published_or_final_version / abstract / toc / Statistics and Actuarial Science / Master / Master of Philosophy

Pang Suet-Yee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 85-87). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview --- p.1 / Chapter 1.1.1 --- Survival Analysis --- p.2 / Chapter 1.1.2 --- Tree、-structured Statistical Method --- p.2 / Chapter 1.1.3 --- Mortality Study --- p.3 / Chapter 1.2 --- Motivation --- p.3 / Chapter 1.3 --- Background Information --- p.4 / Chapter 1.4 --- Data Content --- p.7 / Chapter 1.5 --- Thesis Outline --- p.8 / Chapter 2 --- Imputation and File Splitting --- p.10 / Chapter 2.1 --- Imputation of Missing Values --- p.10 / Chapter 2.1.1 --- Purpose of Imputation --- p.10 / Chapter 2.1.2 --- Procedure of Hot Deck Imputation --- p.11 / Chapter 2.1.3 --- List of Variables for Imputation --- p.12 / Chapter 2.2 --- File Splitting --- p.14 / Chapter 2.2.1 --- Splitting by Gender --- p.14 / Chapter 2.3 --- Splitting for Validation Check --- p.1G / Chapter 3 --- Cox Hazard Model --- p.17 / Chapter 3.1 --- Basic Idea --- p.17 / Chapter 3.1.1 --- Survival Analysis --- p.17 / Chapter 3.1.2 --- Survivor Function --- p.18 / Chapter 3.1.3 --- Hazard Function --- p.18 / Chapter 3.2 --- The Cox Proportional Hazards Model --- p.19 / Chapter 3.2.1 --- Kaplan-Meier Estimate and Log-Rank Test --- p.20 / Chapter 3.2.2 --- Hazard Ratio --- p.23 / Chapter 3.2.3 --- Partial Likelihood --- p.24 / Chapter 3.3 --- Extension of the Cox Proportional Hazards Model for Time-dependent Variables --- p.25 / Chapter 3.3.1 --- Modification of the Cox's Model --- p.25 / Chapter 3.4 --- Results of Model Fitting --- p.26 / Chapter 3.4.1 --- Extract the Significant Covariates from the Models --- p.31 / Chapter 3.5 --- Model Interpretation --- p.32 / Chapter 4 --- CART --- p.37 / Chapter 4.1 --- CART Procedure --- p.38 / Chapter 4.2 --- Selection of the Splits --- p.39 / Chapter 4.2.1 --- Goodness of Split --- p.39 / Chapter 4.2.2 --- Type of Variables --- p.40 / Chapter 4.2.3 --- Estimation --- p.40 / Chapter 4.3 --- Pruning the Tree --- p.41 / Chapter 4.3.1 --- Misclassification Cost --- p.42 / Chapter 4.3.2 --- Class Assignment Rule --- p.44 / Chapter 4.3.3 --- Minimal Cost Complexity Pruning --- p.44 / Chapter 4.4 --- Cross Validation --- p.47 / Chapter 4.4.1 --- V-fold Cross-validation --- p.47 / Chapter 4.4.2 --- Selecting the right sized tree --- p.49 / Chapter 4.5 --- Missing Value --- p.49 / Chapter 4.6 --- Results of CART program --- p.51 / Chapter 4.7 --- Model Interpretation --- p.53 / Chapter 5 --- Model Prediction --- p.58 / Chapter 5.1 --- Application to Test Sample --- p.58 / Chapter 5.1.1 --- Fitting test sample to Cox's Model --- p.59 / Chapter 5.1.2 --- Fitting test sample to CART model --- p.61 / Chapter 5.2 --- Comparison of Model Prediction --- p.62 / Chapter 5.2.1 --- Misclassification Rate --- p.62 / Chapter 5.2.2 --- Misclassification Rate of Cox's model --- p.63 / Chapter 5.2.3 --- Misclassification Rate of CART model --- p.64 / Chapter 5.2.4 --- Prediction Result --- p.64 / Chapter 6 --- Conclusion --- p.67 / Chapter 6.1 --- Comparison of Results --- p.67 / Chapter 6.2 --- Comparison of the Two Statistical Techniques --- p.68 / Chapter 6.3 --- Limitation --- p.70 / Appendix A: Coding Description for the Health Factors --- p.72 / Appendix B: Log-rank Test --- p.75 / Appendix C: Longitudinal Plot of Time Dependent Variables --- p.76 / Appendix D: Hypothesis Testing of Suspected Covariates --- p.78 / Appendix E: Terminal node report for both gender --- p.81 / Appendix F: Calculation of Critical Values --- p.83 / Appendix G: Distribution of Missing Value in Learning sample and Test Sample --- p.84 / Bibliography --- p.85

This thesis analyses the COVID-19 pandemic’s effects on Swedish mortality during 2020 by investigating whether it has resulted in excess mortality. This is done using a stochastic mortality projection model from the Lee-Carter framework and by assuming the number of deaths follows a Poisson distribution. Due to the few confirmed COVID-19 deaths at younger ages, the decision is made to only include 50-to-100-year-olds in the analysis. Models in the Lee-Carter framework are fitted on historical data from 1993–2019 collected from Human Mortality Database and Statistiska Centralbyrån. After evaluating the models, inter alia using residual analysis and backtesting, we ascertain that the classical Lee-Carter model accomplishes a wanted level of fit and forecast accuracy. During the morality projection with the Lee-Carter model, three different sources of uncertainty are accounted for by constructing prediction intervals using bootstrap. The results show that the large age group 67–94-year-olds have suffered from statistically significant excess mortality during 2020. The level of excess mortality differs between ages, with the ages 70–90-year-olds having the highest number of excess deaths. Comparing the number of confirmed COVID-19 deaths to our forecasted number of excess deaths indicates the COVID-19 virus likely caused the surge in deaths.

The performance of three methods of mortality modelling and forecasting are compared. These include the basic Lee–Carter and two functional demographic models; the basic Hyndman–Ullah and the weighted Hyndman–Ullah. Using age-specific data from the Human Mortality Database of two developed countries, France and the UK (England&Wales), these methods are compared; through within-sample forecasting for the years 1999-2018. The weighted Hyndman–Ullah method is adjudged superior among the three methods through a comparison of mean forecast errors and qualitative inspection per the dataset of the selected countries. The weighted HU method is then used to conduct a 32–year ahead forecast to the year 2050.

對於商業保險公司和政府單位而言，死亡率的改善和未來死亡率的預估一直是一大重要議題。特別是對於退休金相關的社會保險、勞退或是商業年金、壽險等等，如何找尋一個準確的預估模式對未來的死亡率改善情況進行預測，並釐訂合理的保費及提列適當的準備金，是對於一個保險制度能否永續經營的重要因素。過去所使用的配適方法，大多僅以單一群體的過去資料輔助未來的預測，例如 Li and Carter (1992)所提出的 Lee-Carter Model，或是 Bell (1997)使用主成分分析法 (Principal Component Analysis, PCA)等僅針對單一群體本身變數進行分析之方式。然而綜觀全球死亡率改善趨勢，可發現國與國間、組與祖間雖有不同，但仍具備共同的趨勢。因此在考慮未來的死亡率配適方面，應加入組與組間的共同因子 (common factors) 進行考量。 Li and Lee (2005)曾提出 Augmented Lee-Carter Model，即對原本的Lee-Carter Model進行修正，加入共同因素項，並且得到更好的預測效果。 本文則採用考慮共同因子之主成分分析原理建構多重群體死亡率模型，即透過主成分分析法，同時考慮不同群體間的死亡率，並以台灣男性和女性1970年至2010年的死亡率資料，做為兩個子群體進行分析。本文使用之主成分分析法模式，和 Lee-Carter Model (Li and Carter, 1992) 和 Augmented Lee-Carter Model (Li and Lee, 2005)，以MAPE法對個別的預測能力進行分析，並得出採用PCA的模式，在預測男性短年期(5年)內的預估能力屬精確(MAPE 介於10%~20%之間)，然而在長期預估下容易失準，且所有使用的模型，在配適台灣資料時皆發生無法準確預估嬰幼兒期(0~3歲)和老年期(80歲以上)之情形。本文並以所有模型預估之死亡率計算保險公司之準備金與保費提列，並與第五回經驗生命表進行比較。 / For governments and life insurance companies, mortality rates are one of the key factors in determining premiums and reserves. Ignoring or miscalculating mortality rates might have negative influences in pricing. However, most of the mortality models do not consider the common trends between groups. In this article, we try to construct the mortality structure which considering common trends of multi-groups populations with principal component analysis (PCA) method. We choose 9 factors to set up our model and fit with the actual data in Taiwan’s gender mortality. We also compare the Lee-Carter Model (Lee and Carter, 1992) and the augmented Lee-Carter Model (Li and Hardy, 2012) with our common factors PCA model, and we find that the PCA model has the least MAPE than other model in five years forecasting in both genders. After finishing basic analysis, we use the mortality data of Taiwan (1970 to 2010) from human mortality database to construct the life expectancy model. We adopt the same criteria to choose the components we need. We also compare the level premium and reserves by different forecasting mortality rates. All of the models indicate life insurance companies to provide higher reserves and level premium than using the 5th TSO experience mortality rare. We will do following research by using company-specific data to construct unique life expectancy model.