan and Selected Industrialized Countries. Journal of the Formosan Medical Association, Volume 106, Issue 2, 177-180.Yang, S. S, Yue, J. C., and Huang, H.-C. (2010). Modeling Longevity Risks Using a Principal Component Approach: A Comparison with Existing Stochastic Mortality Models. Insurance: Mathematics and Economics, Volume 46, Issue 1, 254-270.Yue, J. C. (2002). Oldest-Old Mortality Rates and the Gompertz Law: A Theoretical and Empirical Study Based on Four Countries. Journal of Population Studies, Volume 24, 33-58.Yue, J. C., and Huang, H.-C. (2011). A Study of Incidence Experience for Taiwan Life Insurance. The Geneva Papers on Risk and Insurance. Issues and Practice, Volume 36, Issue 4, 718-733.Yue, J. C., Wang, H.-C. and Wang, T.-Y. (2019). Using Graduation to Modify the Estimation of Lee-Carter Model for Small Populations. North American Actuarial Journal, 1-11. 描述 碩士
國立政治大學
風險管理與保險學系
109358010資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109358010 資料類型 thesis dc.contributor.advisor 余清祥 zh_TW dc.contributor.advisor Yue, Ching-Syang en_US dc.contributor.author (作者) 謝靖惟 zh_TW dc.contributor.author (作者) Hsieh, Ching-Wei en_US dc.creator (作者) 謝靖惟 zh_TW dc.creator (作者) Hsieh, Ching-Wei en_US dc.date (日期) 2022 en_US dc.date.accessioned 1-八月-2022 17:31:42 (UTC+8) - dc.date.available 1-八月-2022 17:31:42 (UTC+8) - dc.date.issued (上傳時間) 1-八月-2022 17:31:42 (UTC+8) - dc.identifier (其他 識別碼) G0109358010 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/141074 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 風險管理與保險學系 zh_TW dc.description (描述) 109358010 zh_TW dc.description.abstract (摘要) 死亡率模型可用於推估人們的未來壽命，有助於政府擬定社福政策及產業發展計畫，以及個人安排退休生活的規劃，然而臺灣各地人口特性差異不小，用於全國的方法未必可直接套用至縣市、鄉鎮市區等小區域。以常見的Lee-Carter死亡率模型（Lee and Carter, 1992）為例，這個模型用於全國層級時相當準確，但當人數較少時參數估計值有明顯偏誤，甚至產生不收斂的現象，必須適度調整以取得較為穩定的估計值。有鑑於此，本文以臺灣縣市、鄉鎮市區層級為研究目標，希冀可修正Lee-Carter模型在人數少時的估計偏誤，並將研究結果應用至估計臺灣人數較少之縣市或鄉鎮市區等行政區域層級，解決死亡人數為零的死亡率震盪問題，以利小區域生命表的編製。Lee-Carter模型的參數估計偏誤多半導因於死亡觀察人數為零，通常發生在死亡率較低的年齡層（如5～19歲），傳統修勻方法未必適用，本文引進修勻方法穩定小區域死亡率，藉由人口數較多的參考地區，降低死亡模型的估計偏誤。本文以臺灣鄉鎮市區的人口資料為依據，透過電腦模擬評估修勻方法的優劣，考慮方法包含部分標準死亡比（Partial Standard Mortality Ratio）、Whittaker比值法等方法，藉此調整死亡人數為零（或偏低）的現象。研究發現修勻方法確實可以降低死亡模型的估計偏差，人口數越少時扮演角色越重，以20～49歲年齡層的參數改善最為顯著。此外，參考地區與小區域是否類似、參考地區的人口數都與降低偏誤有關，且兩者彼此會相互影響死亡率估計的準確度。 zh_TW dc.description.abstract (摘要) Mortality models can be used to predict future life expectancy and help governments to design welfare policies and national development plans, as well as providing information for planning individuals’ retirement lives. However, the population characteristics of local areas, such as counties and townships, are very different and the methods used in the national level may not be applicable to small areas. Taking Lee-Carter model as example, it may not be feasible to small populations since its parameters’ estimates are likely to be under-biased. Thus, we aim to explore the possible modifications for reducing the bias of estimates, if the Lee-Carter model is applied to small populations.The bias of parameters’ estimates is likely caused zero number of deaths, which usually occurs in age groups with low mortality rates (e.g., ages 5-19), and the traditional smoothing method may not be applicable. In this paper, we consider a larger population as the reference population, together with graduation methods, such as Partial Standard Mortality Ratio (PSMR) and Whittaker method, to reduce the estimation bias of the mortality model. Based on Taiwan’s data at county and township level to evaluate the proposed approach. We found the proposed approach can reduce the bias of estimates, especially for the cases of smaller populations and age groups 20-49. In addition, both the similarity between the reference population and small population and the size of reference population have a impact on the accuracy of mortality estimation. en_US dc.description.tableofcontents 第一章 緒論 1第一節 研究背景與動機 1第二節 研究目的 5第二章 文獻探討與研究方法 6第一節 文獻回顧 6第二節 生命表 7第三節 修勻方法介紹 10第四節 資料介紹 17第五節 小區域死亡率估計方法 18第三章 電腦模擬 27第一節 納入參考區的小區域死亡率修勻 27第二節 擴大參考樣本對死亡率估計影響 33第三節 不同年齡層死亡率估計效果分析 42第四章 實證分析 45第一節 澎湖縣男性死亡率修勻 45第二節 澎湖縣男性生命表編製 48第五章 結論與建議 57第一節 結論 57第二節 研究限制與建議 59參考文獻 61附錄一 電腦模擬結果 67附錄二 實證分析結果 73 zh_TW dc.format.extent 9478842 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109358010 en_US dc.subject (關鍵詞) 死亡率模型 zh_TW dc.subject (關鍵詞) 小區域估計 zh_TW dc.subject (關鍵詞) 生命表 zh_TW dc.subject (關鍵詞) 修勻 zh_TW dc.subject (關鍵詞) 電腦模擬 zh_TW dc.subject (關鍵詞) Life table en_US dc.subject (關鍵詞) Mortality rate estimation for small area en_US dc.subject (關鍵詞) Population projection en_US dc.subject (關鍵詞) Graduation en_US dc.subject (關鍵詞) Computer simulation en_US dc.title (題名) 小區域生命表編製與死亡率模型估計 zh_TW dc.title (題名) A Study of Life Table Construction and Mortality Model for Small Areas en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) 中文文獻：王信忠、余清祥、王子瑜（2017）。「臺灣原住民死亡率暨生命表編撰研究」。《人口學刊》，55，99-131。王信忠、金碩、余清祥（2012）。「小區域死亡率推估之研究」。《人口學刊》，45，121-154。余清祥（1997）。《修勻：統計在保險的應用》。臺北：雙葉書廊。余清祥、王信忠、陳譽騰（2021）。「年輪變動比用於小區域人口推估的探討」。《人口學刊》，63，99-133。余清祥、梁穎誼、林沛柔（印刷中）。「健康、醫療利用與人口移動的關聯」。《地理學報》。林正祥、張怡陵（2020）。「影響平均餘命增長之生命表特性及其相關死亡率模式分析」。《臺灣公共衛生雜誌》，39-1，74-89。林正祥、劉士嘉、劉于琪（2014）。「臺灣事故傷害對潛在生命年數、工作年數及社會經濟損失影響探討」。《人口學刊》，48，141-171。林志軒（2014）。小區域死亡率模型的探討。國立政治大學商學院統計學系碩士論文。凃明蕙（2020）。臺灣居民健康與壽命之空間分析。國立政治大學商學院統計學系碩士論文。陳芝嘉、余清祥、蔡偉德（2015）。「921震災對中老年人死亡風險的影響」。《人口學刊》，50，61-99。陳政勳、余清祥（2010）。「小區域人口推估研究：臺北市、雲嘉兩縣、澎湖縣的實證分析」。《人口學刊》，41，153-183。曾奕翔（2002）。「台灣地區死亡率推估的實證方法之研究與相關年金問題之探討」。國立政治大學商學院風險管理與保險學系碩士班碩士論文。劉士嘉、林正祥（2017）。「人類壽命上限值探討——以臺灣為例」。《人口學刊》，55，133-163。英文文獻：Alexander, M., Zagheni, E., and Barbieri, M. (2017). A Flexible Bayesian Model for Estimating Subnational Mortality. Demography, Volume 54, 2025-2041.Arias, E., Escobedo, L. A., Kennedy, J., Fu, C., and Cisewki, J. (2018). U.S. Small-area Life Expectancy Estimates Project: Methodology and Results Summary. Vital and Health Statistics. Volume 181, 1-40.Bacaër, N. (2011). A Short History of Mathematical Population Dynamics. London, England: Springer.Bravo, J. M. and Malta, J. (2010). Estimating Life Expectancy in Small Population Areas. Joint Eurostat / UNECE Work Session on Demographic Projections. https://unece.org/fileadmin/DAM/stats/documents/ece/ces/ge.11/2010/wp.10.e.pdfBrouhns, N., M. Denuit, and J. K. Vermunt (2002). A Poisson Log-Bilinear Regression Approach to the Construction of Projected Lifetables. Insurance: Mathematics and Economics, Volume 31, Issue 3, 373-393.Coulomb, J.-B., Salhi, Y., and Thérond, P.-E. (2020). Credibility Adjustment of the Lee-Carter Longevity Model for Multiple Populations. HAL Open Science. https://hal.archives-ouvertes.fr/hal-02557208/file/CredibilityLeeCarter.pdfCopas, J. B. and Haberman, S. (1983). Non-parametric Graduation Using Kernel Methods. Journal of the Institute of Actuaries, Volume 110, Issue 1, 135-156.Forfar, D., McCutcheon, J., and Wilkie, D. (1988). On Graduation by Mathematical Formula. Journal of the Institute of Actuaries. Volume 115, Issue 1, 1-149.Gompertz, B. (1825). On the Nature of the Nature of the Function Expressive of the Law on a New Model of Determining Life Contingencies. Phil. Trans. R. Soc. Volume 115, 513-585.Greville, T.N.E. (1941). Short Methods of Constructing Abridged Life Tables. The American Institute of Actuaries, Part 1. Volume 32, 29, 34-40.Haberman, S. and Renshaw, A. (1996). Generalized Linear Models and Actuarial Science. The Statistician, Volume 45, 407-436.Hamilton, C. H. and J. Perry (1962). A Short Method for Projecting Population by Age from One Decennial Census to Another. Social Forces. Volume 41, Issue 2, 163-170.Henderson, R. (1924). A New Method of Graduation. Transactions of the Actuarial Society of America, Volume 25, 29-40.Henderson, R. (1925). Further Remarks on Graduation. Transactions of the Actuarial Society of America, Volume 26, 52-57.Henderson, R. (1938). Mathematical Theory of Graduation. New York, NY: Actuary Society of America.Kimeldorf, G. S. and Jones, D. A. (1967). Bayesian Graduation. Transactions of the Society of Actuaries, Volume 19, 66-112.Lawson, C. L., and Hanson, R. J. (1974). Solving Least Squares Problems. New York, NY: Prentice-Hall.Lee, R. D. and Carter, L. R. (1992). Modeling and Forecasting US Mortality, Journal of the American Statistical Association, Volume 87, Issue 419, 659-671.Lee, W. C. (2003). A Partial SMR Approach to Smoothing Age-specific Rates, Annals of Epidemiology, Volume 13, Issue 2, 89-99.Lewis, C. D. (1982). Industrial and Business Forecasting Methods: A Practical Guide to Exponential Smoothing and Curve Fitting. London, England: Butterworth Scientific.Li, N. and Lee, R. (2005). Coherent Mortality Forecasts for a Group of Populations – An Extension of the Lee-Carter Method. Demography. Volume 42, Issue 3, 575-594.Macaulay, F. R. (1931). The Smoothing of Time Series. New York, NY: National Bureau of Economic Research.Mantel, N. and Myers, M. (1971). Maximum Likelihood Iterative Procedures in Multiparameter Situations. Volume 66, Issue 335, 484-491.Massimiliano, M., Maria, F. M., and Manuela, S. (2019). Mortality Projections for Small Populations: An Application to the Maltese Elderly. Risks, Volume 7, Issue 2, 35-59.Makeham, W. M. (1860). On the Law of Mortality and the Construction of Annuity Tables. The Assurance Magazine, and Journal of the Institute of Actuaries, Volume 8, Issue 6, 301-310.Ramlau-Hansen, H. (1983). Smoothing Counting Process Intensities by Means of Kernel Functions. The Annals of Statistics, Volume 11, Issue 2, 453-466.Renshaw, A. and Haberman, S. (2006). A Cohort-Based Extension to the Lee-Carter Model for Mortality Reduction Factors. Insurance: Mathematics and Economics, Volume 3, Issue 3, 556-570.Stephens, A. S., Purdie, S., Yang, B., and Moore, H. (2013). Life Expectancy Estimation in Small Administrative Areas with Non-uniform Population Sizes: Application to Australian New South Wales Local Government Areas. British Medical Journal Open, Volume 3.Su, K. C. and Yue, J. C. (2019). A Synthesis Mortality Model for the Elderly. North American Actuarial Journal, 1-25.Tsai, S., and Wen, C. (1989). Mortality Trends in a Rapidly Developing Economy in Taiwan: Part I: Comparison with the USA and Japan 1976-1983. Asia Pacific Journal of Public Health, Volume 3, Issue 1, 41-50.Wang, H.-C., Yue, C. J. (2015). Mortality, Health and Marriage: A Study Based on Taiwan’s Population Data. North American Actuarial Journal, 1-13.Wang, H.-C., Yue, C. J., and Chong, C.-T. (2018). Mortality Models and Longevity Risk for Small Populations. Insurance: Mathematics and Economics, Volume 78, 351-359.Wang, H.-C., Yue, C. J., and Wang, T.-Y. (2019). Do Domestic Immigrants Live Longer? An Approach for Estimating the Life Expectancy of Small Populations. Migration Letters, Volume 16, Issue 3, 399-416.Whittaker, E. T. (1922). On a New Method of Graduation. Proceedings of the Edinburgh Mathematical Society, Volume 41, 63-75.Wu, J.-C and Chiang, T.-L. (2007).Comparing Child Mortality in Taiwan and Selected Industrialized Countries. Journal of the Formosan Medical Association, Volume 106, Issue 2, 177-180.Yang, S. S, Yue, J. C., and Huang, H.-C. (2010). Modeling Longevity Risks Using a Principal Component Approach: A Comparison with Existing Stochastic Mortality Models. Insurance: Mathematics and Economics, Volume 46, Issue 1, 254-270.Yue, J. C. (2002). Oldest-Old Mortality Rates and the Gompertz Law: A Theoretical and Empirical Study Based on Four Countries. Journal of Population Studies, Volume 24, 33-58.Yue, J. C., and Huang, H.-C. (2011). A Study of Incidence Experience for Taiwan Life Insurance. The Geneva Papers on Risk and Insurance. Issues and Practice, Volume 36, Issue 4, 718-733.Yue, J. C., Wang, H.-C. and Wang, T.-Y. (2019). Using Graduation to Modify the Estimation of Lee-Carter Model for Small Populations. North American Actuarial Journal, 1-11. zh_TW dc.identifier.doi (DOI) 10.6814/NCCU202200942 en_US
