The Study And Application Of Functional Principal Component Analysis For Sparse Data To Forecast Mortality Rates

In recent years,with the development of the economy and the improvement of the social medical care system,the mortality rates in the overall world is continuously decreasing,and the aging population phenomenon is to be serious day by day.As a result,the world's state pension system,welfare policy,enterprise annuity and the life insurance business of the insurance company are affected adversely by the unexpected change.Therefore,to improve the estimation accuracy of the mortality rates is of great theoretical and realistic significance.Considering the fact that the mortality rates are the smooth functions of age,Hyndman and Ullah propose functional data analysis to fit and predict the mortality rates.The methodology reveals more functional information about the mortality rates and provides a more accurate estimation of future mortality rates.And the model is a generalization of the Lee-carter model commonly used in mortality forecasting.But the model requires that the data should be complete,dense,and regular,eliminating the fact that the data in some countries are missing and sparse.Moreover,empirical analysis indicates that imitative effect is bad when the number of years when the mortality rates are accessible is short.Traditional functional data analysis implies that the data should be dense and regular,so when the data sparse and irregular,the method will be not available.The functional principal components analysis based on the conditional expectation for the case of sparse longitudinal data only requires that the pooled data are sufficiently dense,allowing for the sparse and irregular design for each subject.Empirical studies show that besides the general application to functional principal component analysis for sparse and irregular data,an application for the proposed PACE method to impute missing data in longitudinal studies is also feasible.Consider a regular design where for some subjects many data are missing.The PACE method can then be used to impute the missing data from predicted curve.Also,there is an interesting finding that the PACE method improves traditionalFPCA analysis even under dense and regular designs.Note that the mortality rates in some countries are missing and regular,and the periods which the rates are accessible is short,the paper introduces the PACE method to the analysis of the mortality rates.And comparisons of mean functions,eigenfunctions,functional principal component scores,goodness of fit and prediction accuracy between HU model and the proposed PACE model is made.Based on the numerical analysis of some developed countries'data,it is found that mean functions,eigenfunctions and functional principal component scores are similar,but the PACE model can use fewer principal components to extract more information compared to HU model.Also,if there are many missing values and the data are sparse and irregular,such as the data of the ages between 90 and 110,the PACE model improves the goodness of fit than the HU model,even in the case that the mortality data are complete and dense,PACE model fits better than the HU model.Then data of all ages of American mortality rates are deleted randomly to simulate sparsity,the analysis based on these data indicates that with the increasing sparsity of the data,the superiority of PACE model becomes more notable.It's known that mortality rates of China is lacking,and there are many missing values in the available data,such as on the ages of 0-20,so in this paper,PACE model is proposed to fit and analysis the China mortality data.The model provides a new idea for Chinese mortality prediction.And the results show that when fitting and predicting the mortality rate of women with more missing values,the accuracy of PACE model is higher than that of the HU model.