This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractIn this paper, we focus on a Multidimensional Data Analysis approach to the Lee-Carter (LC) model of mortality trends . In particular, we extend the bilinear LC model and specify a new model based on a three-way structure, which incorporates a further component in the decomposition of the log-mortality rates. A multi-way component analysis is performed using the Tucker 3 model. The suggested methodology allows us to obtain combined estimates for the three modes: i) time, ii) agegroups and iii) different populations. From the results obtained by the Tucker 3 decomposition, we can jointly compare, in both a numerical and graphical way, the relationships among all three modes and obtain a time series component as a leading indicator of the mortality trend for a group of populations. Further, we carry out a correlation analysis of the estimated trends in order to assess the reliability of the results of the three-way decomposition. The model's goodness of fit is assessed using an analysis of the residuals. Finally, we discuss how the synthesised mortality index can be used to build concise projected life tables for a group of populations. An application which compares ten European countries is used to illustrate the approach and provide a deeper insight into the model and its implementation.

Several approaches have been developed for forecasting mortality using the stochastic model. In particular, the Lee-Carter model has become widely used and there have been various extensions and modifications proposed to attain a broader interpretation and to capture the main features of the dynamics of the mortality intensity. Hyndman-Ullah show a particular version of the Lee-Carter methodology, the so-called Functional Demographic Model, which is one of the most accurate approaches as regards some mortality data, particularly for longer forecast horizons where the benefit of a damped trend forecast is greater. The paper objective is properly to single out the most suitable model between the basic Lee-Carter and the Functional Demographic Model to the Italian mortality data. A comparative assessment is made and the empirical results are presented using a range of graphical analyses. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Statistics, 2011, Vol. 5, No. 2A, 705-724. This reprint differs from the original in pagination and typographic detail. 1 2 V. D'AMATO, G. PISCOPO AND M. RUSSOLILLOits expected values. These changes clearly affect pricing and reserve allocation for life annuities and represent one of the major threats to a social security system that has been planned on the basis of a more modest life expectancy. The risk is of using mortality tables that do not take these trends into account, thus underestimating the survival probability and determining inappropriate premiums. To face this risk, it is necessary to build projected tables including this trend. Thus, reasonable mortality forecasting techniques have to be used to consistently predict the trends [Brouhns, Denuit and Vermunt (2002)]. In that respect, over the years a number of approaches have been proposed for forecasting mortality using the stochastic model, however, the Lee-Carter model [Lee and Carter (1992)] unquestionably represents a milestone in the literature. This methodology has become widely used and there have been various extensions and modifications proposed to attain a broader interpretation and to capture the main features of the dynamics of the mortality intensity [e.g., Booth, Maindonald and Smith (2002); Haberman and Renshaw (2003, 2008); Hyndman and Ullah (2007); Renshaw and Haberman (2003a, 2003b)].The main statistical tool of LC is least-squares estimation via singular value decomposition of the matrix of the log age-specific observed death rates. In fact, the mortality data (death counts and exposures-to-risk) have to fill a rectangular matrix. Henceforth, we will denote with m x,t the observed death rates at age x during calendar year t, obtained by the ratio between the number of deaths, D x,t , recorded at age x during year t, from an exposure-to-risk E x,t , that is, the number of person years from which D x,t occurred. As regards the Italian population data set on the basis of the death rates, classified by gender and individual ...

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The management of National Social Security Systems is being challenged more and more by the rapid ageing of the population, especially in the industrialized countries. In order to chase the Pension System sustainability, several countries in Europe are setting up pension reforms linking the retirement age and/or benefits to life expectancy. In this context, the accurate modelling and projection of mortality rates and life expectancy play a central role and represent issues of great interest in recent literature. Our study refers to the Italian mortality experience and considers an indexing mechanism based on the expected residual life to adjust the retirement age and keep costs at an expected budgeted level, in the spirit of sharing the longevity risk between Social Security Systems and retirees. In order to combine fitting and projections performances of selected stochastic mortality models, a model assembling technique is applied to face uncertainty in model selection, while accounting for uncertainty of estimation as well. The resulting proposal is an averaged model that is suitable to discuss about the gender gap in longevity risk and its alleged narrowing over time.

Abstract.Recently the interest in the development of country and longevity risk models (Njienga and Sherris. 2011) has been growing. The investigation of long-run equilibrium relationships could provide valuable information about the factors driving changes in mortality, in particular across ages and across countries. In order to investigate cross-country common longevity trends, tools to quantify, compare and model the strength of dependence become essential. On one hand, it is necessary to take into account either the dependence for adjacent age groups, or the dependence structure across time in a single population setting: a sort of intradependence structure (D'Amato et al. 2012b). On the other hand, the dependence across multiple populations, which we describe as inter-dependence, can be explored for capturing common long run relationships between countries. The objective of our work is to produce longevity projections by taking into account the presence of various forms of cross-sectional and temporal dependencies in the error processes of multiple populations, considering mortality data from different countries. The algorithm that we propose combines model-based predictions in the Lee Carter (LC) framework (1992) with a bootstrap procedure for dependent data, and so both the historical parametric structure and the intra-group error correlation structure are preserved. We modify the model presented by D' Amato et al. (2012b), which applies a sieve bootstrap to the residuals of the LC model and is able to reproduce, in the sampling, the dependence structure of the data under consideration. In the current paper, the algorithm that we build is applied to a pool of populations by using ideas from panel data; we refer to this new algorithm as the Multiple Lee Carter Panel Sieve (MLCPS). We are interested in estimating the relationship between populations of similar socioeconomic conditions. The empirical results show that the MLCPS approach works well in the presence of dependence.