Mortality,longevity and experiments with the Lee–Carter model

The paper reviews the Lee-Carter modelling framework, illustrated with an application, and then extends the framework through the development of a wider class of generalised, parametric, non-linear models. The choice of error distribution is also generalised. These extensions permit the modelling and extrapolation of age-specific cohort effects as well as the more familiar age-specific period effects: the age-period-cohort version of the model is discussed with a worked example. The paper also provides a comparative study of simulation strategies for assessing risk in mortality rate predictions and the associated forecast estimates of life expectancy and annuity values in both period and cohort perspectives.     

Nonparametric estimation of the lower tail dependence λL in bivariate copulas

The lower tail dependence λ_L is a measure that characterizes the tendency of extreme co-movements in the lower tails of a bivariate distribution. It is invariant with respect to strictly increasing transformations of the marginal distribution and is therefore a function of the copula of the bivariate distribution. λ_L plays an important role in modelling aggregate financial risk with copulas. This paper introduces three non-parametric estimators for λ_L. They are weakly consistent under mild regularity conditions on the copula and under the assumption that the number k = k(n) of observations in the lower tail, used for estimation, is asymptotically k ≈ √n. The finite sample properties of the estimators are investigated using a Monte Carlo simulation in special cases. It turns out that these estimators are biased, where amount and sign of the bias depend on the underlying copula, on the sample size n, on k, and on the true value of λ_L.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

Some new partial orderings on Sn and Sn×Sn

A new class of partial orderings on S_n, the set of permutations of {1,…,n}, is given. Each of these partial orderings is shown to be a subordering of a recently described partial ordering on S_n (Block, Chhetry, Fang and Sampson (1990)) which is related to Schriever's (1987) more associated ordering on bivariate distributions. Also given is an extension of three known partial orders on S_n to partial orders on S_n×S_n. These extensions facilitate the study of functions from S_n×S_n into , which preser these partial orderings, thereby, providing a methodology for extending the notion of arrangement increasing functions.     

Adaptive wavelet empirical Bayes estimation of a location or a scale parameter

Assume that in independent two-dimensional random vectors (X₁,θ₁),…,(X_n,θ_n), each θ_i is distributed according to some unknown prior density function g. Also, given θ_i=θ, X_i has the conditional density function q(x−θ), x,θ(−∞,∞) (a location parameter case), or θ⁻¹q(x/θ), x,θ(0,∞) (a scale parameter case). In each pair the first component is observable, but the second is not. After the (n+1)th pair (X_n+1,θ_n+1) is obtained, the objective is to construct an empirical Bayes (EB) estimator of θ. In this paper we derive the EB estimators of θ based on a wavelet approximation with Meyer-type wavelets. We show that these estimators provide adaptation not only in the case when g belongs to the Sobolev space H with an unknown , but also when g is supersmooth.     

基于VaR方法的长寿风险自然对冲模型

随着死亡率的下降与预期寿命的提高,保险公司面对着不容忽视的长寿风险。基于VaR方法探讨了长寿风险管理中的自然对冲策略,然后在对中国男性人口死亡率预测的基础上,给出了保险公司自然对冲长寿风险所需的最优产品结构,并进一步考查了利率、签单年龄、缴费方式等因素对最优产品结构的影响。     

基于均衡关系的中国人口死亡率预测模型

人口死亡率反映了人口的死亡程度,准确预测死亡率是人口科学及人口经济学研究的重点之一,同时也是长寿风险测量的重要数据基础。基于Lee-Carter模型,探索中国大陆与台湾地区死亡率的相关性,通过协整分析考虑两地死亡率的长期均衡关系,创新性地建立基于相关性的向量误差修正模型(VECM),克服传统自回归移动平均模型(ARIMA)使用有限数据进行预测的局限性;均方预测误差作为检验标准,结果表明:基于VECM模型的预测效果比传统的预测效果更佳;基于中国大陆地区和台湾地区的死亡率长期均衡关系,可以为两地联合长寿债券的定价提供重要参考。     

有限数据下Lee-Carter模型在人口死亡率预测中的应用

Lee-Carter模型是当今世界上最流行的死亡率建模与预测模型,传统的Lee-Carter模型在样本量很大时才能得到较好的效果,而中国的死亡率数据量较少,且部分年限的数据缺失,从而难以达到较好的预测效果。本文基于Li等(2004)提出的有限数据死亡率建模方法,同时考虑样本量不足的影响,采用韩猛等(2010)提出的“双随机过程”建模,构建了有限数据下中国人口死亡率的预测模型,并用于对未来死亡率变动趋势和人口寿命的预测,最后将预测结果与保险公司采用的死亡率改善因子以及社会养老保险个人账户中采用的计发月数进行对比分析,给出了若干相关结论和有关死亡率风险管理的建议。     

Incidence-based estimates of life expectancy of the healthy for the UK: coherence between transition probabilities and aggregate life-tables

Summary.  Will the UK's aging population be fit and independent, or suffer from greater chronic ill health? Life expectancy of healthy people represents the expected number of years of healthy well-being that a life-table cohort would experience if age-specific rates of mortality and disability prevailed throughout the cohort's lifetime. Robust estimation of this life expectancy is thus essential for examining whether additional years of life are spent in good health and whether life expectancy is increasing faster than the decline of rates of disability. The paper examines a means of generating estimates of life expectancy for people who are healthy and unhealthy for the UK that are consistent with exogenous population mortality data. The method takes population transition matrices and adjusts these in a statistically coherent way so as to render them consistent with aggregate life-tables.     

Likelihood ratio ordering of order statistics

This paper provides an improvement on the work of Bapat and Kochar (1994, Linear Algebra Appl., 199, 281–291) and strengthens the literature on the likelihood ratio ordering of order statistics. For independent (but possibly nonidentically distributed) absolutely continuous random variables X₁,…,X_n, it is shown under some weak conditions that

X_1:n^lrlrX_n:n,

where ^lr stands for the likelihood ratio ordering and X_k:n represents the kth-order statistic.     

A comparison of models for dynamic life tables. Application to mortality data from the Valencia Region (Spain)

The graduation of mortality data aims to estimate the probabilities of death at age x, q ( x ), by means of an age-dependent function, whose parameters are adjusted from the crude probabilities that are directly obtainable from the data. However, current life tables have a problem, the need for periodic updates due to changes in mortality over short periods of time. The table containing mortality rates for different ages in different years, q ( xt ), is called a dynamic life table, which captures mortality variation over time. This paper proposes a review of the most commonly used dynamic models and compares the results obtained by each of them when applied to mortality data from the Valencia Region (Spain). The result of the comparison leads us to the conclusion that the Lee-Carter method offers the best results for both sexes, while that based on Heligman and Pollard functions provides the best fit for men alone. Our working method is of additional interest as it may be applied to mortality data for a wide range of ages in any geographical location, allowing the most appropriate dynamic life table to be selected for the case at hand.     

On Berry-Esseen theorem for some functions of U-statistics

The rate of convergence in the central limit theorem and in the random central limit theorem for some functions of U-statistics are established. The theorems refer to the asymptotic behaviour of the sequence {g(U_n),n≥1}, where g belongs to the class of all differentiable functions g such that g′εL(δ) and U_n is a U-statistics.     

Lee–Carter mortality forecasting: a parallel generalized linear modelling approach for England and Wales mortality projections

Summary. The paper presents a reinterpretation of the model underpinning the Lee–Carter methodology for forecasting mortality (and other vital) rates. A parallel methodology based on generalized linear modelling is introduced. The use of residual plots is proposed for both methods to aid the assessment of the goodness of fit. The two methods are compared in terms of structure and assumptions. They are then compared through an analysis of the gender- and age-specific mortality rates for England and Wales over the period 1950–1998 and through a consideration of the forecasts generated by the two methods. The paper also compares different approaches to the forecasting of life expectancy and considers the effectiveness of the Coale–Guo method for extrapolating mortality rates to the oldest ages.     

A method for estimating age-specific reference intervals ('normal ranges') based on fractional polynomials and exponential transformation

The age-specific reference interval is an important screening tool in medicine. Put crudely, an individual whose value of a variable of interest lies outside certain extreme centiles may be suspected of abnormality. We propose a parametric method for constructing such intervals. It provides smooth centile curves and explicit formulae for the centile estimates and for standard deviation (SD) scores (age-standardized values). Each parameter of an exponential–normal or modulus–exponential–normal density is modelled as a fractional polynomial function of age. Estimation is by maximum likelihood. These three- and four-parameter models involve transformations of the data towards normality which remove non-normal skewness and/or kurtosis. Fractional polynomials provide more flexible curve shapes than do conventional polynomials. The method easily accommodates binary covariates facilitating, for example, parsimonious modelling of age- and sex-specific centile curves. A method of calculating precision profiles for centile estimates is proposed. Goodness of fit is assessed by using Q–Q -plots and Shapiro–Wilk W -tests of the SD scores, and likelihood ratio tests of the parameters of an enlarged model. Four substantial real data sets are used to illustrate the method. Comparisons are made with the semiparametric LMS method of Cole and Green.     

Predicting integrals of diffusion processes

Consider predicting the integral of a diffusion process Z in a bounded interval A, based on the observations Z(t_1n),…,Z(t_nn), where t_1n,…,t_nn is a dense triangular array of points (the step of discretization tends to zero as n increases) in the bounded interval. The best linear predictor is generally not asymptotically optimal. Instead, we predict using the conditional expectation of the integral of the diffusion process, the optimal predictor in terms of minimizing the mean squared error, given the observed values of the process. We obtain that, conditioning on the observed values, the order of convergence in probability to zero of the mean squared prediction error is O_p(n⁻²). We prove that the standardized conditional prediction error is approximately Gaussian with mean zero and unit variance, even though the underlying diffusion is generally non-Gaussian. Because the optimal predictor is hard to calculate exactly for most diffusions, we present an easily computed approximation that is asymptotically optimal. This approximation is a function of the diffusion coefficient.     

Identification of local clusters for count data: a model-based Moran's I test

We set out I_DR as a loglinear-model-based Moran's I test for Poisson count data that resembles the Moran's I residual test for Gaussian data. We evaluate its type I and type II error probabilities via simulations, and demonstrate its utility via a case study. When population sizes are heterogeneous, I_DR is effective in detecting local clusters by local association terms with an acceptable type I error probability. When used in conjunction with local spatial association terms in loglinear models, I_DR can also indicate the existence of first-order global cluster that can hardly be removed by local spatial association terms. In this situation, I_DR should not be directly applied for local cluster detection. In the case study of St. Louis homicides, we bridge loglinear model methods for parameter estimation to exploratory data analysis, so that a uniform association term can be defined with spatially varied contributions among spatial neighbors. The method makes use of exploratory tools such as Moran's I scatter plots and residual plots to evaluate the magnitude of deviance residuals, and it is effective to model the shape, the elevation and the magnitude of a local cluster in the model-based test.     

Asymptotic normality of a weighted integrated squared error of kernel regression estimates with data-dependent bandwidth

Let (X, Y) be a bivariate random vector and let be the regression function of Y on X that has to be estimated from a sample of i.i.d. random vectors (X₁, Y₁),…,(X_n, Y_n) having the same distribution as (X, Y). In the present paper it is shown that the normalized integrated squared error of a kernel estimator with data-driven bandwidth is asymptotically normally distributed.     

Biplot models applied to cancer mortality rates

"A graphical method developed by Gabriel to display the rows and columns of a matrix is applied to tables of age- and period-specific cancer mortality rates. It is particularly useful when the pattern of age-specific rates changes with time. Trends in age-specific rates and changes in the age distribution are identified as projections. Three examples [from England and Wales] are given."     

A critical look at Lo's modified R/S statistic

We report on an empirical investigation of the modified rescaled adjusted range or R/S statistic that was proposed by Lo, 1991. Econometrica 59, 1279–1313, as a test for long-range dependence with good robustness properties under ‘extra’ short-range dependence. In contrast to the classical R/S statistic that uses the standard deviation S to normalize the rescaled range R, Lo's modified R/S-statistic V_q is normalized by a modified standard deviation S_q which takes into account the covariances of the first q lags, so as to discount the influence of the short-range dependence structure that might be present in the data. Depending on the value of the resulting test-statistic V_q, the null hypothesis of no long-range dependence is either rejected or accepted. By performing Monte-Carlo simulations with ‘truly’ long-range- and short-range dependent time series, we study the behavior of V_q, as a function of q, and uncover a number of serious drawbacks to using Lo's method in practice. For example, we show that as the truncation lag q increases, the test statistic V_q has a strong bias toward accepting the null hypothesis (i.e., no long-range dependence), even in ideal situations of ‘purely’ long-range dependent data.