best practice life expectancy: an extreme value approach · best practice life expectancy (bple) is...

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References

Best Practice Life Expectancy: An Extreme ValueApproach

Anthony Medford

[email protected]

September 9, 2015


1 IntroductionWhat is Best Practice Life Expectancy?Trends since 1900Breakpoints

2 Why Extreme Value Theory?Empirical motivationTheoretical motivation

3 The GEVDistribution FunctionInference

4 ResultsFitted ModelProjectionsOther Inference

5 ARIMA and the GEVModel ResidualsInnovations Process

6 Conclusions


Some Facts

Best Practice Life Expectancy (BPLE) is the maximum lifeexpectancy observed among nations at a given age.

At birth, has been increasing almost linearly - beginning inScandinavia c. 1840 - at about 3 months per year (Oeppenand Vaupel, 2002).

Life expectancy trends may fit better than individual-countrytrends in age-standardized (log) death rates (White, 2002).


Some Facts

Nations experience more rapid life expectancy gains whenthey are farther below BPLE and tend to converge towardsBPLE (Torri and Vaupel, 2012).

It is sensible to consider national mortality trends in a largerinternational context rather than individual projections (Lee,2006; Wilmoth, 1998).


Females e0

1900 1920 1940 1960 1980 2000

5055

6065

7075

8085

Female Best Practice e0

Year

e0

IcelandJapanNorwayNZ (non−maori)Sweden


Males e0

1900 1920 1940 1960 1980 2000

5055

6065

7075

8085

Male Best Practice e0

Year

e0

AustraliaDenmarkIcelandJapanNetherlands

NZ (non−maori)NorwaySwedenSwitzerland


Females e65

1900 1920 1940 1960 1980 2000

1214

1618

2022

24Female Best Practice e65

Year

e65

CanadaFranceIcelandJapanNorwayNZ (non−maori)Sweden


Males e65

1900 1920 1940 1960 1980 2000

1214

1618

2022

24Male Best Practice e65

Year

e65

AustraliaDenmarkIcelandJapanNorwaySwitzerland


Breakpoints

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

Malese6

5

Figure: Breakpoints in the trend of the highest life expectancies at birthand age 65, males and females separately, from 1900 - 2012.


Empirical motivation

1960 1970 1980 1990 2000 2010

7274

7678

8082

8486

Year

e0


ObservedDetrendedTrend line

−1.0 −0.5 0.0 0.5 1.0 1.5 2.00.

00.

20.

40.

60.

81.

0

N = 58 Bandwidth = 0.1659

Den

sity

Kernel Density and fitted GEV

kernel densityfitted GEV

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


Theoretical motivation

Suppose that X1,X2, . . . ,Xn is a sequence of independent,identically distributed random variates all having a commondistribution function F (x).

Let Mn = max{X1,X2, . . . ,Xn}.

The distribution of the maxima, Mn, converges (for large n) to theGeneralized Extreme Value (GEV) Distribution.


The Generalized Extreme Value Distribution

G (z) = exp{−

[1 + ξ(

z − u

σ)]−1

ξ}

u is the location parameter

σ is the scale parameter

ξ is the shape parameter, which determines the tail behaviourξ > 0: polynomial tail decay and the Frechet Distributionξ = 0: exponential tail decay and the GumbelDistributionξ < 0: bounded upper finite end point and the WeibullDistribution


Inference

QuantilesInverting the GEV distribution function:

zp = µ− σ

ξ

[1− {−log(1− p)}−ξ

],

where p is the tail probability and G (zp) = 1− p

Return Levels

Simply a different way of thinking about the quantiles.

If data are annual the (1− p)th quantile would be exceededon average once every 1/p years.


Fitted Model

GEV (ut = 59.6 + 0.24t, σ = 1.31, ξ = −0.48)

1900 1940 1980

5565

7585

Year

e0

Median50 Year Return Level



Projections, Females e0

1900 1950 2000 2050

6070

8090

100

Year

e0

Median 95% Conf Ints


Projections, Females e0

1900 1950 2000 2050

6070

8090

100

Year

e0

99th Percentile95% Conf Ints


Other Inference

A probability distribution has been fit so the usual tools areavailable.

Year P(emax0 > 90) P(emax

0 > 95)

2020 35% < 0.001%2050 > 99.99% 91%


In Sample Comparison

Fit model using data up to 1980.Compare Observed 10 Year Maxima vs 10 Year return Levels .

1985 1990 1995 2000

0.2

0.4

0.6

0.8

1.0

Year

Abs

olut

e D

iffer

ence

s

Mean Absolute Difference(MAD)= 0.67 yearsMean Absolute Percentage Error = 0.8%


ARIMA model residuals

−2 −1 0 1 2

−6

−4

−2

02

norm quantiles

Res

idua

ls

Residuals

Den

sity

−6 −4 −2 0 2

0.0

0.1

0.2

0.3

0.4

fitted GEV

Figure: Normality tests for residuals of ARIMA(2,1,1) fitted to female e0BPLE. Left panel: QQ Plot; Right panel: histogram.


Innovations Process

Assumption of Gaussian errors is often arbitrary and can bepoorly fitting.

GEV is more flexible and is able to capture the shape ofdifferent error distributions - not just symmetric.

In practice Gaussian often provides a reasonable fit but GEVshould be considered as an alternative for the innovationsprocess.


Conclusion

Method can be used similarly to the Torri and Vaupel (2012)approach to forecasting life expectancy:

Either through projecting BPLE directly, which is preferableOr using the GEV as the innovations process in an ARIMAmodel

EVT can identify in an objective way whether life expectancyis actually at an extreme level rather than just ”high”

EVT can be used to obtain probabilities and/ or levels ofextreme longevity


References

Lee, R. (2006). Perspectives on Mortality Forecasting. III. TheLinear Rise in Life Expectancy: History and Prospects, VolumeIII of Social Insurance Studies. Swedish Social Insurance Agency,Stockholm.

Oeppen, J. and J. W. Vaupel (2002). Broken limits to lifeexpectancy. Science 296(5570), 1029–1031.

Torri, T. and J. W. Vaupel (2012). Forecasting life expectancy inan international context. International Journal ofForecasting 28(2), 519–531.

White, K. M. (2002). Longevity advances in high-income countries,1955–96. Population and Development Review 28(1), 59–76.

Wilmoth, J. R. (1998). Is the pace of Japanese mortality declineconverging toward international trends? Population andDevelopment Review , 593–600.

best practice life expectancy: an extreme value approach · best practice life expectancy (bple) is...

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