BestPracticeLifeExpectancy: AnExtremeValueTheory Approach · 2015-09-22 ·...

Best Practice Life Expectancy: An Extreme Value Theory

Approach

Anthony Medford

Max Planck Odense Centre on the Biodemography of Aging

University of Southern Denmark

September 21, 2015

Abstract

Whereas the rise in human life expectancy has been extensively studied, the evolution of maximumlife expectancies i.e the rise in best practice life expectancy in a group of populations has not beenexamined to the same extent. Extreme Value Theory has been used previously to examine thepotential maximum human life span by studying ages at death. However, it has not yet beenapplied to the analysis of life expectancies. This paper examines best practice life expectanciesdirectly through Extreme Value Statistics. This well established framework enables the fitting ofextreme value distributions to time series of maximum life expectancies and facilitates projectionsand probabilistic inferences based on the fitted distributions. We also explore the potential of usingExtreme Value Distributions as the innovations distribution in classical time series models.

1 Introduction

Mortality projections are crucial in many areas. For example, in fields such as life insurance and pensionswhere financial systems covering the well being of millions are at stake. Globally, governments andother stakeholders depend on reliable mortality projections for management and administration of theirfinancial liabilities.

Oeppen and Vaupel (2002) introduced the term Best Practice Life Expectancy (“BPLE”), referring to themaximum life expectancy observed among national populations at a particular age. Their Best PracticeLife Expectancies at birth have been increasing in a nearly linear fashion, beginning in Scandinaviaaround 1840 and continuing ever since at a pace of about 0.24 years per annum for females and 0.22years per annum for males (Oeppen and Vaupel, 2002). Vallin and Mesle (2009) in a later study thatcovered a longer period of data (1750-2005) indicated instead that life expectancy increased in a piecewiselinear fashion over four distinct periods. Nonetheless, their basic results, especially for recent decades,are generally consistent with the overall pattern of increase of about 3 months per year (Vaupel, 2012).Additionally, Shkolnikov et al. (2011) showed that best-practice cohort female life expectancy at birthincreased across cohorts from 1870 to 1920 by an average of about 0.43 years annually.

There is a strong argument for using life expectancy in forecasting. White (2002) found that lineartrends in life expectancy give a better fit to the experience of individual countries than linear trends inage-standardized (log) death rates in his study of 21 developed countries.

Among those who have forecast life expectancy are Alho and Spencer (2005); Andreev and Vaupel (2006);Lee (2006); Torri and Vaupel (2012). It is argued by Oeppen and Vaupel (2002) that since the increasein best practice life expectancy is linear and regular then it could be used in forecasting by comparingcountry specific performance with the best practice. This approach takes advantage of national mortalitytrends which ought to be considered within a larger international context rather than being analyzedand projected individually Lee (2006).

Predicting future life expectancy trends by extrapolating from only one country’s mortality experiencecould be problematic as the trend toward catch up among countries tends to exaggerate differences in life

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expectancy between countries over time, as the fastest risers outstrip other countries by ever increasingamounts (Wilmoth, 1998). White (2002) finds that nations experience more rapid life expectancy gainswhen they are farther below the international average, and therefore tend to converge toward the average.In addition, the life expectancy levels in different countries tend to be positively correlated. Age-specificdeath rates are highly correlated between males and females, among countries and across ages. However,most forecasts are made for either males or females separately and for a particular country withouttaking trends for the other sex or for other countries into consideration (Vaupel, 2012).

Forecasting best-practice life expectancy; forecasting the gap from this best-practice forecast for malesand females in various countries; and using models to estimate age-specific mortality from life expectancyare ways to overcome these problems. Torri and Vaupel (2012); Andreev and Vaupel (2006); Lee (2006)proposed methods which attempt to do this.

Torri and Vaupel (2012) employed classic univariate ARIMA techniques to model and forecast futurelevels of best-practice life expectancy. In this paper we explore the modelling and forecasting of BestPractice Life Expectancies by using Extreme Value Theory. Extreme value statistical methodology hasbeen used in a demographic context previously, perhaps most well known in the work of Aarssen andDe Haan (1994) in analysing maximum lifespan of humans, but has not been applied previously tomaximum life expectancy.

This paper is structured as follows. Section 2 presents the data used. Section 3 provides some mo-tivation and gives an outline of basic Extreme Value Theory and the Block Maxima approach to theanalysis of extremes. Section 4 presents the results including details of the fitted GEV models and howquantiles of extreme life expectancy can be projected. There is a Discussion in Section 5 and Section 6concludes.

2 Notation and Data

The following abbreviations and notation will be used throughout this paper.BPLE: Best Practice Life ExpectancyEVT: Extreme Value TheoryEVD: Extreme Value DistributionGEV: Generalised Extreme Value Distribution

e∗x: Best Practice Life Expectancy at given age x where ” ∗ ” indicates maximume∗x,f : Best Practice Life Expectancy at given age x for femalese∗x,m: Best Practice Life Expectancy at given age x for males

The data for use in model implementation and testing comes from the Human Mortality Database(HMD, 2015) which is available at www.mortality.org. The Human Mortality Database (HMD) is acollaborative project sponsored by the University of California at Berkeley (United States) and the MaxPlanck Institute for Demographic Research (Rostock, Germany) and is accessible from www.mortality.organd www.humanmortality.de respectively. It contains life tables for 37 countries (plus all the raw dataused in constructing those tables). The specific data used life expectancy at birth and at age 65, forboth males and females and covers the period 1900 to 2012. Many of the eastern European populations -former Soviet Republics - were excluded from the analysis because at some points their life expectanciesappeared over stated. Among these are Poland, Lithuania, Belarus, Russia and Ukraine.

2

3 Methods

3.1 Motivation

The application of Extreme Value Statistics allows us to investigate the behaviour of a stochastic processat very high or very low levels. The extreme value distributions arise as the limiting distribution of themaxima or minima of a set of random variables. In this paper the theory is applied to maximum lifeexpectancies.

Figure 1 presents the BPLE at birth for females. It is well known that life expectancy has been increasingover time at all ages and this is evident in a strong upward trend in the time series of the BPLE. Theleft panel presents this data from 1955 along with another series representing the same data but with thelinear least squares regression trend removed in order to produce a stationary series (explored further insection 5.2). The right panel shows a non-parametric estimate of this stationary data (we use the kerneldensity) and a fitted Generalized Extreme Value “GEV”) distribution. The data has also been scaledby subtracting the intercept term of the fitted regression. Graphically, the fitted distribution appears toprovide a very reasonable fit to the annual maximum data. Note that the GEV captures the assymetricshape whereas a symmetric distribution e.g. the Gaussian, would not to the same extent. Based onthis evidence therefore, one can conclude that Extreme Value Theory may be useful in analysing theBPLE.

1960 1970 1980 1990 2000 2010

72

74

76

78

80

82

84

86

Year

e0

Female Best Practice e0

ObservedDetrendedTrend line

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

N = 58 Bandwidth = 0.1659

De

nsity

Kernel Density and fitted GEV

kernel densityfitted GEV

Figure 1: Left panel: raw and detrended data. Right panel: kernel density and fitted GEV distribution

3.2 Extreme Value Theory: Basics

The application of the statistical theory of extreme values facilitates the study of a random processesat very high or low levels. The limiting distributions of these extremes give rise to the extreme valuedistributions. In this paper, the parameterisations and notation of Coles (2001) will be used.

Formally, suppose thatX1, X2, . . . , Xn is a sequence of independent random variates all having a commondistribution function F (x). Let the maximum of this sequence of n variables be Mn. We would like to

3

find the distribution of Mn as n becomes large. Now,

P (Mn ≤ z) = P (X1 ≤ z,X2 ≤ z, . . . , Xn ≤ z)

= P (X1 ≤ z)P (X2 ≤ z) . . . P (Xn ≤ z)

= Fn(z)

This result however is not particularly useful as the distribution of F (x) is unknown. However, it ispossible to find the distribution of Mn, say G, without any reference to F .

The distribution of Mn is degenerate since as n tends to infinity, the distribution function F convergeswith certainty to a single point. To avoid the difficulty of the degenerate limit a linear rescaling of Mn

is applied - a result known as the Extremal Types Theorem (Fisher and Tippett, 1928; Gnedenko, 1943;Coles, 2001).

If there exists sequences of constants {an > 0} and {bn}, such that as n → ∞,

P(Mn − bn

an≤ z

)

→ G(z) (1)

where G(z) is a non-degenerate distribution function, then G must be a member of the GeneralizedExtreme Value (GEV) family of distributions. This is a remarkable result because regardless of theunderlying distribution, the distribution of the maxima (or minima) converges to one of the GeneralizedExtreme Value family of distributions.

The GEV distribution function is given by,

G(z) = exp{

−[

1 + ξ(z − µ

σ)]

−1

ξ}

, (2)

defined on {z : 1+ξ(z−µ)/σ > 0}. The model is described by three parameters: µ(−∞ < µ < ∞), σ(σ >0) and ξ(−∞ < µ < ∞) referred to as the location, scale and shape parameters respectively. The locationparameter indicates the center of the distribution; the scale parameter the size of deviations around thelocation parameter; and the shape parameter governs the tail behavior of the GEV distribution.

The shape parameter, ξ determines the heaviness of the right tail and this leads to three types ofdistributions. When ξ < 0, the distribution has a bounded upper finite end point and is short-tailedleading to the Weibull Distribution. When ξ > 0, there is polynomial tail decay leading to heavy tailsand the GEV is of the Frechet type. The case where ξ = 0 is taken to be the limit of Eq. 2 as ξ → ∞ andthere is exponential tail decay leading to light tails and the GEV is of the Gumbel type with distributionfunction,

G(z) = exp{

−exp[

− (z− µ

σ)]}

.

In practice, for sufficiently large n, G(z) can be calculated without the need to know the normalisingconstants {an > 0} and {bn} (Coles, 2001). This has motivated an approach to GEV modelling knownas the Block Maxima approach, where for large enough n, P (Mn < z) can be approximated by using anappropriate member of the GEV family.

In summary, the Block Maxima approach works as follows. Suppose we have independent observationsX1, X2, . . .. Let these observations be divided into blocks of length n for sufficiently large n. Then, takethe maximum of each of these blocks to obtain a series of block maxima and fit a GEV distributionto these maxima in order to obtain parameter estimates µ, σ and ξ. Specific to our analysis, we haveobserved life expectancies from various countries over periods of length one year. The largest fromamong these annual period life expectancies is extracted and a GEV distribution is fitted to these annualmaxima in order to obtain parameter estimates.

For inference, estimates of extreme quantiles of the maxima are obtained by solving for z in equa-tion 2:

zp = µ−σ

ξ

[

1− {−log(1− p)}−ξ]

, (3)

4

where the distribution function of the GEV, G(zp) = 1−p and p is the tail probability or the probabilityof realising a value at least as large as zp. In extreme value terminology, the quantiles of the distribution,zp are termed return levels and are associated with the so called return period 1/p. If we are consideringannual maxima, which is usually the case, then on average the quantile zp is expected to be exceededwith probability p or on average once every 1/p years (Coles, 2001). For example, if p = 0.01 then thereturn level, zp is the 99th percentile, corresponds to the 1/(1 − 0.01) = 100-year return period, and isthe amount which one expects to see once every 100 years, on average.

3.2.1 Alternatives to the Block Maxima method

The Block Maxima method is sometimes criticised for being wasteful and not making sufficient use of theavailable data. Hence, there are two main alternatives: the r-largest statistics approach and the peaksover threshold approach. The first one, the r-largest order statistics approach e.g. Pickands (1975), looksat not only the single highest value within each block but rather the r highest values, for suitably ’small’values of r. The Block Maxima Model from section 3.2 is a special case of the r largest statistics modelwhen r = 1.

The second way to model extremes is to consider exceedances over high thresholds or put differently, alldata points over some suitably chosen high value - not just the maxima. The Peak Over Thresholds modele.g. Balkema and De Haan (1974); Pickands (1975); Davison and Smith (1990), adopts this approach.Both the Peak Over Thresholds model and the Largest Order Statistic Model are special cases of the PointProcess representation (Coles, 2001) of extreme values, under which we approximate the exceedancesover a threshold by a two-dimensional Poisson process (e.g. Leadbetter et al. (1983)).

4 Results

4.1 Leading countries

Oeppen and Vaupel (2002) first highlighted the remarkable linearity in the BPLE. Figure 2 presents aplot of these BPLEs, separately for males and females, at birth and age 65, from the year 1900 andidentifies which countries were leaders.

4.2 A segmented view

A closer inspection of the time series of life expectancies in Figure 2 reveals that the pace of increase isnot constant but has slowed slightly for life expectancies at birth with a more pronounced acceleration atage 65. Therefore, rather than assuming a constant linear increase, we formally investigate the presenceof differential rates of increase. This is done by first testing the null hypothesis of a non-zero differencein slope parameter of a segmented relationship using the Davies Test (Davies, 2002) and then finding thebreak points and allowing the slope parameter of a fitted linear regression to vary between these breakpoints.

These segmented relationships are presented in Figure 3. For life expectancy at birth a structural breakpoint occurs around 1955 for females and 1950 for males probably as a result of the introductsion ofantibiotics in the 1940s which brought sharp reductions in infectious diseases at young adult ages (Vallinand Mesle, 2009) and general improvement in living conditions coming out of the second world war. Atage 65 the structural breaks occur later, around 1967 for females and 1984 for males.

4.3 Fitting the GEV

Torri and Vaupel (2012), used data from 1900 to fit BPLE and argued that this period was used becauseit is nearly linear rather than using the longer time series available from 1840 which varies in a piecewise

5

1900 1920 1940 1960 1980 2000

5055

6065

7075

8085


Year

e0

IcelandJapanNorwayNZ (non−maori)Sweden

1900 1920 1940 1960 1980 2000

5055

6065

7075

8085

Male Best Practice e0

Year

e0

AustraliaDenmarkIcelandJapanNetherlands

NZ (non−maori)NorwaySwedenSwitzerland

1900 1920 1940 1960 1980 2000

1214

1618

2022

24


Year

e65

CanadaFranceIcelandJapanNorwayNZ (non−maori)Sweden

1900 1920 1940 1960 1980 2000

1214

1618

2022

24

Male Best Practice e65

Year

e65

AustraliaDenmarkIcelandJapanNorwaySwitzerland

Figure 2: Countries with the highest life expectancies at birth and age 65, males and females separately,from 1900 - 2012.

linear fashion (Vallin and Mesle, 2009). However, as demonstrated in Figure 3 there also have beenstructural breaks during the 20th century therefore we do not fit the BPLE arbitrarily from 1900 butfrom the point of the most recent structural break, which varies depending on the underlying population.By doing this, we ensure that the correct pace of life expectancy increase is attributed to the correcttime period and population. Hence, the fitting periods are from 1950 for e∗

0,m; from 1955 for e∗0,f ; from

1967 for e∗65,f and from 1984 for e∗

65,m representing females at birth, males at birth, females at age 65and males at age 65 respectively.

A Generalized Extreme Value Model is fitted to the respective best practice life expectancies at birthand age 65. The Best Practice life expectancy was found by selecting the maximum observed period lifeexpectancy from among all the countries in the HMD. Since these countries are the developed countries,which generally have higher life expectancies than the developing ones, then the maxima from amongthis subset of countries in the HMD can be viewed as the highest life expectancies in the world.

In Subsection 3.1 the data were detrended and a GEV model fit to an essentially stationary series in orderto motivate and justify its use. In this section no such restriction is imposed and instead of removingthe trend, it is incorporated into the model.

The linear trend over time in e∗x (x = 0, 65) is accounted for by allowing the location parameter of theGEV model to vary linearly with time such that, instead of a fixed location parameter µ, a more flexibleparameter is adopted. Thus, time is introduced as a covariate into the parametrisation of the GEVdistribution so we assume a location parameter of the form, µt = β0 + β1t where t represents calendar

6

1900 1920 1940 1960 1980 2000

5055

6065

7075

8085

Females

e0

1900 1920 1940 1960 1980 2000

5055

6065

7075

8085

Males

e0

1900 1920 1940 1960 1980 2000

1214

1618

2022

24

Females

e65

1900 1920 1940 1960 1980 2000

1214

1618

2022

24

Males

e65

Figure 3: Breakpoints in the trend of the highest life expectancies at birth and age 65, males and femalesseparately, from 1900 - 2012.

time. More specifically, t is an index commencing at 1 in the first year of the data (e.g. 1950 for e∗0,f )

and increases by one unit per consecutive year. The parameter β1 could be interpreted as roughly theannual rate of increase in life expectancy. The parameter β0 is the fitted initial level of the locationparameter for the first year of data and does not have a convenient demographic interpretation.

Since e∗x has been trending upward linearly over time, µ, the location parameter of the GEV is the mostobvious parameter of the GEV to capture this feature but time dependence could also be introduced intothe other parameters provided that the additional complexity is justified and strongly supported by thedata. In general however, the shape parameter is not usually altered as it can be difficult to estimate inpractice. Also, more sophisticated covariate structures could be introduced. For example suppose therewere an annual index of some variable which is highly correlated with life expectancy then this couldplausibly be introduced as a covariate. For example if there were available an annual worldwide smokingprevalance index or some other index which is highly correlated to and/or predictive of life expectancy,then this can be used as a covariate in the model.

In summary, we fit a time-dependent GEV model which can be represented succinctly as:

GEV (µt, σ, ξ)

where the location parameter, µt = β0 + β1t, the scale parameter is σ, the shape parameter is ξ andt = 1 . . . tmax, where tmax is the last year of the data.

7

The parameters of the GEV distribution µt, β0, β1, σ and ξ are found using maximum likelihood estima-tion. Once calculated, the parameter estimates µt, β0, β1, σ and ξ can then used in the computation ofreturn levels (quantiles), probabilities and other items of interest.

Neg. Likelihood β0 β1 σ ξ

Female e0 33.0 74.0 (0.11) 0.22 (0.003) 0.37 (0.030) 0Male e0 65.0 69.4 (0.15) 0.16 (0.003) 0.75 (0.082) -0.46 (0.101)

Female e65 14.7 16.6 (0.11) 0.16 (0.004) 0.36 (0.048) -0.43 (0.129)Male e65 -4.70 15.5 (0.10) 0.12 (0.005) 0.21 (0.028) -0.29 (0.111)

Table 1: Maximized negative log-likelihoods, parameter estimates and standard errors (in parentheses)of the r Block Maxima Model; e0 and e65 for males and females shown separately

Appendix A presents fitted model diagnostics and shows that the models fit the data well. Using theparameter estimates from Table 1 results in the following fitted GEV models for extreme life expectancy.Note that the shape parameter of the GEV for e∗

0,f is zero indicating a Gumbel distribution. Theestimated value for this parameter was about -0.05 but a likelihood ratio test of significance indicatedthat this value was not significantly different from zero. The negative shape parameters for the othersindicate Weibull distributions.

• For female life expectancy at birth: GEV (74.0 + 0.22t, 0.390, 0)

• For male life expectancy at birth: GEV (69.4 + 0.16t, 0.75,−0.46)

• For female life expectancy at 65: GEV (16.6 + 0.16t, 0.36,−0.43)

• For male life expectancy at 65: GEV (15.5 + 0.12t, 0.21,−0.29)

Since EVT is concerned with rare events, it is convenient to make inferences on the extreme quantiles ofthe fitted model. These extreme quantiles are commonly referred to as the return levels of the distributionin the terminology of extreme value theory. Using Equation 3, and the fitted parameter estimates fromTable 1, the return levels are easily calculated. If we use male life expectancy at birth data, in year t, the98th percentile (50-year return level) estimates the highest life expectancy we might expect to encounteron average every 50 years and is given by,

z0.02(t) = (69.4 + 0.16t) + 1.62[

1− {−log(1−1

50)}0.46

]

The quantiles for the other populations are derived in a similar way. Figure 4 presents graphically the20-year and 100-year return levels, (the upper 95% and 99% quantiles) of the fitted GEV distributions.Also presented are the corresponding lower quantiles. The quantiles of the different populations are atdifferent distances from each other reflecting the different tail behaviour of the fitted GEVs. The Gumbelof the e∗

0,f has a heavier tail right and lighter left tail than the Weibull of the other three populationsso that the lines representing the upper quantiles are more spaced out and for the lower quantiles,closer together. The reverse is true for the Weibull. The return levels can also be used in an objectiveway to identify periods of unusually high or low life expectancy because realisations of high or low lifeexpectancy does not necessarily imply extremities. Suppose that the criteria for extreme life expectancyis a realisation higher than the 99th percentile. Then one can immediately identify from Figure 4 thedata of interest. Indeed, if the model-fitting period is at least as long as the return period of interestthen it would be possible to compare the theoretical tail probability with the observed. For example, the98th quantile implies a 2% tail probability or a probability that the quantile would be realised once every50 years. If the 98th percentile is exceeded regularly then one can assume that mortality experience hasbeen unusually light leading to the high life expectancy and investigate further. If we consider the upper95th percentile from Figure 4 we can make these types of comparisons. Over the period 1955-2012 fore∗0,f we would expect to observe a life expectancy in the upper 5 % tail about 3 times over that 60 yearperiod. In reality we observe four.

8

1960 1970 1980 1990 2000 2010

7075

8085

Year

e0

Median

lower / upper 95th percentile


Females

1950 1960 1970 1980 1990 2000 2010

7075

8085

Year

e0

Median



Males

1970 1980 1990 2000 2010

1618

2022

24

Year

e65

Median



Females

1985 1990 1995 2000 2005 2010

1618

2022

24

Year

e65

Median



Males

Figure 4: Best Practice Life expectancies and the fitted GEV Median, 20 and 100 year

return levels.

9

Group Median (z0.5) 20 Yr Level (z0.05) 50 Yr level (z0.02)

2040

Female e0 92.9 93.9 94.2Male e0 84.2 85.2 85.3

Female e65 28.4 28.9 29.0Male e65 22.4 22.8 22.9

2050

Female e0 95.1 96.1 96.4Male e0 85.9 86.8 87.0

Female e65 30.0 30.5 30.6Male e65 23.7 24.0 24.1

Table 2: Projected extreme life expectancy return levels in 2040 and 2050.

4.4 Projections

The return levels from Figure 4, similar to the data, evolve in a linear way. If we assume that the currentrate of increase in e∗x continues, then it is possible to project return levels into the future. Table 2presents the 100-year, 20-year and 2-year (median) return levels in 2040 and 2050. For example in 2040,the median e∗

65,m at age 65 is 22.4 years but the 98th percentile is 22.9 years. Appendix B provides plotsof the quantiles for the years 2013 to 2050. A key assumption is that the linear trend for µ continuesbeyond the range of data we have observed.

4.5 Illustrations

Besides quantiles, another way of making inferences for GEV models is in the calculation of probabilities(see Table 3). Using Equation 2 and the estimated parameters one can, given the quantiles, calculatethe probabilities. For example the probability that the maximum female life expectancy at birth willexceed 90 years in 2025 is approximately 59% and while by 2050 the probability climbs to over 99.9%.Similarly, there is about a one third chance that by 2025 the maximum observed life expectancy of 65year old females will reach 27 years.

Year P (e∗0,f > 90) P (e∗

0,m > 85) P (e∗65,f > 26) P (e∗

65,m > 24)

2025 0.23 < 0.001 0.57 < 0.0012050 > 0.999 0.86 > 0.999 0.05

Table 3: Probability of the maximum female life expectancies at birth, e∗0,f , exceeds certain levels for

the years 2020 and 2050.

Conversely if we have the life expectancy quantile and assume some level of probability then we canestimate the future time when the given life expectancy level will be observed. For example, if we adopta significance level of 95%, then maximum female life expectancy at birth observed in the world wouldreach 100 years around 2070; with 95% probability the country with the highest male life expectancy atage 65 would reach 27 years around 2080.

The foregoing analysis has been based on worldwide maxima. The blocks to which the Block MaximaMethod was applied were groups of countries. However, provided that there is sufficient high quality lifeexpectancy data available then the methods of the previous section can be applied to within countrydata. For example, maxima can be taken over blocks of provinces in Canada, states in the United Statesof America or prefectures in Japan.

Canada

10

GEV models were fit to data for Canada, taken from the Canadian Human Mortality Database. Thedata were for all Canadian Provinces excluding Yukon and the Northwest Territories which were excludeddue to small and highly variable sample sizes. Below are the fitted GEV models and some illustrativecalculations for Canada.

• For female life expectancy at birth: GEV (77.4 + 0.12t, 0.74,−0.46); t = 1, ..., 49

• For male life expectancy at birth: GEV (70.8 + 0.24t, 0.04,−0.34); t = 1, ..., 38

• For female life expectancy at 65: GEV (15.4 + 0.09t, 0.40, 0); t = 1, ..., 72

• For male life expectancy at 65: GEV (15.1 + 0.13t, 0.27,−0.29); t = 1, ..., 30

5 Discussion

5.1 Classical Time Series vs EVT in forecasting e∗

x

Since e∗x is the maximum annual life expectancy at age x from among the different nations, we haveshown that the theory of Extreme Value Statistics can be applied. We used data beginning in 1900,found structural breakpoints and fit GEV distributions from the most recent breakpoint. This was donein order to ensure that the correct pace of life expectancy improvement was captured for each specificsub-population.

It was shown that the underlying trend in e∗x can be used in combination with a fitted GEV to projectquantiles of extreme life expectancy. Our concern here is inference about extreme or very high levels sothat, by definition it is unlikely that the return levels of life expectancy would be realised regularly inpractice but the methodology serves as a useful objective tool in quantifying the possibility of observingextremely high levels of life expectancy.

Projection of the BPLE can be an important component in forecasting of life expectancy. According tothe work on this method introduced by Oeppen and Vaupel (2002) and further elaborated by Torri andVaupel (2012), forecasts of country specific life expectancy can be obtained by first projecting the BPLEand then modelling and forecasting the gap between the BPLE and the life expectancy in any givenpopulation. Torri and Vaupel (2012) used classical univariate time series ARIMA models to forecaste∗0,f and e∗

0,m. While this is an obvious and straightforward approach, there are some advantages to ourmethodology. Firstly there is the strong theoretical justification for using EVT to fit maxima data andsecondly, with our approach it is possible to not only project e∗x but also to straightforwardly obtainprobabilities about future values of e∗x.

We calculated the median of e∗0,f in 2050 to be 95.1 years which is slightly lower than the 96.59 years

found by Torri and Vaupel (2012). For e∗0,m it is 85.9 years versus 88.38 years for Torri and Vaupel

(2012). This is unsurprising, since, as was shown in Section 4.2, the pace of life expectancy increase forfemales has slowed to about 0.21 years per year from around 1955 so that the constant drift term ofabout 0.24 years per year in the ARIMA model overestimates the future value of BPLE. Similarly, formales at birth the pace has slowed to about 0.17 years per year from 1950 versus a constant ARIMAdrift of about 0.20 years per year.

5.2 GEV as an Innovations Process in classical time series

In this section we discuss the use of the GEV as a distribution for the innovations process in classicaltime series models. Based on the discussions from section 3.2 there are strong theoretical arguments formodelling the underlying stochastic process for the e∗x innovations as an Extreme Value Distribution.Since we are analysing maxima, it is natural to assume that the innovations have a GEV distribution.This is demonstrated first for the specific ARIMA model of Torri and Vaupel and then for a more generalregression approach to modelling the e∗x series. Without loss of generality, results are shown for the e∗

0,f

series.

11

−2 −1 0 1 2

−6

−4

−2

02

norm quantiles

Re

sid

ua

ls

Residuals

De

nsi

ty−6 −4 −2 0 2

0.0

0.1

0.2

0.3

0.4

fitted GEV

Figure 5: Normality tests for residuals of ARIMA(2,1,1) fitted to e∗0. Left panel: QQ Plot; right panel:

histogram.

In classical time series analysis, it is not uncommon to assume a Gaussian distribution for the residuals.Moreover, even when no distributional assumption is expressly made for the error process, the assumptionof normality is used explicitly when the model parameters are estimated using standard maximumlikelihood estimation techniques (Hamilton, 1994; Brockwell and Davis, 2002).

While we are concerned with the distribution of the maximum life expectancies, it is instructive to notethat the assumption of normality for the residuals/errors in the ARIMA models used in forecastingLee-Carter type models has been shown to be poor (Dowd et al., 2010).

Torri and Vaupel (2012) fit an ARIMA(2,1,1) model to the e∗0,f time series from 1900 but did not present

normality tests for the residuals. Presented below are selected normality tests of the residuals obtainedfrom a fitted ARIMA(2,1,1) model. Our model has slightly different parameter values from the Torri andVaupel results but for our purposes, this has minor import. In addition, we do not smooth the troughin life expectancy caused by the impact of the Spanish Flu in 1918 or the 2nd World War. Even thoughthese are outliers they are still valid extreme events.

The following tests for normality are performed:

• QQ plot

• Shapiro-Wilk Normality Test (Royston, 1982, 1995)

• D’Agostino test for skewness in normally distributed data (D’Agostino, 1970)

• Anscombe-Glynn test of kurtosis for normal samples (Anscombe and Glynn, 1983)

All of the tests reject the assumption of normality (Table 4) and are further confirmed by the shape ofa histogram of the residuals (Figure 5, 2nd panel). A Gaussian distribution would not capture this leftskewness but the fitted GEV distribution does well.

Now that it has been shown that the innovations of the e∗0,f time series commencing in 1900 and modelled

by an ARMA process are not Gaussian but likely EVD, a natural question would be whether an EVD

12

Test Statistic p-value

Shapiro-Wilk Normality Test 0.69 < .0001D’Agostino Test -8.46 < .0001

Anscombe-Glynn Test 6.90 < .0001

Table 4: Results of normality tests of residuals of ARMA(2,1,1) model fitted to e∗0

is a better assumption in general and if this assumption holds true for different periods i.e how robustis the assumption. In order to do this, data periods are taken which begin in 1900, 1920 and so on indecennial intervals up to 1990.

Life expectancy has been trending upwards in a very obvious way. In their work, Torri and Vaupel (2012)assumed the trend to be stochastic in nature (confirmed by the Dickey-Fuller Test) and removed the trendby differencing the data. Deciding whether a trend is stochastic or deterministic is not straightforwardand although unit root tests may accept (or not reject) hypotheses of the existence of a unit root, forany finite sample size there is both a deterministic and stochastic trend that fits the data equally well(Hamilton, 1994). We therefore make no assumption in this regard concerning the nature of the trend ine∗0,f , and calculate the residuals using both of the usual trend removal techniques viz. differencing andleast squares trend removal.

The two types of trend result in very different long term behaviour, especially to shocks in the timeseries. With a deterministic trend, the series eventually returns to the trajectory of the long term trend,whereas with a stochastic trend, the series generally does not recover from shocks. History would suggestthat the trend in life expectancy is more of a deterministic trend since life expectancy has recovered -sometimes quickly - from shocks. The Spanish Influenza epidemic and the two world wars are examplesof this. In addition, the spike in life expectancy in the 1950’s as a result of the introduction of antibioticsdid not result in a permanent level shift in life expectancy but a “hump” with subsequent reversion tothe long term trend.

Deterministic vs Stochastic Trend Residuals

When a deterministic trend is assumed, the residuals are calculated by subtracting the ordinary leastsquares regression line from the data. Figure 6 shows the distributions of these residuals, for differentperiods.

13

From 1900

Detrended Residual

De

nsi

ty

−6 −4 −2 0 2

0.0

0.1

0.2

0.3

0.4

GEV

Gaussian

From 1920

Detrended Residual

De

nsi

ty−2 −1 0 1

0.0

0.2

0.4

0.6

GEV

Gaussian

From 1930

Detrended Residual

De

nsi

ty

−1.5 −0.5 0.5 1.5

0.0

0.2

0.4

0.6

GEV

Gaussian

From 1940

Detrended Residual

De

nsi

ty

−1.5 −0.5 0.5 1.5

0.0

0.2

0.4

0.6 GEV

Gaussian

From 1950

Detrended Residual

De

nsi

ty

−1.0 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

GEV

Gaussian

From 1960

Detrended Residual

De

nsi

ty

−1.0 −0.5 0.0 0.5 1.00

.00

.40

.81

.2 GEV

Gaussian

From 1970

Detrended Residual

De

nsi

ty

−0.5 0.0 0.5 1.0

0.0

0.4

0.8

1.2

GEV

Gaussian

From 1980

Detrended Residual

De

nsi

ty

−0.8 −0.4 0.0 0.4

0.0

0.4

0.8

1.2

GEV

Gaussian

From 1990

Detrended Residual

De

nsi

ty

−0.8 −0.4 0.0 0.4

0.0

0.4

0.8

1.2

GEV

Gaussian

Figure 6: Residuals assuming trend stationary data with fitted GEV distribution overlayed.

The question remains whether it is reasonable to assume a Gaussian distribution for the residuals ofe∗0.

The GEV that has been fitted to the maxima in all cases has a negative shape parameter indicating aWeibull distribution. Tables 5 and 6 show the value of the shape parameter under both the deterministicand stochastic trend assumptions.

The shape of the Weibull Distribution is very flexible and depending on the value of the shape paramater,ξ, varies as follows. When ξ is less than about -0.28 the Weibull distribution is left skewed; when ξis greater than -0.28 the distribution is right skewed and when ξ is close to -0.28 the distribution issymmetric (Reiss and Thomas, 2001, p. 18).

14

From 1900

Differenced Residual

De

nsi

ty

−1 0 1 2 3

0.0

0.4

0.8 GEV

Gaussian

From 1920


De

nsi

ty−1 0 1 2 3

0.0

0.4

0.8 GEV

Gaussian

From 1930


De

nsi

ty

−0.5 0.0 0.5 1.0

0.0

0.4

0.8

1.2

GEV

Gaussian

From 1940


De

nsi

ty

−0.5 0.0 0.5 1.0

0.0

0.4

0.8

1.2

GEV

Gaussian

From 1950


De

nsi

ty

−0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

GEV

Gaussian

From 1960


De

nsi

ty

−0.5 0.0 0.5 1.00

.00

.51

.01

.5

GEV

Gaussian

From 1970


De

nsi

ty

−0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

GEV

Gaussian

From 1980


De

nsi

ty

−0.6 −0.2 0.2 0.6

0.0

0.4

0.8

1.2

GEV

Gaussian

From 1990


De

nsi

ty

−0.4 0.0 0.4 0.8

0.0

0.5

1.0

1.5

GEV

Gaussian

Figure 7: Residuals assuming difference stationary data with fitted GEV distribution overlayed.

The shape of the residuals over some of the data periods is close to symmetric (shape parameter not farfrom -0.28), and a Gaussian distribution could theoretically be fit, but this does not necessarily implythat the underlying innovations process is Gaussian. This is particularly true for the ST residuals whichare in general closer to -0.28. Indeed, an assumption of normality would be a shaky one since the shapeof the residuals as shown in Figures 6 and 7 can change over time and is likely to change in the future. Itcan be seen that the shape could be skewed in either direction making the Gaussian distribution a poorchoice. By virtue of its flexibility the GEV fits over all data periods.

There is also little underlying theoretical justification for making an arbitrary assumption of Gaussianity.The skewness of the data as shown in tables 5 and 6 is generally close to zero and the kurtosis close to 3;

15

both in the vicinity of the values for the Gaussian Distribution. This provides further spurious evidenceof Normality, resulting in many of the standard tests for normality (which take into major considerationskewness and kurtosis) failing to reject the null hypothesis of Normality based on insufficient evidence tothe contrary. Without understanding the underlying stochastic dynamics it would be easy to erroneouslyconclude Normality but the GEV assumption is theoretically justifiable and because of its flexibile shape(symmetric, right or left skew) is more robust to changing data fitting periods. In addition, excellentdiagnostics confirms the good fit of the GEV distribution.

Data Start Year ξ (DT) Skewness Kurtosis1920 -0.319 -0.342 4.731930 -0.219 0.241 3.231940 -0.229 0.214 3.321950 -0.041 0.736 3.201960 -0.218 0.171 2.691970 -0.176 0.291 2.581980 -0.420 -0.126 2.131990 -0.548 -0.434 2.60

Table 5: Fitted GEV shape parameter, skewness and kurtosis of the residuals for deterministic trendassumption.

Data Start Year ξ (ST) Skewness Kurtosis1920 -0.013 1.07 5.061930 -0.214 0.206 2.831940 -0.161 0.393 3.071950 -0.220 0.187 2.821960 -0.209 0.222 2.851970 -0.226 0.170 2.801980 -0.424 -0.426 2.931990 -0.327 -0.149 2.72

Table 6: Fitted GEV shape parameter, skewness and kurtosis of the residuals for stochastic trendassumption.

Therefore, in the context of classical ARIMA time series models, assuming GEV innovations may bea reasonable assumption if modelling maximum life expectancies. GEV innovations in the context ofARIMA models are not commonplace but see e.g. Hughes et al. (2007) for an application.

6 Concluding Remarks

In this analysis we used Extreme Value Theory to analyse the time series of annual maximum lifeexpectancy observed from 1900 with the aim of making inferences about future life expectancy. We useda new methodology to forecast the BPLE.

This approach could be useful in a number of contexts. Firstly, as an input into the computation ofpopulation specific life expectancies, where the best practice level and the lag between country and bestpractice level are modelled and forecast separately. Perhaps, exact country-specific life expectancies arenot required but interest is in the possibly of realising high (or low) extreme levels. This may be ofinterest to for example insurers or pension funds who may be want an estimate of the probability ofextreme life expectancy exposures which may trigger loss on some product or contract. Secondly, ifcountr-level data split by e.g state or province, say, then the methodology could be applied to makeinferences about country-level maximum life expectancy. Of course, EVT is useful not only for extrememaxima but also extreme minima. For example one might be interested in ascertaining the likelihood of

16

realising some low level of life expectancy which in turn could be driven by war, disease or some othercatastrophic event that affect life expectancy in an adverse way.

We also examined the innovations process for time series of BPLE with the aim of answering the questionwhether the default assumption of Gaussian residuals is a reasonable one. Our analyses concludedthat the residuals from classical time series models should be assumed GEV. However, although theassumption of Gaussian residuals is arbitrary, in many cases Normality tests do not provide sufficientevidence to reject the assumption of Normality. As a result, the default Gaussian assumption, thoughat times tenuous, may still be reasonable, for many practical purposes. However, caution should beexercised with this assumption and its limitations acknowledged.

Further work could involve applying other EVT approaches such as using other order statistics, where forexample one can model the five highest life expectancies or employ peak over threshold methods. Moresophisticated models making use of other covariates besides time and applied to the other parameters tothan the location parameter could also be explored.

17

References

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Shkolnikov, V. M., D. A. Jdanov, E. M. Andreev, and J. W. Vaupel (2011). Steep increase in best-practicecohort life expectancy. Population and Development Review 37 (3), 419–434.

Torri, T. and J. W. Vaupel (2012). Forecasting life expectancy in an international context. InternationalJournal of Forecasting 28 (2), 519–531.

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Appendix A: Diagnostic Tests

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

empirical

mo

de

l

Residual Probability Plot

−1 0 1 2 3 4

−1

01

23

4

empirical

mo

de

l

Residual Quantile Plot (Gumbel Scale)

(a) Female e0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

empirical

mo

de

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−1 0 1 2 3 4

−1

01

23

45

empirical

mo

de

l


(b) Male e0

20

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

empirical

mo

de

l


−1 0 1 2 3 4

−1

01

23

45

empirical

mo

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(a) Female e65

0.0 0.2 0.4 0.6 0.8 1.0

0.0

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0.4

0.6

0.8

1.0

empirical

mo

de

l


−1 0 1 2 3

−1

01

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(b) Male e65

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Appendix B: Projected Quantiles

1960 1980 2000 2020 2040

7580

8590

95

Year

e0

Median

10 Year Level (90th %ile)

20 Year level (95th %ile)



Females, e0

1960 1980 2000 2020 2040

7075

8085

90

Year

e0

Median





Males, e0

22

1960 1980 2000 2020 2040

1820

2224

2628

30

Year

e65

Median





Females, e65

1990 2000 2010 2020 2030 2040 2050

1618

2022

24

Year

e65

Median





Males, e65

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BestPracticeLifeExpectancy: AnExtremeValueTheory Approach · 2015-09-22 ·...

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